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+Learning from What is Already Out There:
+Few-shot Sign Language Recognition with Online Dictionaries
+Maty´aˇs Boh´aˇcek1,2 and Marek Hr´uz1
+1 Department of Cybernetics and New Technologies for the Information Society,
+University of West Bohemia, Pilsen, Czech Republic
+2 Gymnasium of Johannes Kepler, Prague, Czech Republic
+Abstract— Today’s sign language recognition models require
+large training corpora of laboratory-like videos, whose collec-
+tion involves an extensive workforce and financial resources.
+As a result, only a handful of such systems are publicly
+available, not to mention their limited localization capabili-
+ties for less-populated sign languages. Utilizing online text-to-
+video dictionaries, which inherently hold annotated data of
+various attributes and sign languages, and training models
+in a few-shot fashion hence poses a promising path for the
+democratization of this technology. In this work, we collect
+and open-source the UWB-SL-Wild few-shot dataset, the first
+of its kind training resource consisting of dictionary-scraped
+videos. This dataset represents the actual distribution and
+characteristics of available online sign language data. We select
+glosses that directly overlap with the already existing datasets
+WLASL100 and ASLLVD and share their class mappings to
+allow for transfer learning experiments. Apart from providing
+baseline results on a pose-based architecture, we introduce a
+novel approach to training sign language recognition models
+in a few-shot scenario, resulting in state-of-the-art results on
+ASLLVD-Skeleton and ASLLVD-Skeleton-20 datasets with top-
+1 accuracy of 30.97 % and 95.45 %, respectively.
+I. INTRODUCTION
+Sign languages (SLs) are natural language systems based
+on manual articulations and non-manual components, serving
+as the primary means of communication among d/Deaf
+communities. While they allow one to convey identical
+semantics as the written and spoken language, they operate in
+a distinctively more variable gestural-visual modality. There
+are currently over 70 million people worldwide whose native
+language is one of the approximately 300 SLs that exist [1].
+Nevertheless, no publicly available SL translation system
+has been introduced so far. This hinders d/Deaf people’s
+ability to use their natural form of communication when
+working with technology or interacting with people that do
+not sign. Although the problem of automatic SL Recognition
+(SLR) has been addressed for many years, it is far from
+being solved. Modern solutions utilizing deep learning show
+promise, and neural networks might help tear these barriers
+down.
+There are two prevalent topics related to SLs pursued
+in the literature - SL Synthesis and SLR. The first one’s
+objective is to translate written language into SL, typically by
+animating avatars. The second is intended to translate videos
+of performed signs into the written form of a language. It can
+This work has been accepted and scheduled for publication at the Face
+& Gestures 2023 conference. 979-8-3503-4544-5/23/$31.00 ©2023 IEEE
+be further divided into isolated SLR, which recognizes single
+sign lemmas out of a known set of glosses, and continuous
+SLR, translating unconstrained signing utterances. In this
+paper, we attend to the task of few-shot isolated SLR.
+The current methods can be generally divided into two
+main approaches differing in the means of input repre-
+sentations; the appearance-based and the pose-based. The
+first prevalent stream of works uses a sequence of RGB
+images, optionally complemented with the depth channel.
+These methods reach state-of-the-art results but are more
+computationally demanding. The second approach performs
+an intermediate step of first estimating a body pose sequence
+which is then fed into an ensuing recognition model. These
+systems tend to be more lightweight and would thus be
+more suitable for applications on conventional consumer
+technology, e.g., laptops or mobile phones.
+Multiple model training and evaluation datasets have been
+published over recent years. Generally large-scale in size of
+glosses and instances, they vary primarily in the originating
+SL and the manners of data collection. It is essential to con-
+sider that, unlike with many tasks in the Natural Language
+Processing (NLP) domain, no organic sources of potential SL
+training data (such as the internet and printed media in the
+case of NLP) yield vast amounts of training instances daily.
+It hence takes a dedicated, tailored effort to record a SLR
+dataset. Such an operation is costly and requires specialists
+from multiple fields at once, making it strenuous and risky
+to begin with. Accordingly, languages with a smaller user
+base receive less attention.
+Some of the few resources that contain SL data with
+built-in annotations are online text-to-video dictionaries.
+We believe they will be crucial in minimizing barriers
+in constructing future SLR systems, especially for niche
+regional contexts. We thus focus on training models using
+data scraped from such websites. As these services usually
+contain a few repetitions per sign lemma, such a configu-
+ration comprises a few-shot training paradigm. To account
+for the lack of a diverse, high-repetitive dataset, we utilize
+SPOTER [7], a pose-based Transformer [34] architecture
+for SLR. We hypothesize that it will learn faster since it
+considers only pre-selected information necessary for such
+a classification, which is much smaller in dimension than
+raw RGB video. Appearance-based methods, contrastingly,
+glutted by the large volume of additional sensory infor-
+mation, need more data to generalize sturdily, as observed
+arXiv:2301.03769v1  [cs.CV]  10 Jan 2023
+
+by Boh´aˇcek et al. [7]. We further investigate the ability
+of models to learn across different datasets and introduce
+boosting training mechanisms. The main contributions of this
+work include:
+• Introducing and open-sourcing UWB-SL-Wild: a new
+dataset for few-shot SLR obtained from public SL
+dictionary data, provided with class mappings to already
+existing SLR datasets;
+• Proposing Validation Score-Conscious Training proce-
+dure which adaptively augments and re-trains for classes
+that are identified as under-performing during training;
+• Establishing the state-of-the-art results on the ASLLVD-
+Skeleton and ASLLVD-Skeleton-20 datasets.
+II. RELATED WORK
+This section reviews the existing datasets and methods
+for isolated SLR. As low-instance training has not yet been
+explored to a greater extent for this task, we consider
+the overlaps to few-shot or zero-shot gesture and action
+recognition.
+A. Datasets
+Multiple datasets of isolated signs have been published and
+studied in the literature. We summarize the prominent ones in
+Table I. Purdue RVL-SLLL ASL Database [23], containing
+1, 834 videos across 104 classes within the American Sign
+Language (ASL), was one of the first to encompass a larger
+vocabulary. LSA64 [28] for the Argentinian Sign language is
+similar in size, as it contains 3, 200 instances from 64 classes.
+Later on, substantially larger corpora started to emerge.
+DEVISIGN [12], for instance, provides 24, 000 recordings
+spanning 2, 000 glosses from the Chinese sign language. Its
+videos were captured in a laboratory-like environment and
+were, to the best of our knowledge, the first to provide the
+depth information along RGB for this task. MS-ASL [17]
+brings a similar scale for the ASL, as it contains 25, 000 RGB
+videos from 1, 000 classes. Lastly, the AUTSL [31] dataset
+pushed the size and per-class instance ratio even further. It
+holds 38, 366 RGB-D recordings spanning 226 classes from
+the Turkish SL.
+While the available datasets span different geographical
+contexts, most research has centered around ASL. We left
+out recent datasets, which we consider to capture the most
+significant traction within the community, from the introduc-
+tory survey and provide their detailed descriptions below. We
+later utilize these for experiments and for constructing our
+new dataset.
+1) WLASL:
+Word-level
+American
+Sign
+Language
+dataset [21] is a large-scale database of lemmas from
+the
+ASL
+collected
+from
+multiple
+online
+sources
+and
+organizations. The dataset’s gloss totals 2, 000 terms with
+their translations to English. The authors provide training,
+validation, and test splits. There is an average of over 10
+repetitions in the training set for each class. There are
+three primary splits of the dataset depending on the number
+of
+classes
+they
+cover:
+WLASL100,
+WLASL300,
+and
+WLASL2000. In our experiments, we use the WLASL100
+split only.
+TABLE I
+SURVEY OF PROMINENT SLR DATASETS.
+Dataset
+SL
+Gloss
+Instances
+Format
+DEVISIGN [12]
+CN
+2,000
+24,000
+RGB-D
+LSA64 [28]
+AR
+64
+3,200
+RGB
+AUTSL [31]
+TR
+226
+38,336
+RGB-D
+RVL-SLLL [23]
+US
+104
+1,834
+RGB
+ASLLVD [24]
+US
+2,745
+9,763
+RGB/Skelet.
+MS-ASL [17]
+US
+1,000
+25,000
+RGB
+WLASL [21]
+US
+2,000
+21,083
+RGB
+2) ASLLVD: American Sign Language Lexicon Video
+Dataset [24] holds 2, 745 classes of unique terms in the
+ASL. The authors recorded the data in a consistent lab-like
+environment with a handful of protagonists. The authors have
+not defined training and testing splits, resulting in an average
+of nearly 4 repetitions per gloss in the whole set.
+3) ASLLVD-Skeleton: Amorim et al. have later created an
+abbreviation of the ASLLVD dataset focused on evaluating
+pose-based methods. They open-sourced pose estimations of
+all the included videos from OpenPose [10] and proposed
+fixed training and test splits. The authors also introduced
+ASLLVD-Skeleton-20, a smaller subset with only 20 classes,
+enabling computationally resource-lighter and more distinc-
+tive ablations studies.
+B. Sign language recognition
+The primal works in SLR have leveraged shallow sta-
+tistical modeling such as Hidden Markov Models [32],
+[33], which achieved reasonable performance on very small
+datasets. A big leap has been observed with the advent of
+deep learning. Convolutional Neural Networks (CNNs) were
+amidst the first deep architectures employed for this prob-
+lem [9], [20], [26], [29]. These were used to construct unitary
+representations of the input frames that could be thereafter
+used for recognition. Later, various Recurrent Neural Net-
+works (RNNs) have been utilized for input encoding as well -
+namely Long Short-Term Memory Networks (LSTMs) [13],
+[19] or Transformers [8], [29]. The usage of different 3D
+CNNs has also been studied extensively (e.g., with I3D [11],
+[17], [21]). With the advances in pose estimation, another
+stream of approaches has emerged, making use of signer pose
+representations at the input. Unlike the previous methods,
+these models do not process raw RGB/RGB-D data, but
+rather pose representations of the estimated body, hand, and
+face landmarks. V´azquez-Enr´ıquez et al. [35] have been
+the first to use a Graph Convolutional Network (GCN)
+on top of pose sequences, following Yan et al. [38] who
+earlier proposed using GCNs for action recognition. Trans-
+formers have been recently employed in this regard, as
+Boh´aˇcek et al. [7] introduced Pose-based Transformer for
+SLR (SPOTER). While the architecture does not surpass
+the existing appearance-based approaches in general bench-
+marks, the authors have shown that when trained only on
+small splits of a training set, SPOTER outperforms even the
+appearance-based approaches significantly. Lastly, multiple
+
+Fig. 1.
+Illustrative examples of videos from the used datasets: ASLLVD, WLASL, and our new UWB-SL-Wild. ASLLVD contains videos from a
+homogeneous lab environment with few repetitions for each class. WLASL consists of videos captured in multiple settings with a larger instance repetition.
+UWB-SL-Wild, on the other hand, contains videos from an online dictionary with only a handful of examples for each class and both inconsistent signers
+and recording settings.
+ensemble models combining the raw visual data with the
+pose estimates [16] have also transpired.
+C. Few-shot gesture and action recognition
+Both few-shot gesture and action recognition have not
+gained extensive traction in literature and are hence not
+greatly investigated. Most methods have employed metric
+learning, where the similarity between input videos is learned
+to classify unfamiliar classes at inference using nearest
+neighbors. Bishay et al. [6] have proposed the TARN ar-
+chitecture, being the first to incorporate attention mechanism
+for this task. More recently, Generative Adversarial Networks
+(GANs) have also been studied in this regard [15].
+D. Few- and Zero-shot SLR
+Zero-shot SLR has been studied by Bilge et al. [4], [5].
+In both works, the authors propose a pipeline consisting
+of multiple RNNs and CNNs exploiting the BERT [14]
+representations of given SL lemmas’ textual translations in
+corresponding primary written language. This has enabled
+zero-shot SLR to a limited, yet promising extent, supposing
+the BERT embeddings are available. To the best of our
+knowledge, the only work addressing few-shot SLR specif-
+ically is introduced in [36]. Therein, Wang et al. leverage
+a Siamese Network [18] for feature extraction followed by
+K-means and a custom matching algorithm.
+III. UWB-SL-WILD
+Online SL dictionaries and learning resources are an
+excellent fit for in-the-wild training data, as they inherently
+dispose of a gloss annotation. However, since the primary
+intention with such platforms is not the training of neural
+networks, only a limited amount of repetitions can be found
+for each gloss (often 2 − 3). To the best of our knowledge,
+no available benchmark in the literature can simulate such a
+training paradigm, and we thus decided to create one. We
+collected a custom dataset called UWB-SL-Wild and are
+introducing it in this paper.
+There are numerous text-to-video dictionaries available on
+the internet1. We decided to use the Sign ASL dictionary as it
+1As
+an
+example,
+let
+us
+mention
+Spread
+the
+Sign
+(www.
+spreadthesign.com), Signing Savvy (www.signingsavvy.com),
+Handspeak (www.handspeak.com), and Sign ASL (www.signasl.
+org) websites.
+Fig. 2.
+Distribution of video repetitions per class in the UWB-SL-Wild
+dataset.
+introduces the largest variability of signer identities and video
+settings due to gathering videos from multiple providers.
+The first three websites either contain laboratory-like videos
+with a single signer (similar to the already existing datasets),
+have a limited vocabulary, or hold other unsuitable video
+properties (such as only possessing black-and-white footage).
+To allow for transfer learning experiments with the
+already-existing datasets, we decided that our dataset’s vo-
+cabulary would be equivalent to that of WLASL100. We
+then scraped the dataset structure from the Sign ASL portal.
+This yielded 307 videos from 100 classes (corresponding to
+lemmas in ASL), leaving us with a mean of under 3 repeti-
+tions per class. The total distribution of repetitions per class
+is depicted in Figure 2. There are 25 unique signers in the
+set. Each goes hand-in-hand with a different setting: video
+quality, distance and angle from the camera, and background.
+While some stand in front of a wall, many sit casually on a
+sofa or at a table. We manually annotated the signer identity
+in each video and are providing this information along with
+the dataset. The 100 classes in UWB-SL-Wild represent
+lemmas of frequent terms in ASL, including ordinary objects
+(e.g., book, candy, and hat), verbs (e.g., play, enjoy, go), and
+other adjectives or particles (e.g., thin, who, blue). Given that
+certain signs in ASL dispose of different variations, it may
+almost seem as if the signs gathered under a single class
+were sometimes completely different. Despite distinctively
+unalike in appearance, they still convey identical or highly
+similar meanings. This further enlarges the difficulty of
+learning on this dataset since some glosses’ sign variations
+
+ASLLVD (lab recording)
+UWB Wild Dataset (wild low-shot data)
+WLASL (uniform sources, large-scale)
+class mapping
+class mapping25
+23
+22
+20
+20
+19
+Number of classes
+15
+10
+7
+5
+5
+2
+1
+1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Number of training instanceseventually ended up with only a single instance in the entire
+set (supposing each of the 2 − 3 videos in a given class
+depicts a different variation). While this is not the case for
+most classes with just a single variant, a considerable part
+of the dataset’s glosses hold at least two versions. We thus
+provide manual annotations identifying different variations in
+each class. We created a mapping schema of classes between
+UWB-SL-Wild, WLASL100, and ASLLVD datasets2. This
+enables future researchers to train on and evaluate using these
+three datasets. Examples of videos from all three sources can
+be seen in Figure 1. We are open-sourcing the UWB-SL-
+Wild dataset, including the cross-datasets mappings and pose
+estimates of signers in all videos at https://github.
+com/matyasbohacek/uwb-sl-wild.
+IV. METHODS
+This section presents a method that can learn in a few-
+shot scenario. We build upon SPOTER [7], as it has shown
+substantial promise for training on smaller sets of data, fitting
+our few-shot use case. According to the authors, it should
+require lower amounts of training data because it is a pose-
+based method. We review the pipeline’s key elements and the
+changes we have made below. Any unmentioned attributes or
+configurations were kept identical. We hence refer the reader
+to the original publication for details.
+Preprocessing: We first estimate the signer’s pose in all
+input video frames. 2-D coordinates of key landmarks are
+obtained for the upper body (9), hands (2 × 21), and face
+(70).
+Augmentations and normalization: We follow the aug-
+mentation and normalization procedures from [7] to the full
+extent.
+A. Architecture
+SPOTER is a moderate abbreviation of the Transformer
+architecture [34]. The input to the network is a sequence
+of normalized and flattened skeletal representations with a
+dimension of 242. Learnable positional encoding is added to
+the sequence before it is processed further by the standard
+Encoder module. The input to the Decoder module is a
+single classification query. It is decoded into corresponding
+class probabilities by a multi-layer perceptron on top of the
+Decoder.
+TABLE II
+PERFORMANCE COMPARISON ON ASLLVD-SKELETON DATASET
+ASLLVD-S
+ASLLVD-S-20
+Model
+top-1
+top-5
+top-1
+top-5
+HOF [22]
+–
+–
+70.0
+–
+BHOF [22]
+–
+–
+85.0
+–
+ST-GCN [2]
+16.48
+37.15
+61.04
+86.36
+SPOTER [7]
+30.77
+52.05
+93.18
+97.72
+SPOTER + VSCT
+30.97
+52.87
+95.45
+100.00
+2There were no related videos for 3 classes of WLASL100 split in
+SignASL.org. We thus took the following 3 classes from the full WLASL
+to compensate for this.
+B. Validation score-conscious training
+In an attempt to adapt the SLR pipeline for the few-
+shot training environment, we propose the Validation Score-
+Conscious Training (VSCT). It aims to minimize the classi-
+fication error on the fly by identifying the bottleneck classes,
+i.e., the classes that get misclassified the most. VSCT adds
+the following steps at the end of each epoch of batch gradient
+descent:
+1) Validation accuracy is calculated for every class within
+the set. If a validation split is unavailable, the accuracy
+is computed on the training split.
+2) The classes are sorted by their performance. A set of
+classes Wvsct is found as a proportion of γvsct × c
+worst-performing ones, where c is the total number of
+classes.
+3) Next, a mini-batch is constructed as a random τvsct
+share of the training set with classes from Wvsct.
+4) Backpropagation is performed yet again on the above-
+described mini-batch. However, the parameters of aug-
+mentations are drawn from a different distribution. This
+allows us to target the problematic classes with better-
+suited representations.
+γvsct, τvsct, and all VSCT-specific augmentation parame-
+ters are constant hyperparameters of a training run.
+V. EXPERIMENTS
+In this section, we report our results compared to the
+already existing methods. We also evaluate our approach on
+a newly proposed benchmark leveraging the class mappings
+from UWB-SL-Wild and ASLLVD datasets to WLASL100.
+A. Implementation details
+The SPOTER architecture with VSCT has been imple-
+mented in PyTorch [25]. The model’s weights were initial-
+ized from a uniform distribution within [0, 1). We trained it
+for 130 epochs with an SGD optimizer. The learning rate
+was set to 0.001 with no scheduler and both momentum and
+weight decay set to 0, following the original implementation.
+VSCT hyperparameters differ based on the examined dataset.
+For body pose estimation, we used the HRNet-w48 [37]
+complemented by a Faster R-CNN [27] for person detec-
+tion within the MMPose library [30]. We also leveraged
+the Sweep functionality (hyperparameter search) within the
+Weights and Biases library [3] to find augmentation and
+VSCT hyperparameters. We namely employed the Bayesian
+hyperparameter search method 3 and conducted this proce-
+dure for each dataset individually.
+B. Quantitative results
+The results on the ASLLVD-Skeleton dataset, along with
+a comparison to the already available methods, are shown
+in Table II. We establish an overall state-of-the-art on this
+benchmark by achieving 30.97% top-1 and 52.87% top-5
+accuracy on the primary dataset. Our method surpasses the
+3For details on this search method, we refer the reader to the official
+Weights and Biases documentation available at https://docs.wandb.
+ai/guides/sweeps/.
+
+TABLE III
+RESULTS OF THE TRANSFER LEARNING EXPERIMENTS WHERE TRAINING AND EVALUATION WERE PERFORMED ON DIFFERENT DATASETS
+ASLLVD → WLASL
+UWB-SL-Wild → WLASL
+Norm.
+Aug.
+Bal. sample
+VSCT
+test
+val
+test
+val
+
+
+
+
+10.51
+5.94
+8.56
+7.41
+
+
+
+
+19.07
+16.62
+14.79
+16.62
+
+
+
+
+19.84
+15.73
+15.18
+16.91
+
+
+
+
+20.23
+15.13
+16.73
+14.54
+
+
+
+
+22.96
+13.95
+18.68
+16.02
+pose-based ST-GCN by a significant margin, almost doubling
+the top-1 performance. When evaluated on the much smaller
+20-class subsplit, SPOTER+VSCT achieves 95.45% top-
+1 and 100.0% top-5 accuracy, which exceeds the so far
+best BHOF by more than absolute 10%. Note that all the
+models listed in rows 1-5 of Table II use appearance-based
+representations. BHOF, for instance, builds upon a block-
+based histogram of the incoming videos’ optical flow.
+The latter of our evaluation settings makes use of
+the class mappings introduced in Section III. We trained
+SPOTER+VSCT on ASLLVD or UWB-SL-Wild dataset but
+calculated the accuracy on the WLASL100 testing set. We
+made the WLASL100 validation split available to the training
+procedure for the purposes of per-class statistics computa-
+tion within VSCT. The results are presented in Table III.
+SPOTER+VSCT achieves a top-1 accuracy of 22.96% when
+trained on ASLLVD and 18.68% when trained using UWB-
+SL-Wild.
+TABLE IV
+ABLATION STUDY ON ASLLVD-SKELETON DATASET
+ASLLVD-S
+Norm.
+Aug.
+Bal. sample
+VSCT
+Full
+20 cls.
+
+
+
+
+5.13
+47.73
+
+
+
+
+29.18
+86.36
+
+
+
+
+30.77
+88.64
+
+
+
+
+30.77
+90.91
+
+
+
+
+30.97
+95.45
+To provide context to these values, let us consider the
+results of Boh´aˇcek et al. [7] who trained and evaluated
+SPOTER (without VSCT) on WLASL100. They achieved
+63.18%, roughly three times greater accuracy. Their training
+set averaged 10.5 repetitions per class, whereas ASLLVD
+and UWB-SL-Wild have a mean of 3.6 and 2.9 per-class
+instances, respectively. Moreover, UWB-SL-Wild is signifi-
+cantly more variable as opposed to the other two datasets
+in both unique protagonists and camera settings. While
+these cross-dataset results are not nearly comparable to the
+standard methods applied for WLASL100 benchmarking, we
+believe they attest to the pose-based methods’ ability to
+generalize on characteristically distinct few-shot data.
+C. Ablation study
+We have conducted an ablation study of the individual con-
+tributions of normalization, augmentations, and the VSCT
+to the above-presented results. We also compare VSCT to
+the balanced sampling of classes, which counterbalances the
+disproportion of per-class samples in the training set. We
+summarize the ablations on the ASLLVD-Skeleton dataset
+and its 20-class subset in Table IV. Norm., Aug., and Bal.
+sample refer to using normalization, augmentations, and
+balanced sampling, respectively, in the given model variant.
+The baseline models achieved an accuracy of 5.13% and
+46.73%, respectively. We can observe that normalization
+itself provides the most significant improvement to 29.18%
+and 86.36%, while augmentations provide a slight boost on
+top of that, resulting in an accuracy of 30.77% and 88.64%.
+With all the previous modules fixed, we test the advan-
+tages of using either balanced sampling or VSCT. As for
+the complete dataset, balanced sampling does not provide
+any performance benefits, whereas VSCT brings a slight
+improvement resulting in 30.97% testing accuracy. When
+examined on the smaller subset, the balanced sampling
+improves the result by a relative 2.6% to 90.91%. VSCT,
+nevertheless, still outperforms it by enhancing the result with
+a relative 7.7% to the final 95.45% testing accuracy.
+The outturn of ablations on the cross-dataset training
+experiments is shown in Table III. For both ASLLVD and
+UWB-SL-Wild, we conduct the same ablations. The results
+mimic the tendencies commented on in the previous exper-
+iment. This study suggests that VSCT provides merits to
+training on such low-shot data, evincing itself more beneficial
+than a standard balanced sampling of classes.
+VI. CONCLUSION
+We collected and open-sourced a new dataset for SLR
+with footage from online text-to-video dictionaries. We con-
+structed it with the already-available datasets in mind and
+created class mappings to WLASL100 and ASLLVD. To
+reflect the attained problem’s few-shot setting, we proposed a
+novel procedure of training a neural pose-based SLR system
+called Validation Score-Conscious Training. This procedure
+analyzes intermediate training results on a validation split
+and adaptively selects samples from the worst-performing
+classes to create additional mini-batches for training. We
+demonstrated VSCT’s merits in several experiments of few-
+shot learning tasks utilizing the SPOTER model, resulting in
+a state-of-the-art result on the ASLLVD-Skeleton dataset.
+ACKNOWLEDGEMENT
+This work was supported by the Ministry of Educa-
+tion, Youth and Sports of the Czech Republic, Project
+No. LM2018101 LINDAT/CLARIAH-CZ. Computational
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+

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+arXiv:2301.00063v1  [math.PR]  30 Dec 2022
+The Sticky L´evy Process as a solution to a Time Change Equation
+Miriam Ram´ırez & Ger´onimo Uribe Bravo
+Instituto de Matem´aticas
+Universidad Nacional Aut´onoma de M´exico
+ABSTRACT. Stochastic Differential Equations (SDEs) were originally devised by Itˆo to provide a path-
+wise construction of diffusion processes. A less explored approach to represent them is through Time
+Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential
+Equations driven by random functions. We present a simple example where TCEs have some advantage
+over SDEs.
+We represent sticky L´evy processes as the unique solution to a TCE driven by a L´evy process with
+no negative jumps. The solution is adapted to the time-changed filtration of the L´evy process driving
+the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have
+no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for
+SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in
+probability, even though they do converge weakly. Instead, we provide strong approximation schemes for
+the solution of our TCE (by adapting Euler’s method for ODEs), whenever the driving L´evy process is
+strongly approximated.
+1. INTRODUCTION AND STATEMENT OF THE RESULTS
+Feller’s discovery of sticky boundary behavior for Brownian motion on [0,∞) (in [Fel52, Fel54])
+is, undoubtedly, a remarkable achievement. The discovery is inscribed in the problem of describing
+every diffusion processes on [0,∞) that behaves as a Brownian motion up to the time the former first
+hits 0. See [EP14] for a historical account and [IM63] for probabilistic intuitions and constructions.
+We now consider a definition for sticky L´evy processes associated L´evy processes which only jump
+upwards (also known as Spectrally Positive L´evy process and abbreviated SPLP). General information
+on SPLPs can be consulted in [Ber96, Ch. VII].
+Definition 1. Let X be a SPLP and X0 stand for X killed upon reaching zero. An extension of X0 will
+be c`adl`ag a strong Markov process Z with values in [0,∞) such that X and Z have the same law if
+killed upon reaching 0. We say that Z is a L´evy process with sticky boundary at 0 based on X (or a
+sticky L´evy process for short) if Z is an extension of X0 for which 0 is regular and instantaneous and
+which spends positive time at zero. In other words, if Z0 = 0 then
+0 = inf{t > 0 : Zt = 0} = inf{t > 0 : Zt ̸= 0}
+and
+� ∞
+0 I(Zs = 0)ds > 0
+almost surely.
+It is well known that sticky Brownian motion satisfies a stochastic differential equation (SDE) of the
+form
+(1)
+Zt = z+
+� t
+0 I(Zs > 0)dBs +γ
+� t
+0 I(Zs = 0)ds,
+t ≥ 0,
+2010 Mathematics Subject Classification.
+60G51, 60G17, 34F05.
+Research supported by UNAM-DGAPA-PAPIIT grant IN114720.
+1
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+2
+where B is a standard Brownian motion, the stickiness parameter γ is strictly positive and I denotes
+the indicator function. This equation has no strong solutions, which means that any process satisfying
+(1) involves some extra randomness to that of Brownian motion B. This result was conjectured by
+Skorohod and initially proved by R. Chitashvili in [Chi89] (later published as [Chi97]) and [War97].
+More recent proofs can be found in [EP14, Bas14] and [HCA17]. In contrast to the representation of
+the sticky Brownian motion as a solution to an SDE, we propose a representation of any SPLP with
+a sticky boundary as a solution to a TCE. The particularity of our representation is that it does not
+require any extra randomness to that generated by the L´evy process driving the equation. In the L´evy
+process case, a fundamental hypothesis to construct sticky L´evy processes will be that the sample paths
+have unbounded variation on any interval. Equivalently, we can assume that either there is a Gaussian
+component or the sum of jumps is absolutely divergent (i.e. ∑s≤t |Xs −Xs−| = ∞ almost surely for some
+t > 0).
+Theorem 1. Let X be a SPLP adapted to a right-continuous and complete filtration (Ft,t ≥ 0). Assume
+that the sample paths of X have unbounded variation. Given a parameter γ > 0 and a point z ≥ 0, there
+exists a unique pair of stochastic processes Z = (Zt,t ≥ 0) and C = (Ct,t ≥ 0) satisfying
+(2)
+Zt = z+XCt +γ
+� t
+0 I(Zs = 0)ds,
+where
+Ct =
+� t
+0 I(Zs > 0)ds,
+for every t ≥ 0. For the unique pair (Z,C) verifying Equation (2), it holds that C is a (Ft)-time change
+and that Z is adapted to the time-changed filtration ( �
+Ft,t ≥ 0) given by �
+Ft = FCt. Furthermore, Z is
+a sticky L´evy process based on X.
+This result attempts to honor the memory of Wolfgang Doeblin, the pioneer of TCEs, because for
+historical reasons that can be consulted in [BY02], the representation of diffusion processes suggested
+by Doeblin using TCEs is less known than the one given by Kiyosi Itˆo via SDEs. In particular, the
+region of applicability of TCEs has not been as carefully delineated as the one for SDEs. Note, however,
+that TCEs a priori do not even need the notion of a stochastic integral to be stated and, as showed in
+[CPGUB17, CPGUB13], TCEs have much better stability properties than SDEs.
+To explain the unbounded variation assumption, it implies that the Dini derivatives of X are infinite
+(as proved originally in [Rog68]; see [AHUB20] for an extension and further applications). In other
+words, at any given stopping time T (such as the hitting time of zero), we have
+−liminf
+h→0+
+XT+h −XT
+h
+= limsup
+h→0+
+XT+h −XT
+h
+= ∞.
+This will aid in proving that 0 is regular and instantaneous for Z. The following (counter)example also
+indirectly shows its relevance: the equation
+h(t) = β
+� t
+0 I(h(s) > 0)ds+γ
+� t
+0 I(h(s) = 0)ds
+does not admit solutions if β < 0 < γ. The difficulty with a time-change equation such as (2) is the
+discontinuity of the indicator functions of (0,∞) and of {0}. The success in its analysis follows from
+an explicit description of a solution in terms of reflection in the sense of Skorohod. This is done for a
+deterministic version of (2) in Proposition 3 of Section 2.2.
+Sticky L´evy processes are a one parameter family of processes built from the trajectories of X and
+are part of the notion of recurrent extensions of X0 analyzed in [RUB22] in terms of three non-negative
+constants and a measure on (0,∞). Such processes are called SPLP (with values) in [0,∞). As in Feller’s
+result, these parameters describe the domain of the infinitesimal generator L of the corresponding
+recurrent extension. A possible boundary condition describing such a domain is given by
+f ′(0+) = γ−1L f(0+)
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+3
+for some constant γ > 0. In the Brownian case, this condition corresponds to the so-called sticky
+Brownian motion with stickiness parameter γ. Generalizing the Brownian case, we will compute the
+boundary condition for the generator of the sticky L´evy process of Theorem 1 in Section 3.3. Gen-
+erator considerations are also relevant to explain the assumption on X having no negative jumps: The
+generator L of such a L´evy process acts on functions defined on R, but immediately makes sense on
+functions only defined on [0,∞). This last assertion is not true for the generator of a L´evy process with
+jumps of both signs.
+Our second main result exposes a positive consequence of the adaptability of the solution to the TCE
+(2). In [Bas14], an equivalent system to the SDE (1) is studied. In particular, it is showed that the non-
+existence of strong solutions prevent the convergence in probability of certain natural approximations
+to the solutions of the corresponding SDE, even though they converge weakly. In contrast, we present
+a simple (albeit strong!) approximation scheme for the solution to the TCE (2). To establish such a
+convergence result, we start from an approximation to the L´evy process X which drives the TCE (2).
+Theorem 2. Let X be a SPLP with unbounded variation. Let (Z,C) denote the unique solution to the
+TCE (2). Consider (Xn,n ≥ 1) a sequence of processes with c`adl`ag paths, such that each Xn is the
+piecewise constant extension of some discrete-time process defined on N/n and starts at 0. Suppose
+that Xn → X in the Skorohod topology, either weakly or almost surely. Let (zn,n ≥ 1) be a sequence
+of non-negative real numbers converging to a point z. Consider the processes Cn and Zn defined by
+Cn(0) = Cn(0−) = 0,
+Cn(t) = Cn(⌊nt⌋/n−)+(t −⌊nt⌋/n)I(Zn(t) > 0)
+(3)
+and
+Zn(t) = (zn +Xn −γ Id)(Cn(⌊nt⌋/n))+γ⌊nt⌋/n.
+(4)
+Then Cn →C uniformly on compact sets and Zn → Z in the Skorohod topology. The type of convergence
+will be weak or almost sure, depending on the type of convergence of (Xn,n ≥ 1).
+Observe that the above procedure corresponds to an Euler-type approximation for the solution to
+the TCE (2). If we consider the same equation but now driven by a process for which we could not
+guarantee the existence of a solution, our approximation scheme might converge but the limit might not
+be solution, as shown in the following simple but illustrative example. Let X = −Id, z = 0 and γ = 1.
+Then the approximations proposed in (3) and (4) reduce to
+Cn
+�2k −1
+n
+�
+= Cn
+�2k
+n
+�
+= k
+n
+and
+Zn
+�k
+n
+�
+=
+�
+0
+if k is even
+− 1
+n
+if k is odd
+for each k ∈ N. These sequences converge to C∗(t) = t/2 and Z∗ = 0, but clearly such processes do
+not satisfy TCE (2). In general, TCEs are very robust under approximations; the failure to converge is
+related to the fact that the equation that we just considered actually admits no solutions, as commented
+in a previous paragraph.
+Weak approximation results for sticky Brownian motion or of L´evy processes of the sticky type have
+been given in [Yam94] and [HL81]. In the latter reference, reflecting Brownian motion is used, while in
+the former, an SDE representation is used. In [BRHC20], the reader will find an approximation of sticky
+Brownian motions by discrete space Markov chains and by diffusions in deep-well potentials as well a
+numerical study and many references regarding applications. In particular, we find there the following
+phrase which highlights why Theorem 2 is surprising: ... there are currently no methods to simulate
+a sticky diffusion directly: there is no practical way to extend existing methods for discretizing SDEs
+based on choosing discrete time steps, such as Euler-Maruyama or its variants ... to sticky processes...
+It is argued that the Markov chain approximation can be extended to multiple sticky Brownian motions.
+In the setting of multiple sticky Brownian motions, one can consult [BR20] and [RS15]. We are only
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+4
+aware of a strong approximation of sticky Brownian motion, in terms of time-changed embedded simple
+and symmetric random walks, in [Ami91].
+The rest of this paper is structured as follows. We split the proof of Theorem 1 into several parts. In
+Section 2 we explore a deterministic version of the TCE (2), which is applied in Section 2.1 to show
+a monotonicity property, the essential ingredient to show uniqueness and convergence of the proposed
+approximation scheme (Section 2.3). In Section 2.2, we obtain conditions for the existence of the
+unique solution to the deterministic version of the TCE (2). The purpose of Section 3.1 is to apply
+the deterministic analysis to prove existence and uniqueness of the solution to the TCE (2) and the
+approximation Theorem 2. Then in Section 3.2, we verify that the unique process satisfying the TCE
+(2) is is measurable with respect to the time-changed filtration and that it is a sticky L´evy process.
+Finally in Section 3.3, using stochastic calculus instead of Theorem 2 from [RUB22], we analyze the
+boundary behavior of the solution to the proposed TCE to describe the infinitesimal generator of a
+sticky L´evy process.
+2. DETERMINISTIC ANALYSIS
+Following the ideas from [CPGUB13] and [CPGUB17], we start by considering a deterministic
+version of the TCE (2).
+We will prove that every solution to the corresponding equation satisfies a monotonicity property,
+which will be the key in the proof of uniqueness. Assume that Z solves almost surely the TCE (2).
+Hence, its paths satisfy an equation of the type
+(5)
+h(t) = f(c(t))+g(t),
+c(t) =
+� t
+0 I(h(s) > 0)ds.
+where f : [0,∞) → R is a c`adl`ag function without negative jumps starting at some non-negative value
+and g is an non-decreasing c`adl`ag
+function. (Indeed, we can take as f a typical sample path of
+t �→ z + Xt − γt and g(t) = γt.) Recall that, f being c`adl`ag , we can define the jump of f at t, denoted
+∆ f(t), as f(t) − f(t−). By a solution to (5), we might refer either to the function h (from which c is
+immediately constructed), or to the pair (h,c).
+We first verify the non-negativity of the function h.
+Proposition 1. Let f and g be c`adl`ag and assume that ∆ f ≥ 0, g is non-decreasing and f(0)+g(0) ≥
+0. Then, every solution h to the TCE (5) is non-negative. Furthermore, if g is strictly increasing, the
+function c given by c(t) =
+� t
+0 I(h(s) > 0)ds is also strictly increasing.
+Proof. Let h be a solution to (5) and suppose that it takes negative values. Note that h(0) = f(0) +
+g(0) ≥ 0 and that h is c`adl`ag without negative jumps. Hence, h reaches (−∞,0) continuously. The
+right continuity of f (and then of h) ensures the existence of some non-degenerate interval on which h
+is negative. Fix ε > 0 small enough to ensure that τ defined by
+τ = inf{t ≥ 0 : h < 0 on (t,t +ε)}
+is finite. (Note that, with this definition and the fact that f decreases continuously, we have that h(τ) =
+0. ) Given that h is negative on a right neighborhood of τ, then
+� τ
+0 I(h(s) > 0)ds =
+� τ+ε
+0
+I(h(s) > 0)ds,
+which leads us to a contradiction because
+0 = h(τ) = f
+�� τ
+0 I(h(s) > 0)ds
+�
++g(τ) ≤ f
+�� τ+ε
+0
+I(h(s) > 0)ds
+�
++g(τ +ε) = h(τ +ε) < 0.
+Hence, h is non-negative.
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+5
+Assume now that g is strictly increasing. By definition, c is non-decreasing. We prove that c is
+strictly increasing by contradiction: assume that c(t) = c(s) for some s < t. Then, h = 0 on (s,t) and,
+by working on a smaler interval, we can assume that h(s) = h(t) = 0. However, we then get
+0 = h(s) = f ◦c(s)+g(s) < f ◦c(s)+g(t) = f ◦c(t)+g(t) = h(t) = 0.
+The contradiction implies that c is strictly increasing.
+□
+If f−(t) = f(t−), note that the above result and (a slight modification of) its proof also holds for
+solutions to the inequality
+� t
+s I(h(r) > 0)dr ≤ c(t)−c(s) ≤
+� t
+s I(h(r) ≥ 0)dr
+where h(r) = f− ◦c(r)+g−(r) and f and g satisfy the hypotheses of Proposition 1. These inequalities
+are natural when studying the stability of solutions to (5) and will come up in the proof of Theorem 2.
+2.1. Monotonicity and Uniqueness. The following comparison result for the solutions to Equation
+(5) will be the key idea in the uniqueness proof of Theorem (1). Moreover, we pick up it in Section 2.3,
+where it also plays an essential role in the approximation of sticky L´evy processes.
+Proposition 2. Let ( f 1,g1) and ( f 2,g2) be pairs of functions satisfying that f i and gi are c`adl`ag ,
+∆ f i ≥ 0, gi is strictly increasing and f i(0)+gi(0) ≥ 0. Suppose that f 1 ≤ f 2 and g1 ≤ g2. If h1 and h2
+satisfy
+hi(t) = f i(ci(t))+gi(t),
+ci(t) =
+� t
+0 I(hi(s) > 0)ds,
+for i = 1,2, then we have the inequality c1 ≤ c2. In particular, Equation (5) admits has at most one
+solution when g is strictly increasing.
+Proof. Fix ε > 0 and define cε(t) = c2(ε +t). Set
+τ = inf{t > 0 : c1(t) > cε(t)}.
+To get a contradiction, suppose that τ < ∞. The continuity of c1 and cε guarantees that c1(τ) = cε(τ)
+and c1 is bigger than cε at some point t of every right neighborhood of τ. At such points, the inequality
+cε(t)−cε(τ) < c1(t)−c1(τ) is satisfied. Applying a change of variable, this is equivalent to
+(6)
+� t
+τ I(h2(ε +s) > 0)ds <
+� t
+τ I(h1(s) > 0)ds.
+The assumpions about g1 and g2 imply that g1(τ) < g2(ε +τ). Therefore
+0 ≤ h1(τ) = f 1(c1(τ))+g1(τ) < f 2(cε(τ))+g2(ε +τ) = h2(ε +τ).
+Thanks to the right continuity of h2, we can choose t close enough to τ such that h2(ε +s) > 0 for every
+s ∈ [τ,t). Going back to the inequality (6), we see that
+t −τ =
+� t
+τ I(h2(ε +s) > 0)ds <
+� t
+τ I(h1(s) > 0)ds ≤ t −τ,
+which is a contradiction. Therefore τ = ∞ and we conclude the announced result by letting ε → 0.
+In particular, if (h1,c1) and (h2,c2) are two solutions to (5) (driven by the same functions f and
+g), then the above monotonicity result (applied twice) implies c1 = c2 and therefore h1 = f ◦c1 +g =
+f ◦c2 +g = h2.
+□
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+6
+2.2. Existence. The following variant of a well-known result of Skorohod (cf. [RY99, Chapter VI,
+Lemma 2.1]) will be helpful to verify the existence of the unique solution to the TCE (5).
+Lemma 1. Let f : [0,∞) → R be a c`adl`ag function with non-negative jumps and f(0) ≥ 0. Then there
+exists a unique pair of functions (r,l) defined on [0,∞) which satisfies: r = f + l, r is non-negative,
+l is a non-decreasing continuous function that increases only on the set {s : r(s) = 0} and such that
+l(0) = 0. Moreover, the function l is given by
+l(t) = sup
+s≤t
+(− f(s)∨0).
+Note that the lack of negative jumps of f is fundamental to obtain a continuous process l.
+With the above Lemma, we can give a deterministic existence result for equation (5).
+Proposition 3. Assume that f is c`adl`ag , ∆ f ≥ 0 and f(0) ≥ 0. Let (r,l) be the pair of processes of
+Lemma 1 applied to f. If {t ≥ 0 : r(t) = 0} has Lebesgue measure zero, then, for every γ > 0 there
+exists a solution h to
+(7)
+h = f
+�� t
+0 I(h(s) > 0)ds
+�
++γ
+� t
+0 I(h(s) = 0)ds.
+Equivalently, in terms of Equation (5), the function h satisfies
+(8)
+h = f γ ◦c+γ Id,
+c(t) =
+� t
+0 I(h(s) > 0)ds.
+where f γ(t) = f(t)−γt.
+Proof. Applying Lemma 1 to f, we deduce the existence of a unique pair of processes (r,l) satisfying
+r(t) = f(t) + l(t) with r is a non-negative function and l a continuous function with non-decreasing
+paths such that l(0) = 0 and
+(9)
+� t
+0 I(r(s) > 0)l(ds) = 0.
+To construct the solution to the deterministic TCE (7), let us consider the continuous and strictly in-
+creasing function a defined by a(t) = t + l(t)/γ for every t ≥ 0. Denote its inverse by c and consider
+the composition h = r ◦c. The hypothesis on f implies that
+� t
+0 I(r(s) = 0)ds = 0 for all t. Therefore,
+since r is non-negative, then
+t =
+� t
+0 I(r(s) > 0)ds =
+� t
+0 I(r(s) > 0)(ds+γ−1l(ds)).
+Substituting the deterministic time t for c(t) in the previous expression and using that c is the inverse
+of a, we have
+c(t) =
+� c(t)
+0
+I(r(s) > 0)a(ds) =
+� t
+0 I(h(s) > 0)ds.
+Finally, the definition of a and its continuity imply l(t) = γ(a(t)−t), so that
+l(c(t)) = γ(t −c(t)) = γ
+� t
+0 I(h(s) = 0)ds.
+Hence, the identity h(t) = r(c(t)) can be written as
+h(t) = f
+�� t
+0 I(h(s) > 0)ds
+�
++γ
+� t
+0 I(h(s) = 0)ds,
+as we wanted.
+□
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+7
+2.3. Approximation. It is our purpose now to discuss a simple method to approximate the solution
+to the TCE (7). Among the large number of existing discretization schemes, we choose a widely used
+method, an adaptation of that of Euler’s. Again, the key to the proof relies deeply on our monotonicity
+result.
+Proposition 4. Let f be c`adl`ag and satisfy ∆ f ≥ 0, and f(0) ≥ 0. Assume that Equation (7), or
+equivalently (8), admits a unique solution denoted by (h,c). Let ˜f n be a sequence of c`adl`ag functions
+which converge to f and let f n = ˜f n −γ⌊n·⌋/n. Let cn and hn be given by cn(0) = cn(0−) = 0,
+cn(t) = cn(⌊nt⌋/n−)+(t −⌊nt⌋/n)I(hn(t) > 0)
+(10)
+and
+hn(t) = f n(cn(⌊nt⌋/n))+γ⌊nt⌋/n.
+(11)
+Then hn → h in the Skorohod J1 topology and cn → c uniformly on compact sets.
+Note that Propositions 2 and 3 give us conditions for the existence of a unique solution, which is
+the main assumption in the above proposition. Also, hn is piecewise on [(k −1)/n,k/n) and, therefore,
+cn is piecewise linear on [(k − 1)/n,k/n] and, at the endpoints of this interval, cn takes values in N/n.
+Hence, cn(⌊tn⌋/n) = ⌊ncn(t)/n⌋.
+The proof of Proposition 4 is structured as follows: we prove that the sequence (cn,n ≥ 1) is
+relatively compact. Given (cnj, j ≥ 1) a subsequence that converges to certain limit c∗, we see that
+((cnj,hnj), j ≥ 1) also converges and its limit is given by (c∗,h∗), where h∗ = f γ ◦c∗ +γ Id and we re-
+call that f γ = f −γ Id. A slight modification of the proof of Proposition 2 implies that the limit (c∗,h∗)
+does not depend on the choice of the subsequence (nj, j ≥ 1) and consequently the whole sequence
+((cn,hn),n ≥ 1) converges.
+Proof of Proposition 4. Since γ Id is continuous, then our hypothesiss ˜f n → f implies that f n → f −
+γ Id. (Since addition is not a continuous operation on Skorohod space as in [Bil99, Ex. 12.2], we need
+to use Theorem 4.1 in [Whi80] or Theorem 12.7.3 in [Whi02].)
+Fix t0 > 0. Note that Equation (10) can be written as
+cn(t) =
+� t
+0 I(hn(s) > 0)ds.
+This guarantees that the functions cn are Lipschitz continuous with Lipschitz constant equal to 1. Hence
+they are non-decreasing, equicontinuous and uniformly bounded on [0,t0]. It follows from Arzel`a-
+Ascoli Theorem that (cn,n ≥ 1) is relatively compact. Let (cnj, j ≥ 1) be a subsequence which con-
+verges uniformly in the space of continuous function on [0,t0], let us call c∗ to the limit, which is
+non-decreasing and continuous. Actually, c∗ is 1-Lipschitz continuous, so that c∗(t)−c∗(s) ≤ t −s for
+s ≤ t. This is a fundamental fact which will be relevant to proving that c = c∗. Since cnj(⌊njt⌋/nj) =
+⌊njcnj(t)⌋/nj for every t ≥ 0, we can write hnj = f nj ◦cnj +γ⌊nj·⌋/nj. We now prove that: as j → ∞:
+(cnj, f nj ◦cnj) → (c∗, f γ ◦c∗). Indeed, the convergence f n → f γ implies that liminfn→∞ f n(tn) ≥ f γ
+−(t)
+whenever tn → t. (If a proof is needed, note that Proposition 3.6.5 in [EK86] tells us that the accumu-
+lation points of f n(tn) belong to {f γ
+−(t), f γ(t)}.) Then,
+I( f γ
+− ◦c∗(s)+γs > 0) ≤ liminf
+j
+I( f nj ◦cnj(s)+γ⌊ns⌋/n > 0),
+so that, by Fatou’s lemma,
+� t
+s I( f γ
+− ◦c∗(r)+γr > 0)dr ≤ c∗(t)−c∗(s).
+But now, arguing as in Proposition 1, we see that f γ
+− ◦c∗ +γ Id is non-negative and that c∗ is strictly in-
+creasing. Since c∗ is continuous and stricly increasing, Theorem 13.2.2 in [Whi80, p. 430] implies that
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+8
+the composition operation is continuous at ( f γ,c∗), so that f nj ◦cnj → f γ ◦c∗. Since γ Id is continuous,
+we see that hnj → h∗ := f γ ◦c∗ +γ Id, as asserted.
+Another application of Fatou’s lemma gives
+� t
+s I( f γ ◦c∗(r)+γr > 0)dr ≤ c∗(t)−c∗(s).
+Now, arguing as in the monotonicity result of Proposition 2, we get c ≤ c∗.
+Let us obtain the converse inequality c∗ ≤ c by a small adaptation of the proof of the aforementioned
+proposition, which then finishes the proof of Theorem 2. Let ε > 0, define ˜c(t) = c(ε + t) and let
+τ = inf{t ≥ 0 : c∗(t) > ˜c(t)}. If τ < ∞, note that c∗(τ) = ˜c(τ) and, in every right neighborhood of τ,
+there exists t such that c∗(t) > ˜c(t). At τ, observe that
+0 ≤ h∗(τ) = f γ ◦c∗(τ)+γτ < f γ ◦ ˜c(τ)+γ(τ +ε) = h(τ +ε).
+Thanks to the right continuity of the right hand side, there exists a right neighborhood of τ on which
+h(· + ε) is strictly positive and on which, by definition of c, ˜c grows linearly. Let t belong to that
+right-neighborhood and satisfy c∗(t) > ˜c(t). Since c∗ is 1-Lipschitz continuous, we then obtain the
+contradiction:
+(t −τ) =
+� t
+τ I(h(ε +r) > 0)dr = ˜c(t)− ˜c(τ) < c∗(t)−c∗(τ) ≤ t −τ.
+Hence, τ = ∞ and therefore c∗ ≤ ˜c. Since this inequality holds for any ε > 0, we deduce that c∗ ≤ c.
+The above implies that c∗ = c and consequently h∗ = h. In other words, the limits c∗ and h∗ do not
+depend on the subsequence (nj, j ≥ 1) and then we conclude the convergence of the whole sequence
+((cn,hn),n ≥ 1) to the unique solution to the TCE (8).
+□
+3. APPLICATION TO STICKY L´EVY PROCESSES
+The aim of this section is to apply the deterministic analysis of the preceeding section to prove
+Theorems 1 and 2. The easy part is to obtain existence, uniqueness and approximation, while the
+Markov property and the fact that the solution Z to Equation (2) is a sticky L´evy process require some
+extra (probabilistic) work. We tackle the existence and uniqueness assertions in Theorem 1 and prove
+Theorem 2 in Subsection 3.1. Then, we prove the strong Markov property of solutions to Equation 2 in
+Subsection 3.2. This allows us to prove that solutions are sticky L´evy processes, thus finishing the proof
+of Theorem 1, but leaves open the precise computation of the stickiness parameter (or, equivalently,
+the boundary condition for its infinitesimal generator). We finally obtain the boundary condition in
+Subsection 3.3. We could use the excursion analysis of [RUB22] to obtain the boundary condition but
+decided to also include a different proof via stochastic analysis to make the two works independent.
+3.1. Existence, Uniqueness and Approximation. We now turn to the proof of the existence and
+uniqueness assertions in Theorem 1.
+Proof of Theorem 1, Existence and Uniqueness. Note that uniqueness of Equation (2) is immediate
+from Proposition 2 by replacing the c`adl`ag function f by the paths of x+X −γ Id and taking g = γ Id.
+To get existence, note that applying Lemma 1 to the paths of X, we deduce the existence of a unique
+pair of processes (R,L) satisfying Rt = z + Xt + Lt with R a non-negative process and L a continuous
+process with non-decreasing paths such that L0 = 0 and
+� t
+0 I(Rs > 0)dLs = 0. In fact, we have an explicit
+representation of L as
+(12)
+Lt = sup
+s≤t
+((−z−Xs)∨0) = −inf
+s≤t((z+Xs)∧0).
+Note that R corresponds to the process X reflected at its infimum which has been widely studied as a
+part of the fluctuation theory of L´evy processes (cf. [Ber96, Ch. VI, VII], [Bin75] and [Kyp14]).
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+9
+From the explicit description of the process L given in (12), it follows that P(Rt = 0) = P(Xt = Xt),
+where Xt = infs≤t(Xs ∧ 0). Similarly, we denote Xt = sups≤t(Xs ∨ 0). Proposition 3 from [Ber96, Ch.
+VI] ensures that the pairs of variables (Xt −Xt,−Xt) and (Xt,Xt −Xt) have the same distribution under
+P. Consequently
+P(Xt = Xt) = P((Xt −Xt,−Xt) ∈ {0}×[0,∞)) = P((Xt,Xt −Xt) ∈ {0}×[0,∞)) ≤ P(Xt = 0).
+The unbounded variation of X guarantees that 0 is regular for (−∞,0) and for (0,∞) (as mentioned, this
+result can be found in [Rog68] and has been extended in [AHUB20]). Hence, for any t > 0, Xt > 0.
+We decude that P(Xt = 0) = 1−P(Xs > 0 for some s ≤ t) = 0. Thus,
+E
+�� ∞
+0 I(Rt = 0)dt
+�
+=
+� ∞
+0 P(Xt = Xt)dt = 0.
+Therefore, we can apply Proposition 3 to deduce the existence of solutions to Equation (2).
+□
+Let us now pass to the proof of 2.
+Proof of Theorem 2. As we have stated in Theorem 2, we allow the convergence Xn → X to be weak
+or almost surely. Using Skorohod’s representation Theorem, we may assume that it is satisfied almost
+surely in some suitable probability space. The desired result follows immediately from Proposition 4
+by considering the paths of f = z+X −γ Id and f n = zn +Xn −γ⌊n·⌋/n.
+□
+3.2. Measurability details and the strong Markov property. In order to complete the proof of The-
+orem 1, it remains to verify the adaptability of the unique solution to the TCE (2) to the time changed
+filtration ( �
+Ft,t ≥ 0) and that such a solution is, in fact, a sticky L´evy process based on X. This is the
+objective of the current section, which ends the proof of Theorem 1.
+By construction the mapping t �→ Ct is continuous and strictly increasing. Furthermore, given that C
+is the inverse of the map t �→ t +Lt/γ, we can write
+{Ct ≤ s} = {γ(t −s) ≤ Ls} ∈ Fs,
+for every t ≥ 0. In other words, the random time Ct is a (Fs)-stopping time, since the filtration is
+right-continuous. Therefore the process C is a (Fs)-time change and Z is adapted to the time-changed
+filtration ( �
+Ft,t ≥ 0). In this sense we say that Z exhibits no extra randomness to that of the original
+L´evy process. This contrasts with the SDE describing sticky Brownian motion (cf. [War97, Theorem
+1]).
+Let us verify that the unique solution Z to (2) is an extension of the killed process X0. By construc-
+tion, we see that if Z0 = z > 0, then Z equals X until they both reach zero. Hence Z and X have the
+same law if killed upon reaching zero. Let now Z be the unique solution of (2) with Z0 = z = 0. The
+concrete construction which proves existence to (2) of Section 2.2 shows that
+γ
+� t
+0 I(Zs = 0)ds = L◦C
+where Ct =
+� t
+0 I(Zs > 0)ds, Lt = −infs≤t Xs. We have already argued that the unbounded variation
+hypothesis implies that Lt > 0 for any t > 0 and therefore L∞ > 0 almost surely. As above, recalling
+that C is the inverse of Id+L/γ, we see that C∞ = ∞. We conclude that L◦C∞ > 0 almost surely, so that
+Z spends positive time at zero. We will now use the unbounded variation of X to guarantee the regular
+and instantaneous character of 0 for Z. By construction, the unique solution Z to the TCE (2) is the
+process X reflected at its infimum by applying a continuous strictly increasing time change C to it, that
+is Z = R◦C where R = X −X. Consequently
+P(inf{s > 0 : Zs = 0} = 0) = P(inf{s > 0 : X ◦Cs = X ◦Cs} = 0) = P(inf{s > 0 : Xs = Xs} = 0).
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+10
+Since 0 is regular for (−∞,0) thanks to the unbounded variation hypothesis (meaning that X visits
+(−∞,0) immediatly upon reaching 0), we conclude the regularity of 0. Similarly, given the regularity
+of 0 for (0,∞) for X, we have
+P(inf{s > 0 : Zs > 0} = 0) = P(inf{s > 0 : Xs > Xs} = 0) ≥ P(inf{s > 0 : Xs > 0} = 0) = 1.
+Thus, 0 is an instantaneous point.
+To conclude the proof of Theorem 1, it now remains to prove the strong Markov property. From the
+construction of the unique solution to the TCE (2), we deduce the existence of a measurable mapping Fs
+that maps the paths of the L´evy process X and the initial condition z to the unique solution to the TCE
+(2) evaluated at time s, that is, Zs = Fs(X,z) for s ≥ 0. Let T be a ( �
+Ft)-stopping time. Approximating
+T by a decreasing sequence of ( �
+Ft)-stopping times (T n,n ≥ 1) taking only finitely many values, we
+see that CT is an (Ft)-stopping time. From the TCE (2), we deduce that
+ZT+s = ZT +(XC(T+s) −XC(T))+γ
+� s
+0 I(ZT+r = 0)dr.
+Consider the processes ˜C, ˜X and ˜Z given by ˜Cs =C(T +s)−C(T), ˜Xs = XC(T)+s −XC(T) and ˜Zs = ZT+s
+respectively. We can write the last equation as
+(13)
+˜Zs = ZT + ˜X ˜C(s) +γ
+� s
+0 I( ˜Zr = 0)dr,
+and ˜C satisfies ˜Cs =
+� s
+0 I( ˜Zr > 0)dr for s ≥ 0. In other words, ˜Z is solution to the TCE (2) driven by
+˜X with initial condition ZT. Consequently ˜Zs = Fs( ˜X,ZT). Note that ˜X has the same distribution as X
+and it is independent of �
+FT. Hence, the conditional law of ˜Z given �
+FT is that of F(·,ZT). (One could
+make appeal to Lemma 8.7 in [Kal21, p. 169] if needed.) This allows us to conclude that Z is a strong
+Markov process and concludes the proof of Theorem 1.
+3.3. Stickiness and martingales. In this section we aim at describing the boundary condition of the
+infinitesimal generator of the sticky L´evy process Z of Theorem 1 by proving the following result.
+Proposition 5. Let X be a L´evy process of unbounded variation and no negative jumps and let L be
+its infinitesimal generator. For a given z ≥ 0, let Z be the unique (strong Markov) process satisfying the
+time-change equation (2):
+Zt = z+X� t
+0 I(Zs>0)ds +γ
+� t
+0 I(Zs = 0)ds.
+Then, for every f : [0,∞) → R which is of class C2,b and which satisfies the boundary condition
+γ f ′(0+) = L f(0+), the process M defined by
+Mt = f(Zt)−
+� t
+0 L f(Zs)ds
+is a martingale and
+∂
+∂t
+����
+t=0
+E( f(Zt)) = L f(z).
+Theorem 2 from [RUB22] describes the domain of the infinitesimal generator of any recurrent exten-
+sion of X0 (which is proved to be a Feller process) by means of three non-negative constants pc, pd, pκ
+and a measure µ on (0,∞). To describe such parameters we note a couple of important facts about the
+unique solution to (2). By construction we can see that it leaves 0 continuously. Indeed, if we consider
+the left endpoint g of some excursion interval of Z, then Cg is the left endpoint of some excursion inter-
+val of the process reflected at its infimum R. Thanks to Proposition 2 from [RUB22], such excursions
+start at 0, so Z leaves 0 continuously. Thus, from [RUB22], pc > 0 and µ = 0. Note also that Z has
+infinite lifetime because R has it and C is bounded by the identity function, so pκ = 0. Finally, since Z
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+11
+spends positive time at 0, then pd > 0. Theorem 2 from [RUB22] ensures that every function f in the
+domain of the infinitesimal generator of Z satisfies
+f ′(0+) = pd
+pc
+L f(0+).
+Our proof of Proposition 5 does not require the results from [RUB22]. The main intention is to give
+an application of stochastic calculus, since we recall that a classical computation of the infinitesimal
+generator for L´evy processes is based on Fourier analysis (cf. [Ber96]). Regarding the generator L ,
+recall that it can be applied to C2,b functions such as f and that L f is continuous (an explicit expression
+is forthcoming). The lack of negative jumps implies that L f is defined even if f is only defined and
+C2,b on an open set containing [0,∞).
+Proof of Proposition 5. Let Z be the unique solution to the TCE (2) driven by the SPLP X. Itˆo’s formula
+for semimartingales [Pro04, Chapter II, Theorem 32] guarantees that for every function f ∈ C2
+0[0,∞):
+f(Zt) = f(z)+
+� t
+0 f ′(Z−
+s )dXCs +
+� t
+0 γ f ′(Z−
+s )I(Z−
+s = 0)ds+ 1
+2
+� t
+0 f ′′(Z−
+s )d[Z,Z]c
+s
++∑
+s≤t
+(∆ f(Zs)− f ′(Z−
+s )∆Zs).
+(14)
+In order to analyze this expression, we recall the so-called L´evy-Itˆo decomposition, which describes
+the structure of any L´evy process in terms of three independent auxiliary L´evy processes, each with a
+different type of path behaviour. Consider the Poisson point process N of the jumps of X given by
+Nt = ∑
+s≤t
+δ(s,∆Xs).
+Denote by ν the characteristic measure of N, which is called the L´evy measure of X and fulfills the
+integrability condition
+�
+(0,∞)(1 ∧ x2)ν(dx) < ∞. Then, we write the L´evy-Itˆo decomposition as X =
+X(1) + X(2) + X(3), where X(1) = bt + σBt is a Brownian motion independent of N, with diffusion
+coefficient σ 2 ≥ 0 and drift b = E[X1 −
+�
+(0,1]
+�
+[1,∞) xN(ds,dx)],
+X(2) =
+�
+(0,t]
+�
+[1,∞) xN(ds,dx)
+is a compound Poisson process consisting of the sum of the large jumps of X and finally
+X(3) =
+�
+(0,t]
+�
+(0,1) x(N(ds,dx)−ν(dx)ds)
+is a square-integrable martingale.
+Assuming the L´evy-Itˆo decomposition of X and using the next result, whose proof is postponed, we
+will see that
+� t
+0 f ′(Z−
+s )dXCs is a semimartingale of the form
+(15)
+Mt +
+� t
+0 bf ′(Z−
+s )(1−I(Zs = 0))ds+
+� t
+0 f ′(Z−
+s )dX(2)
+Cs ,
+for some square-integrable martingale M.
+Lemma 2. Let C be a (Ft)-time change whose paths are continuous and locally bounded. Let X be a
+right-continuous local martingale with respect to (Ft,t ≥ 0). Then the time-changed process XC is a
+right-continuous local martingale with respect to the time-changed filtration ( �
+Ft,t ≥ 0).
+Lemma 2 ensures that the time-changed process (σB + X(3)) ◦C remains a local martingale. Ac-
+cording to Theorem 20 from [Pro04, Chapter II], square-integrable local martingales are preserved
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+12
+under stochastic integration provided that the integrand process is adapted and has c`adl`ag paths. Con-
+sequently the stochastic integral1 M = f ′(Z−) · (σBC + X(3)
+C ) is a ( �
+Ft)-local martingale. Thanks to
+Corollary 27.3 from [Pro04, Chapter II], we know that a necessary and sufficient condition for a local
+martingale to be a square-integrable martingale is that its quadratic variation is integrable. Let us verify
+that E[[M,M]t] < ∞ for every t ≥ 0. Theorem 10.17 from [Jac79] implies the quadratic variation of the
+time-changed process coincides with the time change of the quadratic variation
+�
+σBC +X(3)
+C ,σBC +X(3)
+C
+�
+t =
+�
+σB+X(3),σB+X(3)�
+Ct ,
+t ≥ 0.
+Given that the Brownian motion B is independent of X(3), the quadratic variation is σ 2Ct +
+�
+X(3),X(3)�
+Ct,
+which is bounded by σ 2t +
+�
+X(3),X(3)�
+t. Thus
+E[[M,M]t] ≤ ∥f ′∥2
+∞E
+��
+σB+X(3),σB+X(3)�
+Ct
+�
+≤ ∥f ′∥2
+∞
+�
+σ 2t +t
+�
+(−1,1) x2 ν(dx)
+�
+< ∞.
+This verifies the decomposition (15). Later we will deal with the last term of this decomposition.
+Coming back to Itˆo’s formula (14), we need to calculate the term corresponding to the integral with
+respect to the continuous part of the quadratic variation of Z. First, we decompose the variation as
+[Z,Z]s = [XC,XC]s +2[XC,γ(Id−C)]s +γ2[Id−C,Id−C]s,
+for every s ≥ 0. The first term is [X,X]Cs. Given the finite variation of γ(Id−C) and the continuity
+of C, Theorem 26.6 from [Kal02] implies that almost surely the other two terms are zero. Thereby
+[Z,Z]s = [X,X]Cs for every s ≥ 0 and
+1
+2
+� t
+0 f ′′(Z−
+s )d[Z,Z]c
+s = 1
+2
+� t
+0 σ 2 f ′′(Z−
+s )(1−I(Zs = 0))ds.
+Now we analyze the last term on the right-hand side from (14), which corresponds to the jump part.
+Let us note that the discontinuities of f ◦Z derive from the discontinuities of Z, which are caused by
+the jumps of X ◦C, in other words
+{s ≤ t : |∆ f(Zs)| > 0}⊆{s ≤ t : ∆Zs > 0} = {s ≤ t : ∆(X ◦C)s > 0}.
+Making the change of variable r = Cs, the sum of the jumps in (14) can be written as
+(16)
+∑
+r≤Ct
+(∆ f(Z ◦Ar)− f ′(Z− ◦Ar)∆(Z ◦Ar)),
+where A denotes the inverse of C. We claim that A is a ( �
+Ft)-time change. Indeed, splitting in the cases
+r < t and r ≥ t, we see that {At ≤ s}∩{Cs ≤ r} = {t ≤Cs ≤ r} ∈ Fr for any r ≥ 0. Exercise 1.12 from
+[RY99, Chapter V] ensures that the time-changed filtration ( �
+FAt,t ≥ 0) is in fact (Ft,t ≥ 0). Thus, for
+any continuous function g, the process (g(Z−
+At),t ≥ 0) is (Ft)-predictable.
+We return to (15) to put together the sum of the jumps in (16) and the stochastic integral ( f ′ ◦Z−)·
+(X(2) ◦C). For this purpose, it is convenient to rewrite the last integral as ( f ′ ◦Z− ◦A ◦C) · (X(2) ◦C)
+and apply Lemma 10.18 from [Jac79] to deduce that ( f ′ ◦Z−) · (X(2) ◦C) = (( f ′ ◦Z− ◦A) · X(2)) ◦C.
+1We use both notations
+� Hs dXs and H ·X to refer to the stochastic integral.
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+13
+Consequently
+� t
+0 f ′(Z−
+s )dX(2)
+Cs + ∑
+s≤Ct
+(∆ f(Z ◦As)− f ′(Z− ◦As)∆(Z ◦As))
+=
+� Ct
+0
+�
+(0,∞)
+�
+f(Z−
+As +x)− f(Z−
+As)− f ′(Z−
+As)xI(x ∈ (0,1))
+�
+(N(ds,dx)−ν(dx)ds)
++
+� Ct
+0
+�
+(0,∞)
+�
+f(Z−
+As +x)− f(Z−
+As)− f ′(Z−
+As)xI(x ∈ (0,1))
+�
+ν(dx)ds.
+(17)
+Define the process M by
+Mt =−
+� t
+0
+�
+[1,∞)
+�
+f(Z−
+As +x)− f(Z−
+As)
+�
+ν(dx)ds
++
+� t
+0
+�
+[1,∞)
+�
+f(Z−
+As +x)− f(Z−
+As)
+�
+N(ds,dx)
++
+� t
+0
+�
+(0,1)
+�
+f(Z−
+As +x)− f(Z−
+As)− f ′(Z−
+As)x
+�
+(N(ds,dx)− ds).
+Since ν is a L´evy measure, then
+E
+�� t
+0
+�
+[1,∞)
+�� f(Z−
+As +x)− f(Z−
+As)
+�� ds
+�
+≤ ∥f∥2
+∞tν([1,∞)) < ∞.
+We develop the first degree Taylor polynomial of f(Z−
+As +x) to obtain
+f ′(Z−
+As)x = f(Z−
+As +x)− f(Z−
+As)−R(x),
+x ∈ (0,1),
+where the remainder R satisfies |R(x)| ≤ 1
+2∥f ′′∥∞x2. Therefore
+E
+�� t
+0
+�
+(0,1)
+�
+f(Z−
+As +x)− f(Z−
+As)− f ′(Z−
+As)x
+�
+ν(dx)ds
+�
+≤ 1
+2∥f ′′∥∞tE
+��
+(0,1) x2 ν(dx)
+�
+< ∞.
+Theorem 5.2.1 from [App09] ensures that M is a (Ft)-local martingale and Lemma 2 implies that MC
+is a ( �
+Ft)-local martingale. Furthermore, for t ≥ 0 it holds that
+E
+�
+sup
+s≤t
+|MCs|
+�
+≤ E
+�
+sup
+s≤t
+|Ms|
+�
+≤
+�
+2∥f∥∞ + 1
+2∥f ′′∥2
+∞
+�
+t
+�
+(0,∞)(1∧x2)ν(dx) < ∞.
+It follows from Theorem 51 from [Pro04, Chapter I] that MC is a true martingale.
+Gathering all the expressions involved in Itˆo’s formula (14), we get the semimartingale decomposi-
+tion
+f(Zt)− f(z) =Mt +
+� t
+0 bf ′(Z−
+s )(1−I(Zs = 0))ds+
+� t
+0 γ f ′(0+)I(Zs = 0)ds
++ 1
+2
+� t
+0 σ 2 f ′′(Z−
+s )(1−I(Zs = 0))ds+MCt
++
+� Ct
+0
+�
+(0,∞)
+�
+f(Z−
+As +x)− f(Z−
+As)− f ′(Z−
+As)xI(x ∈ (0,1))
+�
+ν(dx)ds.
+Recall that the extended generator of X (as in [RY99, Ch. VII]) is given by
+L f(z) = bf ′(z)+ σ 2
+2 f ′′(z)+
+�
+R+
+�
+f(z+x)− f(z)− f ′(z)xI(x ∈ (0,1))
+�
+ν(dx)
+
+The Sticky L´evy Process as a solution to a Time Change Equation
+14
+on C2,b functions and that the extended generator of X0 is given by L f on C2,b functions f on [0,∞)
+which vanish (together with its derivatives) at 0 and ∞. Note that L f(z) is bounded. Define
+˜
+L f(0) by
+˜
+L f(0) = (b−γ) f ′(0+)+ σ 2
+2 f ′′(0+)+
+�
+R+
+�
+f(x)− f(0+)− f ′(0+)xI(x ∈ (0,1))
+�
+ν(dx).
+Given that
+˜
+L f(0) = L f(0+)−γ f ′(0+), we can write the martingale M +MC as
+M +MCt = f(Zt)− f(z)−
+� t
+0 L f(Z−
+s )ds+
+� t
+0
+˜
+L f(0)I(Zs = 0)ds.
+We deduce that if a function f ∈ C2[0,∞) satisfies the boundary condition
+˜
+L f(0) = 0 or equivalently
+γ f ′(0+) = L f(0+), then f(Zt) − f(z) −
+� t
+0 L f(Zs)ds is a martingale. By hypothesis, the last term
+is bounded by a linear function of t, so that E[f(Zt)] is differentiable at zero and the derivative equals
+L f(z).
+□
+We conclude this section with the proof of Lemma 2.
+Proof. (Lemma 2) Let (βn,n ≥ 1) be localizing sequence for X, then βn → ∞ as n → ∞ and for each
+n ≥ 1, the process XβnI(βn > 0) is a uniformly integrable martingale. Keeping the notation A for the
+inverse of C, we will prove that (A(βn),n ≥ 1) is a sequence of ( �
+Ft)-stopping times that localizes to XC.
+The property of being (Ft)-stopping time is deduced by observing that {βn ≤ Ct} ∈ Fβn ∩FCt ⊂ �
+Ft,
+which implies that
+{A(βn) ≤ t}∩{Ct ≤ s} = {βn ≤ Ct}∩{Ct ≤ s} ∈ Fs.
+Since C ◦A = Id, then
+(Z ◦C)A(βn)
+t
+= ZCt∧βn = Zβn
+Ct .
+Given that Zβn is a (Ft)-martingale, Optional Stopping Theorem guarantees that
+E
+�
+Zβn
+Ct
+���FCs
+�
+= Zβn
+Cs ,
+0 ≤ s ≤ t.
+Hence (Z ◦C)A(βn) is a ( �
+Ft)-martingale. Moreover A(βn) → ∞ as n → ∞ since C ≤ Id.
+□
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+Stochastics of DNA Quantification
+Abdoelnaser M Degoot and Wilfred Ndifon∗
+African Institute for Mathematical Sciences, Next Einstein Initiative, Rwanda
+January 6, 2023
+1
+Abstract
+A common approach to quantifying DNA involves repeated cycles of DNA amplification.
+This approach, employed by the polymerase chain reaction (PCR), produces outputs that
+are corrupted by amplification noise, making it challenging to accurately back-calculate
+the amount of input DNA. Standard mathematical solutions to this back-calculation prob-
+lem do not take adequate account of such noise and are error-prone. Here, we develop a
+parsimonious mathematical model of the stochastic mapping of input DNA onto experi-
+mental outputs that accounts, in a natural way, for amplification noise. We use the model
+to derive the probability density of the quantification cycle, a frequently reported exper-
+imental output, which can be fit to data to estimate input DNA. Strikingly, the model
+predicts that a sample with only one input DNA molecule has a <4% chance of testing
+positive, which is >25-fold lower than assumed by a standard method of interpreting PCR
+data. We provide formulae for calculating both the limit of detection and the limit of quan-
+tification, two important operating characteristics of DNA quantification methods that are
+frequently assessed by using ad-hoc mathematical techniques. Our results provide a math-
+ematical foundation for the rigorous analysis of DNA quantification.
+2
+Introduction
+The quantification of genomic targets is of interest in a large variety of applications in
+biology, biotechnology and medicine, from determining an individual’s disease status to
+detecting minute changes in gene expression profiles occurring across space and time (eg.
+[1, 2]). This is typically achieved by converting non-DNA genomic targets into DNA, which
+is then amplified to enable its quantification. In principle, this allows even small numbers
+∗Address for correspondence: wndifon@aims.ac.za
+1
+arXiv:2301.02149v1  [q-bio.QM]  5 Jan 2023
+
+of genomic targets to be accurately measured. However, in practice, the DNA amplifica-
+tion process, being stochastic, generates outputs that contain noise. Accurate measure-
+ment, therefore, requires an adequate, quantitative understanding of this noise. Thus far,
+this has proved challenging to achieve.
+A specific and very popular instance of a DNA quantification method is the real-time
+polymerase chain reaction (PCR) [3, 4]. In PCR, DNA molecules are repeatedly amplified
+in a cyclic manner. As they are amplified, fluorescently labeled nucleotides are incorpo-
+rated into the newly formed DNA molecules, increasing the overall fluorescence emitted.
+The resulting fluorescence profile is used to determine the quantification cycle (denoted
+Cq or Ct value), at which the number of molecules exceeds a defined threshold, called
+the quantification threshold. A PCR reaction is considered to be positive if its Ct value
+is less than or equal to the maximum possible cycle. Despite the fact that the Ct value is
+only an indirect readout of the number of input DNA molecules, it is often the only re-
+ported output of PCR experiments. A variant of conventional PCR, called digital PCR [5],
+uses the fraction of positive reactions to estimate the number of input DNA molecules. To
+this end, it assumes that a reaction is positive if and only if it contains at least one target
+molecule. It is unclear under what conditions this assumption is valid, and when it must
+be discarded in favor of a more realistic alternative.
+Here we describe a parsimonious mathematical model that is useful for analysing the
+DNA quantification process, and for guiding the interpretation of experimental outputs.
+We use PCR as an example, although our analysis is applicable to other methods such as
+loop-mediated isothermal amplification of DNA [6]. Experiments indicate that the PCR
+process exhibits different phases, characterized by different efficiencies of DNA amplifi-
+cation. Therefore, we construct a mathematical model of a PCR process with an arbitrary
+number of phases, each with its own amplification efficiency. We use this model to obtain
+the following results:
+• We derive the generating function for the probability distribution of the number of
+molecules found in a PCR experiment at an arbitrary time t. We also derive the
+probability density function (pdf), mean, variance, and cumulative density function
+(cdf) of the Ct values produced by such an experiment. Either the pdf or the cdf can
+be fit to PCR data to estimate the number of input DNA molecules.
+• In the simplest instance of our model – a single-phase PCR model that accounts for
+amplification noise but not for (upstream) DNA sampling noise – the mean Ct value,
+given by (ψ(x +1)−ψ(n))/r, is well approximated by ln(x/n)/r [7] when n ≫ 1, where
+n is the number of input molecules, r (defined on a base-e scale) is the amplification
+efficiency, x is the quantification threshold, and ψ(·) denotes the digamma function.
+• We provide a formula for calculating the limit of detection (LoD) of a PCR experi-
+2
+
+ment, that is, the smallest number of input molecules that can be detected with a
+failure rate not exceeding α. Using a single-phase PCR model, we find that when
+α = 0.05, the LoD increases from 3, the value determined while accounting for sam-
+pling noise only, to ≈10 when both sampling noise and amplification noise (with r
+set to 95% of the maximum possible efficiency, m.p.e.) are considered. The LoD
+increases as r decreases, doubling to ≈20 at 90% m.p.e. This illustrates the under-
+appreciated, dramatic effect that amplification efficiency has on the LoD.
+• We provide a formula for calculating the limit of quantification (LoQ) of a PCR ex-
+periment, that is, the smallest number of molecules that can be quantified with a
+defined level of precision and a given maximum failure rate α. Counter-intuitively,
+the single-phase PCR model predicts that the LoQ does not depend on amplifica-
+tion efficiency. When α = 0.05, the LoQ increases from 10, obtained when up to a
+two-fold deviation from the expected number of input molecules is allowed, to 820,
+when at most a 10% deviation is allowed. This indicates that 10 or fewer molecules
+cannot be measured with a better than 2-fold error more than 95% of the time.
+• The model indicates that a key assumption commonly used when interpreting digital
+PCR data – that a PCR experiment with only one input molecule will always produce
+a positive outcome – is invalid under a wide range of conditions. Even when the
+amplification efficiency is set to a high value of 95% m.p.e, the probability that such
+an experiment will yield a positive outcome is predicted to be <4%. We describe
+two different approaches by which accurate estimates of the number of input DNA
+molecules may be obtained from digital PCR data.
+It should be noted that there have been previous attempts to improve the interpretation
+of PCR data through mathematical modeling. The classical approach to estimating the
+amount of DNA found in a focal sample involves comparing data generated by that sample
+versus data obtained from a reference sample containing either a known or an unknown
+amount of DNA [8]. The need for a reference sample with a known amount of DNA, the
+determination of which is itself subject to experimental error, makes accurate absolute
+quantification of DNA found in the focal sample challenging. An alternative approach
+involves fitting mathematical models, mostly phenomenological in their construction, to
+PCR data generated by the focal sample alone [4, 8, 9, 10, 11]. See [12] for a comparison
+of various methods based on this approach. None of these methods provides an adequate
+accounting of how amplification noise shapes PCR data.
+The remainder of this paper is organized as follows: We provide an overview of the
+model’s structure in Section 3.1 and present our main mathematical results in Sections
+3.2 and 3.3. We apply these results to compute the LoD and LoQ in Section 3.4, and
+we investigate how amplification noise complicates the accurate interpretation of digital
+3
+
+PCR data in Section 3.5. We summarize the results and discuss other applications of our
+methods in Section 4. To improve readability, we only present mathematical proofs and
+detailed calculations in the Appendix (Section 5.1).
+3
+Results
+3.1
+Preliminaries
+We model the PCR process as a continuous-time, discrete-state Markov jump process [13]
+evolving up to time T . This representation of the PCR process is based on the facts that
+(1) the primary products of PCR reactions, DNA molecules, are countable, and (2) what
+happens in the next cycle of the reaction is conditionally independent of what happened
+in the past given the present state of the reaction. Our decision to make time continuous
+(rather than discrete) is based on the fact that experimentally measured Ct values are
+positive real numbers. As a consequence, reaction rates are defined in base e instead of
+base 2 (expected for a discrete-time PCR process), but it is straightforward to convert
+between these two bases.
+We divide the time interval [0,T ] of the PCR process into p non-overlapping subin-
+tervals Ii, each one corresponding to a distinct phase of the process and associated with
+the probabilistic state transition rate ri, i = 1,2,...,p. These transition rates govern the ef-
+ficiency of DNA amplification. We derive the probability generating function [14] for the
+number of target molecules found at an arbitrary time t. We use this generating func-
+tion to derive the corresponding probability distribution and, importantly, the probability
+density function (pdf) of the Ct value. We derive the pdf in two different cases, namely
+1. when the initial state of the PCR process is deterministic, and the PCR phase lengths
+and amplification efficiencies are given; and
+2. when the initial state is Poisson-distributed, and the phase lengths and amplification
+efficiencies are given.
+To illustrate the mathematical ideas, we will report calculations and simulations based
+on a single-phase model. We argue that this simpler instance of our model is sufficient
+for analysing a large variety of real-world PCR experiments. In principle, each PCR ex-
+periment can be divided into the following three amplification rate-dependent phases: a
+pre-exponential phase, in which the amplification rate is sub-exponential; an exponential
+phase; and a post-exponential phase where the rate slows down as DNA molecules saturate
+the reagents required for their further amplification. However, in practice, the usual out-
+put of PCR experiments – the Ct value – is determined as soon as the PCR process enters
+the exponential phase, meaning that dynamics occurring in the pre-exponential phase pri-
+marily determine this particular outcome. Therefore, for the purposes of understanding
+4
+
+the factors that shape the Ct value and its statistics, and evaluating related operating char-
+acteristics of PCR, a single-phase model appears sufficient. Accordingly, when applicable,
+we highlight the forms taken by our mathematical equations in the case of a single-phase
+model. In addition, we estimate the LoD and LoQ using a single-phase model (Section
+3.4), which we also apply to critique the standard method of interpreting digital PCR data
+(Section 3.5).
+3.2
+Case 1: A PCR process with a deterministic initial state
+3.2.1
+Probability generating function for the number of molecules
+Theorem 1. Let {X(t),t ∈ R} be a continuous-time Markov process with p phases, a countable
+state space S ⊂ N+, phase-specific transition rates ri, i ∈ 1,2,...,p, and state transition proba-
+bility given by
+P (X(t′ + ∆t) = x|X(t′) = x′) = δ(x′ − x + 1)
+p
+�
+i=1
+ri1Ii(t′),
+(1)
+where 1 denotes the indicator function and δ(.) denotes the Kronecker delta function. If the pro-
+cess starts with n molecules, then the probability generating function for the number of molecules
+present at time t ∈ Ik, k ≤ p, is given by
+G(n,⃗r,t,⃗τ;s) =
+�
+se−z
+1 − s(1 − e−z)
+�n
+,
+(2)
+where
+z = rkt +
+k−1
+�
+i=1
+(ri − rk)τi,
+(3)
+Ii denotes the i’th phase and τi = |Ii|, i < k, is its length.
+The proof of this theorem is given in Section 5.1.1. We will now use the theorem to
+derive the probability distribution of the number of molecules found at time t.
+3.2.2
+Probability distribution of the number of molecules
+Corollary 1. The probability that there are x molecules at time t ∈ Ik in the PCR process de-
+scribed in Theorem 1 is given by the following negative binomial distribution:
+P(x|n,⃗r,t,⃗τ) =
+�x − 1
+n − 1
+�
+e−nz × (1 − e−z)x−n ,
+(4)
+where z is given by (3).
+The proof of this corollary is given in Section 5.1.2. We will now use this corollary to
+derive the pdf, mean, variance and cdf of the Ct value.
+5
+
+3.2.3
+pdf, mean and variance of the Ct value
+Let t be the Ct value of the PCR process described in Theorem 1. By definition, t is the time
+at which the number of molecules reaches the quantification threshold, which we denote
+by x. Let t ∈ Ik. In the Appendix [Section 5.1.5], we show that, given n,⃗r = (r1,r2,...,rk−1),
+and ⃗τ = (τ1,τ2,...,τk−1), the pdf of t has the following form:
+P(t|n,⃗r,⃗τ,x)
+=
+rke−nz (1 − e−z)x−n
+Bθ(n,x − n + 1) ,
+(5)
+where Bθ(n,x − n + 1) is the incomplete Beta function, z is given by (3), and
+θ = e−�k−1
+i=1 riτi.
+(6)
+For the single-phase PCR process, θ = 1, so the pdf is given by
+P(t|n,r1,x)
+=
+r1e−nz (1 − e−z)x−n
+B(n,x − n + 1)
+.
+(7)
+The mean Ct value is given by (see Section 5.1.5)
+E(t)
+=
+k−1
+�
+i=1
+τi + Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ)
+rkBθ(n,x − n + 1)
+,
+(8)
+where 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) is the regularized generalized hypergeometric function.
+For the single-phase process, the mean is given by
+E(t)
+=
+ψ(x + 1) − ψ(n)
+r1
+.
+(9)
+Observe that when n ≫ 1, the right-hand-side of (9) is well-approximated by ln(x/n)/r1.
+The latter expression is commonly used to approximate the mean Ct value. For example,
+it was used in [7] to estimate PCR amplification efficiency from data.
+The variance of the Ct value is given by E(t2) − E(t)2, where E(t2) is given by (85). For
+the single-phase process, the variance is given by
+Var(t) = ψ1(n) − ψ1(x + 1)
+r2
+1
+,
+(10)
+where ψ1(·) is the second polygamma function (also called the trigamma function).
+Finally, the cdf of the Ct value is given by [see Section 5.1.5]
+F(t|n,⃗r,⃗τ,x)
+=
+1 − Be−z(n,x − n + 1)
+Bθ(n,x − n + 1) .
+(11)
+6
+
+For the single-phase process, the cdf is given by
+F(t|n,r1,x)
+=
+1 − Ie−r1t(n,x − n + 1),
+(12)
+where Ie−r1t(n,x − n + 1) = Be−rt (n,x − n + 1)/B(n,x − n + 1) is the regularized incomplete Beta
+function. Sampling from this cdf is relatively straightforward: A random Ct value t is
+obtained as follows:
+t
+=
+−lnI−1
+1−u(n,x − n + 1)
+r1
+,
+(13)
+where u is sampled uniformly at random from the interval (0,1) and I−1
+1−u is the inverse of
+the regularized incomplete Beta function. To find a Ct value that corresponds to a quantile
+q ∈ (0,1), q is substituted for u.
+To estimate n, either the pdf or the cdf of t can be fit to data. Alternatively, the posterior
+density of n conditioned on t can be computed. In Section 5.1.5, we show that, for the
+single-phase process, it is given by
+P (n|r1,t,x)
+=
+e−(n−1)r1t(1 − e−r1t)x−n
+xB(n,x − n + 1)
+.
+(14)
+In Figure 1, we illustrate the shape of the single-phase pdf for different numbers of
+input molecules and different amplification efficiencies. To this end, we set T = 35 (a com-
+mon upper-bound for the duration of real-world PCR experiments) and x = 2T , which is
+equal to the number of molecules expected after T cycles under perfect amplification con-
+ditions (a sample that contains only one, perfectly amplified input molecule is expected
+to reach the quantification threshold, x, at time t ≤ T). We vary the efficiency from 60%
+m.p.e. (equivalent to setting r1 = 0.6 × ln2) to 100% m.p.e. The pdf has a bell shape, the
+location and width of which are governed by both the efficiency and the number of input
+molecules (Figure 1, left panel). Higher efficiencies or larger numbers of input molecules
+produce smaller mean Ct values, smaller variances, and narrower pdfs. In contrast, lower
+efficiencies or smaller numbers of input molecules produce larger mean Ct values, larger
+variances, and wider pdfs (Figure 1, left panel). In fact, Equation (5) predicts that in the
+limit as the efficiency goes to 0, the pdf will become flat as it will map every Ct value to 0.
+3.3
+Case 2: A PCR process with a Poisson-distributed initial state
+3.3.1
+Probability generating function for the number of molecules
+Theorem 2. Let {X(t),t ∈ R} be the continuous-time Markov process described in Theorem 1.
+If, instead of starting with a precisely known number of input DNA molecules, the initial state
+of the process is Poisson-distributed with mean λ, then the probability generating function for
+7
+
+Figure 1: pdf of the quantification cycle for the single-phase process. Processes with
+either a deterministic (left panel) or a Poisson-distributed (right panel) initial state were
+considered. The pdf was calculated using Equation (5) for the former case, and Equation
+(18) for the latter case. The quantification threshold, x, was set to 235. The mean µ and
+variance σ2 corresponding to different amplification efficiences r are shown. For ease of
+comprehension, r, which in our model is defined on a base-e scale, is shown as a percentage
+of its maximum possible value of ln(2).
+8
+
+n =1
+入 =1
+r=1.0,μ=35.83,α2=3.424
+r=1.0,μ=35.12,o2=3.257
+0.0010-
+r=0.9,μ=39.81,α2=4.227
+r=0.9,μ=39.02,2=4.021
+0.2-
+=0.8,μ=44.79,2=5.35
+r=0.8,μ=43.9,g2=5.089
+0.7,μ=51.19,02=6.987
+r=0.7,50.17,o2=6.647
+0.6,μ59/72,2=9.51
+=0.6，=58.54,2=9.048
+0.0005-
+0.1-
+0.0000
+0.0
+30
+40
+50
+60
+70
+30
+40
+50
+60
+70
+n=10
+入=10
+0.6-
+0.004
+r=1.0,μ=31.75,2=0.219
+r=1.0,μ=31.84,2=0.54
+Probability density
+r=0.9.μ=35.28,g2=0.27
+0.5-
+r=0.9,μ=35.38,o2=0.666
+0.003-
+r=0.8,μ=39.69,g2=0.342
+r=0.8,μ=39.8,α2=0.843
+0.4-
+r=0.7，μ=45.36,?元0.447
+r=6|7,μ=45.49.@2=1.101
+0.002
+r=06/μ=52.92,=0.608
+0.3-
+r=0.6，μ=53.07/=1.499
+0.2-
+0.001
+0.1-
+0.000
+0.0-
+30
+40
+50
+30
+40
+50
+n=100
+入=100
+2.0-
+r=1.0,μ=28.36,2=0.021
+r=1.0,μ=28.37o2=0.042
+r=0.9,μ=31.51,2=0.026
+r=0.9,μ=31.52,2=0.052
+1.5
+0.010
+r=0.8,μ=35.45,2=0.033
+r=0.8,μ=35.46,o2=0.066
+r=0.7
+μ=40.52,2=0.@43
+r=0.7μ=40.53,2=0.087
+r=0.6.
+μ=47.27, 2=0.958
+1.0-
+r=0.6,μ=47.28,o2=q.118
+0.005
+0.5-
+0.000-
+0.0
+30
+35
+40
+45
+30
+35
+40
+45
+Cycles
+Cyclesthe state of the process at a future time t ∈ Ik is given by
+G(λ,⃗r,t,⃗τ;s) = e
+λ(s−1)
+1−s(1−e−z) ,
+(15)
+where z is given by (3).
+The proof of Theorem 2 is given in Section 5.1.3. We now use Theorem 2 to derive the
+probability distribution of the number of molecules found in the PCR process at time t.
+3.3.2
+Probability distribution of the number of molecules
+Corollary 2. The probability that there are x molecules at cycle t ∈ Ik in the PCR process
+described in Theorem 2 is given by:
+P(x|λ,⃗r,t,⃗τ) = e−λ (1 − e−z)x
+x
+�
+i=1
+�x−1
+i−1
+�
+i!
+� λe−z
+1 − e−z
+�i
+,
+(16)
+where z is given by (3).
+The proof of this corollary is provided in Section 5.1.4. It is interesting to note that
+from the proof emerged the following combinatorial triangle, which is related to the well-
+known Narayana triangle [15]:
+x
+1
+1
+2
+1
+2
+3
+1
+6
+6
+4
+1
+12
+36
+24
+5
+1
+20
+120
+240
+120
+1
+2
+3
+4
+5
+k
+.
+The entries of this triangle, given by
+T(x,k) =
+� x
+k − 1
+��x − 1
+k − 1
+�
+(k − 1)!, x ∈ Z+,k = 1,2,...,x,
+(17)
+count the number of ways of obtaining x molecules by replicating a randomly selected
+subset of k molecules. T (x,k) is related to the Narayana numbers N(x,k) by
+T (x,k) = k! N(x,k).
+We will now use this corollary to derive the pdf of the Ct value together with the mean,
+variance and cdf.
+9
+
+3.3.3
+pdf, mean and variance of the Ct value
+Let t be the Ct value of the PCR process described in Theorem 2. As noted earlier, the Ct
+value t is the time at which the number of DNA molecules found in the process reaches
+the quantification threshold, which we denote by x. In the Appendix [Section 5.1.6], we
+show that the pdf of t is given by
+P(t|λ,⃗r,⃗τ,x)
+=
+rkλe−z(1 − e−z)x−1 1F1
+�
+1 − x;2; −λe−z
+1−e−z
+�
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+,
+(18)
+where θ is given by (6), z is given by (3), 1F1(a;b;c) is the hypergeometric function (also
+called the Kummer confluent hypergeometric function of the first kind).
+For the single-phase process, the pdf is given by [see Section 5.1.6]
+P(t|λ,r1,x) =
+r1xλe−r1t(1 − e−r1t)x−1 1F1
+�
+1 − x;2; −λe−r1t
+1−e−r1t
+�
+eλ − 1
+.
+(19)
+The mean Ct value is given by [see Section 5.1.6]
+E(t)
+=
+�x
+j=1
+(x−1
+j−1)λj
+j!
+�
+rkBθ(j,x − j + 1)�k−1
+i=1 τi + Γ(j)2θj 3 ˜F2(j,j,j − x;j + 1,j + 1;θ)
+�
+rk
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+, (20)
+while the second moment is given by (119).
+For the single-phase process, the mean and variance are, respectively, given by [see
+Section 5.1.6]
+E(t)
+=
+ψ(x + 1)
+r1
+−
+�x
+j=1
+λj
+j! ψ(j)
+r1
+�
+eλ − 1
+�
+and
+(21)
+Var(t) =
+�
+eλ − 1
+��x
+j=1
+λj
+j!
+�
+ψ1(j) + ψ(j)2
+�
+−
+��x
+j=1
+λj
+j! ψ(j)
+�2
+�
+r1(eλ − 1)
+�2
+− ψ1(x + 1)
+r2
+1
+.
+(22)
+The cdf of the Ct value is given by [see Section 5.1.6]
+F(t|λ,⃗r,⃗τ,x)
+=
+1 −
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Be−z(j,x − j + 1)
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+.
+(23)
+10
+
+For the single-phase process, the cdf is given by
+F(t|λ,r1,x)
+=
+1 −
+x�x
+j=1
+(x−1
+j−1)λj
+j!
+Be−r1t(j,x − j + 1)
+eλ − 1
+.
+(24)
+To estimate λ, either the pdf or the cdf of t can be fit to data. Alternatively, the posterior
+density of λ conditioned on t can be computed. In Section 5.1.6, we show that, for the
+single-phase process, it is given by
+P(λ|r1,t,x) =
+λw 1F1(1 − x;2; −λw
+1−w )
+(eλ − 1)(1 − w)�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+,
+(25)
+where w = e−r1t and ζ(·) denotes the Riemann zeta function.
+Note that, because they result from calculating expectations over the Poisson distribu-
+tion, the summations found in Equations (18) - (25) can be truncated at any value of j ≫ λ
+without a loss of accuracy.
+In Figure 1, we illustrate the shape of the single-phase pdf for different values of λ
+and different amplification efficiencies. As was the case for the PCR process with a deter-
+ministic initial state (Figure 1, left panel), the pdf also has a bell shape (Figure 1, right
+panel). Its location and width are governed by both the efficiency and λ. Consistent with
+expectations, higher efficiencies or larger values of λ produce smaller mean Ct values,
+smaller variances, and narrower pdfs (Figure 1, right panel). In contrast, lower efficiencies
+or smaller values of λ produce larger mean Ct values, larger variances, and wider pdfs.
+3.4
+Limit of detection and limit of quantification
+We will now demonstrate theoretically how the mathematical framework we have devel-
+oped can be applied to achieve certain operationally important objectives. In particular,
+it is often of interest to quantify the limit of detection (LoD) of a particular instance of
+the PCR method (henceforth referred to as “PCR protocol”). The LoD of a PCR protocol
+is the smallest number of molecules that it can detect with a failure rate not exceeding a
+defined threshold α (α is also called the significance level). Protocols with smaller LoDs
+are in general preferred to those with larger LoDs. Ideally, the LoD should be either equal
+to or smaller than the number of input DNA molecules expected in the considered sample.
+Another important operational objective is to determine a PCR protocol’s limit of quan-
+tification (LoQ) – i.e. the smallest number of molecules that it can estimate with a given
+level of precision (measured here using the parameter β) and a given maximum failure
+rate α. When a ≥ β-fold change in the number of target molecules needs to be detected,
+the protocol should have an LoQ with precision ≤ β. The methods available for estimat-
+ing LoD and LoQ are laborious [16] and frequently rely on certain ad-hoc mathematical
+11
+
+approximations [16, 17], which we would like to circumvent by developing and executing
+mathematically precise statements of the estimation problem.
+We begin with the LoD estimation problem. For the PCR process with a deterministic
+initial state, the LoD can be expressed as follows:
+LoD =
+min n
+s.t. F(T |n,⃗r,⃗τ,x) > 1 − α,
+(26)
+where F(T|n,⃗r,⃗τ,x) is given by (11) and T is the maximum practical duration of the PCR
+process. For the process with a Poisson-distributed initial state, n is replaced by λ.
+In Supplementary Figure 5.1, we show how the LoD varies with amplification effi-
+ciency in a single-phase process with either a deterministic or a Poisson-distributed initial
+state. In the former case, the process contains amplification noise but no sampling noise
+while in the latter case it contains both sampling noise and amplification noise. For com-
+parison, we also show the LoD in a process with sampling noise, modeled by using the
+Poisson distribution, but without amplification noise. The LoD is lowest in a process with
+sampling noise alone (LoD = 3 molecules) and it is highest when both sampling noise and
+amplification noise are present (LoD ranges from 6 molecules, at 100% of maximum pos-
+sible efficiency or m.p.e, to 157 molecules, at 80% m.p.e). A process with amplification
+noise but without sampling noise has an intermediate LoD. In these computational exam-
+ples, the parameters of the equation used to estimate the LoQ are perfectly known, and
+this makes it possible to obtain perfect knowledge of the LoD. In real-world applications,
+the parameter values will be associated with uncertainty, which will, in a quantifiable way,
+make uncertain the LoD estimates.
+We now turn our attention to the problem of estimating the LoQ in a PCR process
+with a deterministic initial state. Suppose that a Ct value t is generated by such a process
+and then used to obtain an estimate, denoted ˆn, of the number of input molecules. Let n
+be the actual number of input molecules. We want to calculate the probability that, for
+any data t generated by the same process, ˆn will not differ from n by more than a factor
+β,β ≥ 1. We define the LoQ as the smallest value of n for which this probability exceeds
+1 − α. Specifically, the LoQ is given by
+LoQ =
+min n
+s.t. P
+��n/β� ≤ ˆn ≤ �βn� | n,⃗r,⃗τ,x
+�
+> 1 − α,
+(27)
+where ⌊v⌋ (respectively ⌈v⌉) denotes the largest (respectively smallest) integer less than
+(respectively greater than) or equal to v.
+Focusing on the single-phase process, to obtain P (�n/β� ≤ ˆn ≤ �βn� | n,r1,x), we marginal-
+ize the right-hand-side of (14) with respect to t and then take the sum from �n/β� to �βn�,
+12
+
+yielding [see Section 5.1.5]
+P (�n/β� ≤ ˆn ≤ �βn� | n,r1,x)
+=
+⌈βn⌉
+�
+ˆn=⌊n/β⌋
+B( ˆn + n − 1,2x − ˆn − n + 1)
+xB( ˆn,x − ˆn + 1)B(n,x − n + 1)
+=
+P (�n/β� ≤ ˆn ≤ �βn� | n,x).
+(28)
+Strikingly, while Equation (28) depends on both n and x, it does not depend on the ampli-
+fication efficiency r1. In the Appendix [see Equation (130)], we follow a similar procedure
+to obtain P
+��λ/β� ≤ ˆλ ≤ �βλ� | λ,r1,x
+�
+, the probability that, for any data t generated by a
+single-phase PCR process with a Poisson-distributed initial state, the estimated value ofλ,
+denoted ˆλ, will not differ from the actual value by more than a factor β.
+Setting α = 0.05 and allowing at most a 10% deviation of ˆn from n (corresponding to
+setting β = 1.1) results in an LoQ of 820 molecules. The LoQ decreases to 146 molecules
+when a deviation of up to 25% from expectation is allowed (corresponding to β = 1.25),
+and to 43 molecules when the allowable deviation increases to 50% (corresponding to β =
+1.5). The analysis suggests that at the considered 5% failure rate, 10 input molecules can be
+detected with an error of at least ≈200%, that is, a ≈ 2 fold deviation from n (corresponding
+to β = 2). Because these calculations do not account for sampling noise, they provide only
+a lower-bound for the LoQ that is achievable at the considered failure rate and level of
+precision (β). As noted earlier, in real-world applications the parameters of the equation
+used to estimate LoQ will be imperfectly known, and this will determine the amount of
+uncertainty associated with the estimated LoQ.
+3.5
+Amplification noise determines digital PCR outcomes
+As noted earlier, digital PCR is a variant of conventional PCR that was developed to im-
+prove the quantification of DNA. In digital PCR, a sample master mix (containing an un-
+known number of input DNA molecules together with all the reagents required for DNA
+replication) is uniformly distributed into hundreds (and sometimes thousands) of phys-
+ical partitions, which may take the form of droplets or microwells [5]. Each partition is
+expected to receive zero, one, or more DNA molecules following a Poisson distribution
+with mean λ = CV /D, where C is the concentration of the DNA in the original sample, V
+is the partition volume, and D is the (known) dilution factor applied to the sample during
+preparation of the mastermix. PCR reactions are independently and simultaneously run
+inside each partition, and positive partitions are identified. The resulting data – i.e. the
+positive or negative outcome of each PCR reaction – are thus digital. The standard method
+of interpreting these data assumes that partitions that receive at least one target molecule
+will test positive, and their fraction, ˆf , is approximated by ˆf ≈ 1 − e−λ, from which λ is
+estimated as ˆλ = −ln(1 − ˆf ) and then used to estimate C.
+13
+
+Figure 2: Under-estimation of λ by the standard method of interpreting digital PCR
+data. We varied both the amplification efficiency and the amount of input DNA λ and
+used Equation (29) to estimate λ. The resulting estimate, denoted ˆλ, was plotted against
+efficiency, expressed as a percentage of the maximum possible efficiency. At efficiencies
+lower than 95%, only a very small amount of the input DNA is detected. At 95% efficiency,
+the amount detected ranges from 12%, when λ = 1, to 69%, when λ = 100. The amount
+detected increases to ≈80% when the efficiency equals 100%.
+However, according to our model, the assumption that the fraction of positive parti-
+tions equals the Poisson probability that a partition receives one or more target molecules
+is untenable due to the effects of PCR amplification noise. Indeed, setting the amplifi-
+cation efficiency to a reasonably high value of 95% m.p.e (i.e. r1 = 0.95 ln2) and us-
+ing T = 35,x = 2T in Equation (24), we predict that <4% of partitions that contain only
+one molecule will test positive, which is >25-fold smaller than assumed by the Poisson
+method. In Supplementary Figure 5.2, we compare the fraction of positive partitions cal-
+culated using our model [Equation(24)] versus the positive fraction calculated by the Pois-
+son method, for different values of λ and different amplification efficiencies. We find that
+the Poisson method over-estimates the fraction of positive partitions for small values of λ,
+including the value (1.61) at which the method is expected [18] to produce its most precise
+estimates of λ. Only for a relatively large value of λ (10) do we find the Poisson method’s
+estimate of the fraction of positive partitions to agree with the noise-adjusted expectation
+calculated using our model (Supplementary Figure 5.2).
+We use our model to investigate how this over-estimation of the fraction of positive
+partitions affects the accuracy of the estimate of λ (denoted ˆλ) produced by the Poisson
+14
+
+入=1
+入 =1.61
+0.8-
+1.0-
+0.6
+0.4-
+0.5
+0.2-
+0.0
+0.0
+80
+85
+90
+95
+100
+80
+85
+90
+95
+100
+>
+入=10
+入=100
+6-
+80
+5 
+60
+4
+3-
+40
+2-
+20
+1
+0
+0.
+80
+85
+90
+95
+100
+80
+85
+90
+95
+100
+Amplificationefficiency,r(%)method. Setting t = T in Equation (24), we find that
+ˆλ
+=
+−ln
+�
+1 − ˆf
+�
+=
+ln
+�
+eλ − 1
+�x
+j=1
+(x
+j)
+(j−1)!λjBe−r1T (j,x − j + 1)
+�
+.
+(29)
+According to Equation (29), ˆλ depends strongly on the amplification efficiency r1. ˆλ equals
+0 in the limit as r1 tends to 0. As r1 increases to its maximum possible value of ln(2), ˆλ
+also increases, approaching λ. In Figure 2, we illustrate the relationship between ˆλ/λ and
+amplification efficiency. Strikingly, for efficiencies lower than 95% m.p.e, ˆλ/λ is very small,
+indicating that λ is markedly under-estimated by the Poisson method. When the efficiency
+equals 95% m.p.e, ˆλ/λ increases from ≈ 12% (at λ = 1) to ≈69% (at λ = 100). Increasing the
+efficiency to the maximum possible value of 100% m.p.e causes ˆλ/λ to increase to ≈80%
+(at λ = 100). These results indicate that the Poisson method is expected to under-estimate
+λ because it does not account for amplification noise. Indeed, experimental data show a
+strong tendency by the method to under-estimate the number of input DNA molecules
+(eg. see Supplementary Table 6 in [19]).
+4
+Discussion
+The outputs of DNA quantification experiments, including those based on the polymerase
+chain reaction (PCR), tend to vary within and across different experimental instances,
+making the results difficult to interpret and limiting their utility beyond the particular
+contexts in which they are generated. Indeed, various factors are known to contribute
+to the variability of PCR outputs [20, 21, 22] including the varying complexity of DNA
+templates and the random distribution of target molecules in the reaction environment;
+the type of PCR machine and buffer components used; the durations and temperatures of
+the three thermal cycles of PCR; the binding kinetics of oligonucleotide primers to target
+DNA; and the stability of DNA polymerase and other PCR reagents. Taylor et. al [22]
+reviewed the sources of variability in PCR experiments and proposed a stepwise process
+to minimize such variability in practice.
+A common output of a PCR experiment is the quantification cycle (denoted Ct or Cq
+value), the PCR cycle at which the number of DNA molecules exceeds a defined threshold,
+called the quantification threshold. The Ct value varies with both the number of input
+DNA molecules and the PCR amplification efficiency, which in turn varies with the afore-
+mentioned experimental variables. It is desirable to deconvolute such variable outputs to
+estimate the number of input DNA molecules, which is of greatest interest in experiments,
+by applying mathematical methods that account for the stochasticity that is inherent in the
+15
+
+underlying generative process.
+We have developed a mathematical approach to modeling DNA quantification that
+takes into account the underlying stochasticity. We used PCR, the most widely used class
+of DNA quantification process, to illustrate our mathematical ideas, which are also ap-
+plicable to a broader class of such processes (eg. [6]). Using the model, we derived the
+probability generating function for the number of molecules found in a PCR process with
+either a deterministic or a Poisson-distributed number of input molecules as well as the
+probability density function (pdf), mean, variance and cumulative density function (cdf)
+of the Ct value produced by such a process. In contrast to the deterministic case, in which
+PCR outputs are contaminated only by amplification noise, in the latter case the outputs
+also contain sampling noise. The equations we derived for these important statistical prop-
+erties of the PCR process revealed functional relationships between the Ct value and un-
+derlying variables that could previously only be accessed by empirical means.
+To illustrate our mathematical ideas, we focused on the single-phase PCR process, for
+which our modeling results take relatively simple mathematical forms. We found that
+the common assumption that the mean Ct value is a simple logarithmic function of the
+number of input DNA molecules n is correct when n is large. For small n, corrections
+are required. An exact mean Ct value is given by (ψ(x + 1) − ψ(n))/r, where x is the quan-
+tification threshold, r is the amplification efficiency and ψ(·) denotes the first polygamma
+function. The variance has the elegant form (ψ1(n) − ψ1(x + 1))/r2, where ψ1(·) denotes the
+second polygamma function. Therefore, the variance is strongly dependent on amplifica-
+tion efficiency. This effect is illustrated in Figure 1, which shows that the pdf of the Ct
+value becomes wider as the amplification efficiency decreases. Interestingly, in this simple
+case, the coefficient of variation of the Ct value (i.e. the ratio of the standard deviation to
+the mean) does not depend on amplification efficiency.
+Two important numbers that characterize the performance of a PCR process are the
+limit of detection (LoD) and the limit of quantification (LoQ). The LoD is the smallest
+number of molecules that can be detected with a failure rate not exceeding a threshold
+α, while LoQ is the smallest number of molecules that can be quantified with a given
+level of precision (i.e. allowing a defined maximum fold deviation from the true value)
+and a given maximum failure rate α. We provided mathematical formulae for calculating
+both LoD and LoQ. Close examination of these formulae in the context of a single-phase
+PCR process revealed that a small reduction of the amplification efficiency may cause a
+large increase of LoD. For example, when α = 5%, reducing the efficiency from 95% of the
+maximum possible efficiency (m.p.e) to 90% m.p.e. caused the LoD to double, from ≈10
+input molecules to ≈20 molecules. In contrast to LoD, we found that LoQ is independent
+of efficiency. Allowing up to a 2-fold difference between n and its estimate results in an
+LoQ of ≈10 molecules. Reducing the allowed fold difference to 10% increases LoQ to 820
+16
+
+molecules. Our methods may be used to improve significantly the current approaches to
+estimating LoD and LoQ, which are laborious [16] and frequently rely on certain crude
+mathematical approximations [16, 17] that can be avoided by using our methods.
+Furthermore, we applied our methods to shed light on the effects that amplification
+noise has on estimates of the expected number of input DNA molecules λ obtained by the
+standard method of interpreting digital PCR data. A key assumption of this method is
+that a PCR reaction will be positive if it contains at least one input DNA molecule. We
+showed that this assumption is in general invalid because of the stochastic nature of PCR
+amplification. Stochastic effects are particularly large when λ is small, which is the regime
+in which digital PCR preferentially operates. At a high amplification efficiency of 95%
+m.p.e, we find that the ratio of the fraction of positive digital PCR reactions calculated by
+the standard method versus the value obtained after accounting for amplification noise is
+only 18.3% when λ = 1 and it increases to ≈100% when λ = 10 (Figure 5.2). Accordingly,
+stochastic effects were found to cause a significant under-estimation of λ by the standard
+method. Indeed, at a high efficiency of 95% m.p.e., the standard method is predicted
+to under-estimate λ by factors of ≈8.1, ≈3.1, and ≈1.4 when λ equals 1, 10, and 100,
+respectively (Figure 2). This is in the same range as empirically observed (eg. [19]).
+Using our mathematical methods, the following two different approaches may be used
+to obtain much more accurate estimates of λ. Firstly, if Ct values are available from pos-
+itive digital PCR reactions, then Equation (19) can be fit to those Ct values, using either
+a likelihood-based or a Bayesian statistical approach, to estimate the most probable value
+of λ together with a confidence (or credible) interval for it. Secondly, if only binary (ie.
+positive or negative) outcomes are available from individual reactions, then the variability
+of such outcomes can still be exploited to estimate λ. Specifically, suppose there are N
+different reactions. These can be randomly distributed into groups of N′ reactions each.
+Assuming a binomial distribution of the number of positive reactions found in each group,
+their first and second moments are given by F(T )N′ and F(T)N′ (1 + F(T )(N′ − 1)), respec-
+tively, where F(T ) is calculated using (24). These moments contain information about the
+two free parameters of F(T ) (i.e. λ and r1), which can be readily extracted to estimate λ
+together with a confidence (or credible) interval.
+The Ct value is estimated from the fluorescence profiles produced by DNA molecules
+as they are amplified during PCR. Our mathematical analysis can be straightforwardly ex-
+tended to obtain a time-dependent probability density of the fluorescence intensity Pt(y),
+which can then be fit to fluorescence profiles as an alternative approach to estimating the
+number of input DNA molecules. Using standard results from probability theory (eg. see
+[23]), Pt(y) can be derived from both the cumulative distribution function of the number
+of molecules found in the PCR process at time t, Ft(x) [calculated based on Equation (4)
+or (16)] and the linear relation expected [24] between y and x. Specifically, Pt(y) can be
+17
+
+expressed as
+Pt(y) = 1
+α ht(g−1(y)),
+(30)
+where
+ht(x) = d
+dxFt(x),
+(31)
+y = αx + β = g(x), and α,β > 0. We will explore in detail this alternative approach to
+estimating the number of input DNA molecules in a future paper.
+5
+Supporting Information
+5.1
+Appendix
+This section contains mathematical proofs and detailed calculations supporting the results
+presented in Section 3.
+5.1.1
+Proof of Theorem 1
+Proof. We will prove Theorem 1 by mathematical induction on k.
+• k = 1:
+The Chapman-Kolmogorov forward equation corresponding to the single-phase pro-
+cess is given by:
+∂P(X = x,t|X = x′,t′)
+∂t
+= r1(x−1)P(X = x−1,t|X = x′,t′)−r1xP(X = x,t|X = x′,t′), (32)
+where we have set t = t′ +∆t, and r1 is the amplification efficiency associated with the
+process. To simplify our notation, we will abbreviate P(X = x,t|X = x′,t′) by P(x,t).
+We will solve (32) by using a powerful combinatorial device called the probability
+generating function (pgf) [14]. Recall that the pgf of P(x,t) is defined as:
+G(s,t) =
+∞
+�
+x=0
+sxP(x,t),
+where s is a book-keeping variable.
+18
+
+Multiplying both sides of (32) by sx and summing over all possible values of x yields:
+∞
+�
+x=0
+sx ∂P(x,t)
+∂t
+=
+r1
+∞
+�
+x=0
+(x − 1)sxP(x − 1,t) − r1
+∞
+�
+x=0
+xsxP(x,t)
+=
+r1s2
+∞
+�
+x=0
+(x − 1)sx−2P(x − 1,t) − r1s
+∞
+�
+x=0
+xsx−1P(x,t)
+=
+= r1s
+�
+������s
+∞
+�
+x=0
+(x − 1)sx−2P(x − 1,t) −
+∞
+�
+x=0
+xsx−1P(x,t)
+�
+������.
+(33)
+Using
+∂G(s,t)
+∂s
+=
+∞
+�
+x=0
+xsx−1P(x,t) and
+∂G(s,t)
+∂t
+=
+∞
+�
+x=0
+sx ∂P(x,t)
+∂t
+,
+(34)
+we simplify (33) to obtain
+∂G(s,t)
+∂t
+= r1s(s − 1)∂G(s,t)
+∂s
+,
+(35)
+which is a partial differential equation (pde) in G(s,t).
+We will solve (35) by the method of characteristics. To this end, we define new
+variables
+u = u(s,t) and v = v(s,t),
+which will transform (35) into the simpler equation
+∂W(u,v)
+∂u
++ H(u,v)W(u,v) = F(u,v),
+(36)
+which has the solution
+W(u,v) = e−
+�
+H(u,v)du
+��
+F(u,v)e
+�
+H(u,v)du + Ψ(v)
+�
+,
+where
+W(u,v) = G(s(u,v),t(u,v)).
+This requires that v(s,t) = c, where c is an arbitrary constant. The resulting charac-
+teristic equation is given by
+ds
+dt = −r1s(s − 1),
+19
+
+which has the solution
+s − 1
+s
+er1t = c = v(s,t).
+Setting u(s,t) = t, we obtain
+∂G
+∂t
+=
+∂W
+∂t = ∂W
+∂u
+∂u
+∂t + ∂W
+∂v
+∂v
+∂t
+=
+∂W
+∂u + r1(s − 1)
+s
+er1t ∂W
+∂v
+(37)
+and
+∂G
+∂s
+=
+∂W
+∂u
+∂u
+∂s + ∂W
+∂v
+∂v
+∂s
+=
+1
+s2 er1t ∂W
+∂v .
+(38)
+Substituting (37) and (38) into (35) gives
+∂W
+∂u = 0,
+(39)
+which has the same form as (36). The solution to (39) is given by
+W(u,v)
+= Ψ(v)
+=⇒
+G(s,t)
+= Ψ
+�s − 1
+s
+er1t�
+.
+(40)
+If there are n molecules at the start of the process (t = 0), then p(x,0) = 1 if x = n and
+p(x,0) = 0 otherwise. Therefore,
+G(s,0) = Ψ
+�s − 1
+s
+�
+=
+∞
+�
+x=0
+sxP(x,0) = sn.
+(41)
+In (41), the argument y of Ψ (y) maps onto ( 1
+1−y )n, implying that
+G(s,t) = Ψ
+�s − 1
+s
+er1t�
+=
+�
+�����
+1
+1 − s−1
+s er1t
+�
+�����
+n
+=
+sne−nr1t
+[1 − s(1 − e−r1t)]n .
+(42)
+Equation (42) matches (2) when k = 1.
+Corollary 3. Equation (42) solves (35).
+20
+
+Proof. The right-hand-side of (35) is
+∂G
+∂s
+=
+nsn−1e−nr1t �
+1 − s(1 − e−r1t)
+�−n + nsne−nr1t �
+1 − s(1 − e−r1t)
+�−(n+1) �
+1 − e−r1t�
+=
+nsn−1e−nr1t
+[1 − s(1 − e−r1t)]n
+�
+1 +
+s(1 − e−r1t)
+1 − s(1 − e−r1t)
+�
+=
+nsn−1e−nr1t
+[1 − s(1 − e−r1t)](n+1)
+�
+1 − s(1 − e−r1t) + s(1 − e−r1t)
+�
+=
+nsn−1e−nr1t
+[1 − s(1 − e−r1t)](n+1) ,
+(43)
+and the left hand-side is
+∂G
+∂t
+=
+−nr1sne−nr1t �
+1 − s(1 − e−r1t)
+�−n + nsn+1r1e−(n+1)r1t �
+1 − s(1 − e−r1t)
+�−(n+1)
+=
+nr1sne−nr1t
+[1 − s(1 − e−r1t)]n
+�
+se−r1t
+1 − s(1 − e−r1t) − 1
+�
+=
+nr1sne−nr1t
+[1 − s(1 − e−r1t)](n+1)
+�
+se−r1t − 1 + s − se−r1t�
+=
+nr1sne−nr1t(s − 1)
+[1 − s(1 − e−r1t)](n+1)
+=
+r1s(s − 1)
+this matches (43)
+������������������������������������������������
+�
+�����
+nsn−1e−nr1t
+[1 − s(1 − e−r1t)](n+1)
+�
+����� = r1s(s − 1)∂G
+∂s .
+(44)
+• k = 2:
+There are two amplification phases with rates r1 and r2, respectively. The first one
+runs from time t = 0 to t = τ1, and the second one runs from t = τ1 to t = τ1 +
+τ2. In the second phase, the probability generating function takes exactly the same
+general functional form as in the first phase, albeit with a different initial condition.
+Specifically, we have
+G(s,t) = Ψ
+�s − 1
+s
+er2(t−τ1)�
+,
+with the initial condition (at time t = τ1)
+G(s,τ1) = Ψ
+�s − 1
+s
+�
+=
+sne−nr1τ1
+[1 − s(1 − e−r1τ1)]n .
+21
+
+Using the same procedure as in the case when k = 1, we obtain
+G(s,t)
+=
+Ψ
+�s − 1
+s
+er2(t−τ1)�
+=
+�
+1
+1− s−1
+s er2(t−τ1)
+�n
+e−nr1τ1
+�
+1 −
+�
+1
+1− s−1
+s er2(t−τ1)
+�
+(1 − e−r1τ1)
+�n
+=
+sne−n[r2t+(r1−r2)τ1]
+�
+1 − s
+�
+1 − e−[r2t+(r1−r2)τ1]��n .
+(45)
+The right side of (45) equals (35) when k = 2, as expected.
+• We assume the statement is true for t ∈ Ik, that is
+G(s,t) =
+�
+se−z
+1 − s(1 − e−z)
+�n
+,
+where z = rkt + �k−1
+i=1(ri − rk)τi, and we prove it for t ∈ Ik+1. As before, in phase k + 1,
+the generating function has the functional form
+G(s,t) = Ψ
+�s − 1
+s
+erk+1(t−�k
+i=1 τi)�
+.
+At time t = �k
+i=1 τi, by the induction step, we have
+G(s,t) = Ψ
+�s − 1
+s
+�
+=
+�
+se−z
+1 − s(1 − e−z)
+�n
+.
+Using the same arguments as before, we find that, for t ∈ Ik+1,
+G(s,t)
+=
+Ψ
+�s − 1
+s
+erk+1(t−�k
+i=1 τi)�
+=
+�
+���������
+1
+1− s−1
+s erk+1(t−�k
+i=1 τi ) e−z
+1 −
+1
+1− s−1
+s erk+1(t−�k
+i=1 τi ) (1 − e−z)
+�
+���������
+n
+=
+sne−n[rk+1t+�k
+i=1(ri−rk)τi]
+�
+1 − s
+�
+1 − e−[rk+1t+�k
+i=1(ri−rk)τi]
+��n ,
+(46)
+and this ends the proof of Theorem 1.
+22
+
+5.1.2
+Proof of Corollary 1
+Proof. From Theorem 1, we know that the generating function for P(x|n,⃗r,t,⃗τ) is given by
+G(s,t) =
+�
+se−z
+1 − s(1 − e−z)
+�n
+.
+Let p = e−z and q = 1 − p. Then,
+G(s,t) =
+�
+sp
+1 − sq
+�n
+= (sp)n [1 − sq]−n .
+P(x|n,⃗r,t,⃗τ) is the coefficient of sx in the power series expansion of G(s,t), given by
+G(s,t)
+=
+(sp)n [1 − sq]−n = (sp)n �
+i=0
+�−n
+i
+�
+(−sq)i = (sp)n �
+i=0
+�−n
+i
+�
+(−1)i(sq)i.
+(47)
+But
+�−n
+i
+�
+=
+−n(−n − 1)(−n − 2)(−n − 3)...(−n − (i − 2))(−n − (i − 1))
+1.2.3....(i − 1)i
+=
+(−1)i n(n + 1)(n + 2)(n + 3)...(n + (i − 2))(n + (i − 1))
+i!
+=
+(−1)i
+�n + i − 1
+i
+�
+=⇒
+(−1)i
+�−n
+i
+�
+=
+�n + i − 1
+i
+�
+.
+(48)
+Substituting (48) into (47) gives
+G(s,t)
+=
+(sp)n �
+i=0
+�−n
+i
+�
+(−1)i(sq)i = (sp)n �
+i=0
+�n + i − 1
+i
+�
+(sq)i.
+(49)
+Let x = n + i. Then,
+G(s,t)
+=
+(sp)n
+∞
+�
+x=n
+�x − 1
+x − n
+�
+(sq)x−n =
+∞
+�
+x=n
+�x − 1
+x − n
+�
+pnqx−nsx.
+(50)
+The probability of having x molecules at time t, P(x,t), is therefore given by
+P(x,t) =
+�x − 1
+x − n
+�
+pnqx−n =
+�x − 1
+n − 1
+�
+e−nz (1 − e−z)x−n .
+(51)
+Corollary 4. The probability distribution given in Theorem 1 solves the Chapman-Kolmogorov
+23
+
+equation given by (32).
+Proof.
+∂P(x,t)
+∂t
+= �x−1
+x−n
+��
+−nrke−z (1 − e−z)x−n + rk(x − n)e−2z (1 − e−z)x−n−1�
+= rk
+�x−1
+x−n
+�e−z (1 − e−z)x−n �
+−n + (x − n) e−z
+1−e−z
+�
+= rk
+�x−1
+x−n
+�e−z (1 − e−z)x−n �
+−n + (x − n) e−z
+1−e−z − x−n
+1−e−z + x−n
+1−e−z
+�
+= rk
+�x−1
+x−n
+�e−z (1 − e−z)x−n � x−n
+1−e−z − n + (x−n)
+1−e−z (e−z − 1)
+�
+= rk
+�x−1
+x−n
+�e−z (1 − e−z)x−n � x−n
+1−e−z − x
+�
+= rk
+�
+(x − n)�x−1
+x−n
+��
+e−z (1 − e−z)x−n−1 − rkx�x−1
+x−n
+�e−z (1 − e−z)x−n
+= rk(x − 1)
+�� x−2
+x−n−1
+�e−z (1 − e−z)x−n−1�
+− rkx
+��x−1
+x−n
+�e−z (1 − e−z)x−n�
+= rk(x − 1)P(x − 1,t) − rkxP(x,t),
+(52)
+which equals the right-hand side of (32).
+5.1.3
+Proof of Theorem 2
+Proof. We prove Theorem (2) by mathematical induction on k.
+• k = 1:
+There is only one phase with amplification efficiency r1. Recall that the Chapman-
+Kolmogorov forward equation for the dynamics of P (x,t) is given by (32), with the
+initial condition
+P (x,0) = e−λλx
+x!
+.
+Using the same arguments as above, we can write the generating function for P (x,t)
+as
+G(s,t) = Ψ
+�s − 1
+s
+er1t�
+,
+(53)
+with the initial condition
+G(s,0)
+=
+Ψ
+�s − 1
+s
+�
+=
+∞
+�
+x=0
+sxP(x,0) = eλ(s−1).
+(54)
+Therefore,
+G(s,t)
+=
+Ψ
+�s − 1
+s
+er1t�
+=
+e
+λ
+�
+1
+1− s−1
+s
+er1t −1
+�
+=
+e
+�
+λ(s−1)
+1−s(1−e−r1t)
+�
+.
+(55)
+24
+
+Equation (55) matches (15) when k = 1.
+Corollary 5. The generating function given by (55) solves Equation (35).
+Proof. Differentiate the right hand side of (55) with respect to s.
+• k = 2:
+There are two phases with amplification efficiencies r1 and r2, respectively. The first
+phase runs from time t = 0 to t = τ1, and the second one runs from t = τ1 to t = τ1+τ2.
+As before, for t ∈ I2 the probability generating function has the form
+G(s,t) = Ψ
+�s − 1
+s
+er2(t−τ1)�
+,
+with the initial condition (at time t = t1)
+G(s,τ1) = Ψ
+�s − 1
+s
+�
+= e
+�
+λ(s−1)
+1−s(1−e−r1t)
+�
+.
+Therefore,
+G(s,t)
+=
+Ψ
+�s − 1
+s
+er2(t−τ1)�
+=
+e
+�
+�������
+λ(
+1
+1− s−1
+s
+er2(t−τ1) −1)
+1−
+1
+1− s−1
+s
+er2(t−τ1) (1−e−r1t)
+�
+�������
+=
+e
+�
+λ(s−1)
+1−s(1−e−(r2t+(r1−r2)τ1))
+�
+.
+(56)
+The right side of (56) equals (15) for the case k = 2.
+• We assume the statement is true for t ∈ Ik, that is
+G(s,t) = e
+�
+λ(s−1)
+1−s(1−e−z)
+�
+,
+where z = rkt + �k−1
+i=1(ri − rk)τi, and we prove it for t ∈ Ik+1.
+In phase k + 1, the probability generating function has the form
+G(s,t) = Ψ
+�s − 1
+s
+erk+1(t−�k
+i=1 τi)�
+.
+By the induction step, the initial condition (at time t = �k
+i=1 τi) is given by
+G(s,t) = Ψ
+�s − 1
+s
+�
+= e
+�
+λ(s−1)
+1−s(1−e−z)
+�
+.
+25
+
+Therefore, for t ∈ Ik+1, we have
+G(s,t)
+=
+Ψ
+�s − 1
+s
+erk+1(t−�k
+i=1 τi)�
+=
+e
+�
+����������
+λ(
+1
+1− s−1
+s
+erk+1(t−�k
+i=1 τi )
+−1)
+1−
+1
+1− s−1
+s
+erk+1(t−�k
+i=1 τi )
+(1−e−z)
+�
+����������
+=
+e
+�
+λ(s−1)
+1−s(1−e−z′ )
+�
+,
+(57)
+where z′ = rk+1t + �k
+i=1(ri − rk+1)τi, and this ends the proof of Theorem 2.
+5.1.4
+Proof of Corollary 2
+Proof. Recall that P(x|λ,⃗r,t,⃗τ) is the coefficient of sx in the power series expansion of the
+probability generating function given in Theorem 2, that is
+P(x|λ,⃗r,t,⃗τ)
+=
+1
+x!
+∂xG(s,t)
+∂sx
+���s=0.
+(58)
+Now, let us compute the partial derivatives of G(s,t) with respect to s. For brevity, we set
+a = e−z,v = (1 − e−z) = (1 − a),u = λe−z = aλ, and Q(s,t) = 1 − s(1 − e−z) = 1 − sv.
+Observe that
+G(s,t) = e
+λ(s−1)
+Q ,Q(0,t) = 1,G(0,t) = e−λ, ∂Q(s,t)
+∂s
+= ∂Q(s,t)
+∂s
+���s=0 = −v
+26
+
+and
+∂
+∂sG(s,t)
+=
+u G(s,t)
+Q2
+=⇒ ∂
+∂sG(s,t)
+���s=0 = ue−λ,
+∂2
+∂s2 G(s,t)
+=
+ue−λ
+Q4
+�Q2uG(s,t)
+Q2
++ 2vQG(s,t)
+�
+= u [u + 2vQ] G(s,t)
+Q4
+=⇒
+∂2
+∂s2 G(s,t)
+���s=0 = e−λ �
+u2 + 2uv
+�
+,
+∂3
+∂s3 G(s,t)
+=
+u [u + 2v]
+Q8
+�
+uG(s,t)(u + 2vQ)Q2 − 2v2G(s,t)Q4 + 4vG(s,t)(u + 2vQ)Q3�
+=
+u G(s,t)
+Q6
+�
+u2 + 6uvQ + 6v2Q2�
+=⇒
+∂3
+∂s3 G(s,t)
+���s=0 = e−λ(u3 + 6u2v + 6uv2),
+∂4
+∂s4 G(s,t)
+=
+u G(s,t)
+Q8
+�
+u3 + 12u2vQ + 36uv2Q2 + 24v3Q3�
+=⇒
+∂4
+∂s4 G(s,t)
+���s=0 = e−λ �
+u4 + 12u3v + 36u2v2 + 24uv3�
+∂5
+∂s5 G(s,t)
+=
+u G(s,t)
+Q8
+�
+u4 + 20u3vQ + 120u2v2Q2 + 240uv3Q3 + 120v4Q4�
+,
+=⇒
+∂4
+∂s5 G(s,t)
+���s=0 = e−λ �
+u5 + 20u4v + 120u3v2 + 240u2v3 + 120uv4�
+.
+(59)
+By closely examining the coefficients of powers of the terms u,v and uv in (59), the follow-
+ing combinatorial triangle emerges
+x
+1
+1
+2
+1
+2
+3
+1
+6
+6
+4
+1
+12
+36
+24
+5
+1
+20
+120
+240
+120
+1
+2
+3
+4
+5
+i
+In particular, the (x,i)’th entry of the triangle is given by
+T (x,i) =
+� x
+i − 1
+��x − 1
+i − 1
+�
+(i − 1)!, for i = 1,2,...x.
+(60)
+27
+
+Thus
+P(x|λ,⃗r,t,⃗τ)
+=
+1
+x!
+∂x
+∂sx G(s,t)
+����s=0 = e−λ
+x!
+x
+�
+i=1
+T (x,i)
+=
+e−λ
+x!
+x
+�
+i=1
+� x
+i − 1
+��x − 1
+i − 1
+�
+(i − 1)!(λe−z)x−i+1 (1 − e−z)i−1
+=
+e−λ
+x
+�
+i=1
+1
+(x − i + 1)!
+�x − 1
+i − 1
+�
+(λe−z)x−i+1 (1 − e−z)i���1 .
+(61)
+Setting k = x − i + 1 in (61) gives the desired result.
+Corollary 6. The probability distribution given in Theorem 2 solves (32).
+Proof. Differentiate the right-hand-side of Equation (61) with respect to t.
+We will now derive the probability density function (pdf), mean, variance, and cumu-
+lative density function (cdf) of the Ct value. We will consider two different cases, namely:
+1. when the initial state of the PCR process is deterministic, and the PCR phase lengths
+and amplification efficiencies are given; and
+2. when the initial state is Poisson-distributed, and the phase lengths and amplification
+efficiencies are given.
+5.1.5
+Case 1: The initial state is deterministic, and the phase lengths and amplifica-
+tion efficiencies are given
+• General form of the pdf
+Consider the PCR process described in Theorem 1. The process begins with n cDNA
+molecules, which are amplified across up to p successive phases {Ii} of lengths ⃗τ =
+(τ1,τ2,...,τp) at the corresponding amplification efficiencies ⃗r = (r1,r2,...,rp). By def-
+inition, the Ct value t is the time at which the number of molecules reaches the quan-
+tification threshold, which we denote by x. By Bayes’ theorem, a general expression
+for the pdf of t is given by
+P(t|n,⃗r,⃗τ,x)
+=
+P(n,⃗r,⃗τ,x|t)P(t)
+P(n,⃗r,⃗τ,x)
+.
+(62)
+Since n is independent of ⃗r, ⃗τ, and t, and ⃗r is also independent of t and of the values
+taken by the entries of ⃗τ, we re-write the numerator of the right-hand-side of (62) as
+28
+
+follows:
+P(n,⃗r,⃗τ,x|t)P(t)
+=
+P(x|n,⃗r,⃗τ,t)P(n,⃗r,⃗τ|t)P(t)
+=
+P(x|n,⃗r,⃗τ,t)P(n|⃗r,⃗τ,t)P(⃗r,⃗τ|t)P(t)
+=
+P(x|n,⃗r,⃗τ,t)P(n)P(⃗r|⃗τ,t)P(⃗τ|t)P(t)
+=
+P(x|n,⃗r,⃗τ,t)P(n)P(⃗r)P(t|⃗τ)P(⃗τ).
+(63)
+Similarly, the denominator of the right-hand-side of (62) can be simplified to
+P(n,⃗r,⃗τ,x)
+=
+� ∞
+�k−1
+i=1 τi
+P(n,⃗r,⃗τ,x|t)P(t)dt
+=
+P(n)P(⃗r)P(⃗τ)
+� ∞
+�k−1
+i=1 τi
+P(x|n,⃗r,⃗τ,t)P(t|⃗τ)dt.
+(64)
+Therefore, (62) can be re-written as
+P(t|n,⃗r,⃗τ,x)
+=
+P(x|n,⃗r,⃗τ,t)P(t|⃗τ)
+� ∞
+�k−1
+i=1 τi P(x|n,⃗r,⃗τ,t)P(t|⃗τ)dt
+.
+(65)
+We will derive the pdf, mean, variance, and cdf of the Ct value for a PCR process
+with an arbitrary number of phases p. Without loss of generality, we suppose that
+t ∈ Ik,k ≤ p. We will assume a uniform prior density for t given ⃗τ. As we will
+demonstrate later, this assumption produces very similar results to those we obtain
+by assuming a Jeffreys prior [25]. We will state results for the case of a single-phase
+PCR process whenever these cannot be readily gleaned from the general results.
+We will first consider the case when the lengths of the intermediate phases, recorded
+in the vector ⃗τ, are given. This is useful, for example, when it is of interest to estimate
+the lengths of such phases from data. We will then show how to marginalize ⃗τ out
+of the pdf.
+• pdf
+Using the posterior density given in Equation (65) and the likelihood function given
+in Corollary 1, we obtain the following functional form for the pdf:
+P(t|n,⃗r,⃗τ,x)
+∝
+e−nz (1 − e−z)x−n ,
+(66)
+where
+z = rkt +
+k−1
+�
+i=1
+(ri − rk)τi,
+(67)
+⃗r = (r1,r2,...,rk) is a vector of amplification efficiencies, ⃗τ = (τ1,τ2,...,τk) is a vector of
+29
+
+phase lengths, and we have used a uniform prior for t.
+The normalizing constant is given by
+C =
+� ∞
+�k−1
+i=1 τi
+e−nz(1 − e−z)x−ndt.
+(68)
+Let w = e−z. Then,
+C
+=
+1
+rk
+� θ
+0
+wn−1(1 − w)x−ndw
+=
+Bθ(n,x − n + 1)
+rk
+,
+(69)
+where
+θ = e−�k−1
+i=1 riτi.
+(70)
+Therefore, the pdf is given by
+P(t|n,⃗r,⃗τ,x)
+=
+rke−nz (1 − e−z)x−n
+Bθ(n,x − n + 1) .
+(71)
+For the single-phase process, θ = 1, so the pdf simplifies to
+P(t|n,r1,x)
+=
+r1e−nr1t �
+1 − e−r1t�x−n
+B(n,x − n + 1)
+.
+(72)
+Note that in some cases (eg. when knowledge of the lengths of individual PCR ampli-
+fication phases is not of interest), it may be useful to marginalize ⃗τ out of P(t|n,⃗r,⃗τ,x).
+This can be achieved by using the fact that
+P(t|n,⃗r,x)
+=
+�
+Ω
+P(t,⃗τ|n,⃗r,x)d⃗τ
+=
+�
+Ω
+P(t|n,⃗r,⃗τ,x)P(⃗τ|n,⃗r,x)d⃗τ,
+(73)
+where Ω is the domain of ⃗τ.
+In addition, note that an alternative formulation of the prior for t, based on an ap-
+proach proposed by Jeffreys [25] for generating priors that are invariant to reparametriza-
+tion, is the following:
+p(t|⃗τ) ∝
+�
+|I(t|⃗τ)|,
+(74)
+30
+
+where I(t|⃗τ) is the Fisher information of the likelihood function and is given by
+I(t|⃗τ)
+=
+EX
+�� ∂
+∂t lnP(x|n,⃗r,⃗τ,x)
+�2 �
+=
+EX
+�r2
+k
+�
+w2x2 − 2nwx + n2�
+(1 − w)2
+�
+=
+r2
+k w2
+(1 − w)2 EX
+�
+x2�
+− 2nr2
+k w
+(1 − w)2 EX
+�
+x
+�
++
+n2r2
+k
+(1 − w)2
+=
+r2
+k
+(1 − w)2
+�
+�����w2
+∞
+�
+j=1
+x2
+�x − 1
+j − 1
+�
+wn(1 − w)x−n − 2nw
+∞
+�
+j=1
+x
+�x − 1
+j − 1
+�
+wn(1 − w)x−n + n2
+�
+�����,
+(75)
+where w = e−z.
+Observe that
+∞
+�
+x=1
+x
+�x − 1
+n − 1
+�
+wn(1 − w)x−n
+=
+∞
+�
+x=1
+x!
+(x − n)!(n − 1)!wn(1 − w)x−n
+=
+∞
+�
+y=2
+(y − 1)!
+(y − m)!(m − 2)!wm−1(1 − w)y−m
+=
+m − 1
+w
+∞
+�
+y=1
+(y − 1)!
+(y − m)!(m − 1)!wm(1 − w)y−m
+=
+m − 1
+w
+=
+n
+w
+(76)
+31
+
+and
+∞
+�
+x=1
+x2
+�x − 1
+n − 1
+�
+wn(1 − w)x−n
+=
+∞
+�
+x=1
+x!x
+(x − n)!(n − 1)!wn(1 − w)x−n
+=
+∞
+�
+y=2
+(y − 1)!(y − 1)
+(y − m)!(m − 2)!wm−1(1 − w)y−m
+=
+∞
+�
+y=1
+(y − 1)!y
+(y − m)!(m − 2)!wm−1(1 − w)y−m − m − 1
+w
+=
+∞
+�
+y′=2
+(y′ − 1)!
+(y′ − m′)!(m′ − 3)!wm′−2(1 − w)y′−m′ − m − 1
+w
+=
+(m′ − 1)(m′ − 2)
+w2
+∞
+�
+y′=1
+(y′ − 1)!
+(y′ − m′)!(m′ − 1)!wm′(1 − w)y′−m′ − m − 1
+w
+=
+(m′ − 1)(m′ − 2)
+w2
+− m − 1
+w
+=
+n(n + 1)
+w2
+− n
+w ,
+(77)
+where m = n + 1,m′ = m + 1,y = x + 1,y′ = y + 1.
+Plugging (76) and (77) into (75), we obtain
+I(t|⃗τ)
+=
+nr2
+k
+1 − w
+=⇒
+p(t|⃗τ) ∝
+1
+√
+1 − w
+=
+1
+√
+1 − e−z .
+(78)
+Using this prior, and following the steps we used earlier to derive (71), we find that
+the pdf is given by
+P(t|n,⃗r,⃗τ,x)
+∝
+e−nz (1 − e−z)x−n−1/2
+=⇒ P(t|n,⃗r,⃗τ,x)
+=
+rke−nz (1 − e−z)x−n−1/2
+Bθ(n,x − n + 1/2)
+,
+(79)
+which has a similar form as (71).
+For simplicity, we will continue to use a uniform prior for t.
+• Mean
+The mean Ct value is given by
+E(t)
+=
+rk
+� ∞
+�k−1
+i
+τi te−nz(1 − e−z)x−ndt
+Bθ(n,x − n + 1)
+,
+(80)
+32
+
+where z is given by (67).
+Let w = e−z. Then,
+E(t)
+=
+rk
+� 0
+θ
+�
+− (lnw−lnθ′)
+rk
+�
+wn(1 − w)x−n
+�
+− dw
+rkw
+�
+Bθ(n,x − n + 1)
+=
+� 0
+θ (lnw − lnθ′)wn−1(1 − w)x−ndw
+rkBθ(n,x − n + 1)
+=
+lnθ′ � θ
+0 wn−1(1 − w)x−ndw −
+� θ
+0 lnw wn−1(1 − w)x−ndw
+rkBθ(n,x − n + 1)
+=
+lnθ′
+rk
+−
+�
+∂
+∂n + ∂
+∂x
+�
+Bθ(n,x − n + 1)
+rkBθ(n,x − n + 1)
+=
+ln θ′
+θ
+rk
++ Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ)
+rkBθ(n,x − n + 1)
+=
+k−1
+�
+i=1
+τi + Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ)
+rkBθ(n,x − n + 1)
+,
+(81)
+where θ is given by (70), ψ(·) is the first polygamma function (also called the digamma
+function), and
+θ′ = θerk
+�k−1
+i=1 τi.
+(82)
+Note that for the single-phase process, θ = θ′ = 1. In this case, using
+3 ˜F2(n,n,n − x;n + 1,n + 1;1) = nΓ(x − n + 1)[ψ(x + 1) − ψ(n)]
+Γ(n + 1)Γ(x + 1)
+,
+we find that the mean Ct value is given by
+E(t) = ψ(x + 1) − ψ(n)
+r1
+.
+(83)
+• Variance
+The variance of the Ct value is given by E(t2) − E(t)2, where
+E(t2)
+=
+rk
+� ∞
+�k−1
+i=1 τi t2e−nz(1 − e−z)x−ndt
+Bθ(n,x − n + 1)
+,
+(84)
+and z is given by (67).
+33
+
+Let w = e−z. Then,
+E(t2)
+=
+rk
+� 0
+θ
+�
+lnw−lnθ′
+rk
+�2
+wn(1 − w)x−n
+�
+− dw
+rkw
+�
+Bθ(n,x − n + 1)
+=
+� θ
+0 (lnw − lnθ′)2 wn−1(1 − w)x−ndw
+rk2Bθ(n,x − n + 1)
+=
+� θ
+0 (lnw)2 wn−1(1 − w)x−ndw − 2lnθ′ � θ
+0 lnw wn−1(1 − w)x−ndw
+rk2Bθ(n,x − n + 1)
++
+(lnθ′)2 � θ
+0 wn−1(1 − w)x−ndw
+rk2Bθ(n,x − n + 1)
+=
+�
+∂2
+∂n2 + 2 ∂2
+∂n∂x + ∂2
+∂x2 − 2lnθ′
+�
+∂
+∂n + ∂
+∂x
+��
+Bθ(n,x − n + 1)
+rk2Bθ(n,x − n + 1)
++ (lnθ′)2
+rk2
+=
+�
+∂2
+∂n2 + 2 ∂2
+∂n∂x + ∂2
+∂x2
+�
+Bθ(n,x − n + 1)
+rk2Bθ(n,x − n + 1)
++ 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ)
+r2
+k Bθ(n,x − n + 1)
++
+lnθ′ ln θ′
+θ2
+r2
+k
+=
+�
+∂2
+∂n2 + 2 ∂2
+∂n∂x + ∂2
+∂x2
+�
+Bθ(n,x − n + 1)
+rk2Bθ(n,x − n + 1)
++ 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ)
+r2
+k Bθ(n,x − n + 1)
++
+�
+������
+k−1
+�
+i=1
+τi
+�
+������
+2
+−
+�
+������
+k−1
+�
+i=1
+riτi
+rk
+�
+������
+2
+,
+(85)
+where θ is given by (70) and θ′ is given by (82).
+For the single-phase process, the second moment of the Ct value is given by
+E(t2)
+=
+�
+∂2
+∂n2 + 2 ∂2
+∂n∂x + ∂2
+∂x2
+�
+B(n,x − n + 1)
+rk2B(n,x − n + 1)
+=
+ψ1(n) − ψ1(x + 1) + [ψ(x + 1) − ψ(n)]2
+rk2
+,
+(86)
+where ψ1(·) is the second polygamma function (also called the trigamma function).
+Therefore, the variance is
+Var(t) = ψ1(n) − ψ1(x + 1)
+rk2
+.
+(87)
+• cdf
+34
+
+The cdf of the Ct value is given by
+F(t|n,⃗r,⃗τ,x)
+=
+rk
+Bθ(n,x − n + 1)
+� t
+�k−1
+i=1 τi
+e−nz′(1 − e−z′)x−nds,
+(88)
+where z′ = rks + �k−1
+i=1(ri − rk)τi
+Let w = e−z′. Then,
+F(t|n,⃗r,⃗τ,x)
+=
+� θ
+e−z wn−1(1 − w)x−ndw
+Bθ(n,x − n + 1)
+=
+Bθ(n,x − n + 1) − Be−z(n,x − n + 1)
+Bθ(n,x − n + 1)
+=
+1 − Be−z(n,x − n + 1)
+Bθ(n,x − n + 1) ,
+(89)
+where θ is given by 70.
+For the single-phase process, the cdf is given by
+F(t|n,r1,x)
+=
+1 − Ie−r1t(n,x − n + 1),
+(90)
+where Ie−r1t(n,x−n+1) = Be−rt (n,x−n+1)
+B(n,x−n+1)
+is the regularized incomplete Beta function. Be-
+cause the cdf is in closed analytical form, we can apply the efficient inverse transform
+method to generate random samples of Ct values as follows:
+t
+=
+−lnI−1
+1−u(n,x − n + 1)
+r1
+,
+(91)
+where u is a real number sampled uniformly at random from the interval (0,1) and
+I−1
+1−u is the inverse of the regularized incomplete Beta function. To find a Ct value
+that corresponds to a quantile q ∈ (0,1), simply replace u in Equation (91) by q.
+• Probability distribution of n
+We conclude by deriving the probability distribution of n, denoted P(n|r1,t,x), for the
+single-phase PCR process. This distribution can be used to estimate n from measured
+Ct values. It can also be used to calculate the LoD and LoQ of a PCR assay, as we
+demonstrated in the main text. The steps described below can also be used to derive
+P(n|⃗r,t,⃗τ), for a PCR process with an arbitrary number of phases, although this will
+not yield a closed-form result like we will obtain in the single-phase case.
+By Bayes’ Theorem, we have
+P(n|r1,t,x)
+∝
+wn(1 − w)x−n
+B(n,x − n + 1),
+(92)
+35
+
+where
+w = e−r1t.
+(93)
+The normalizing constant is given by
+C
+=
+x
+�
+n=1
+wn(1 − w)x−n
+B(n,x − n + 1)
+=
+x
+�
+n=1
+x! wn(1 − w)x−n
+(n − 1)! (x − n)! .
+(94)
+Let m = n − 1. Then,
+C
+=
+x−1
+�
+m=0
+x! wm+1(1 − w)x−1−m
+m! (x − 1 − m)!
+=
+xw
+x−1
+�
+m=0
+�x − 1
+m
+�
+wm(1 − w)x−1−m
+=
+xw.
+(95)
+Therefore, we have
+P (n|r1,t,x)
+=
+wn−1(1 − w)x−n
+xB(n,x − n + 1).
+(96)
+Suppose that t is a Ct value generated by a PCR process with n input molecules. If
+we replace n by ˆn in Equation (96), then the equation will give the probability that
+ˆn will be obtained as the estimate of n based on the data t. It is useful – eg. for the
+purpose of determining the LoQ – to calculate the probability that ˆn will be obtained
+as the estimate of n based on any data t that can be generated by a PCR process with
+n input molecules. This probability is given by
+P ( ˆn|n,r1,x)
+=
+� ∞
+0
+P ( ˆn,t|n,r1,x)dt
+=
+� ∞
+0
+P ( ˆn|n,r1,t,x)P (t|n,r1,x)dt
+=
+� ∞
+0
+P ( ˆn|r1,t,x)P (t|n,r1,x)dt
+=
+r1
+� ∞
+0
+w ˆn−1(1 − w)x− ˆn
+xB( ˆn,x − ˆn + 1)
+wn(1 − w)x−n
+B(n,x − n + 1)dt
+=
+� 1
+0
+w ˆn+n−2(1 − w)2x−m−n
+xB( ˆn,x − ˆn + 1)B(n,x − n + 1)dw
+=
+B( ˆn + n − 1,2x − ˆn − n + 1)
+xB( ˆn,x − ˆn + 1)B(n,x − n + 1) = P( ˆn|n,x),
+(97)
+36
+
+where, using (96), we have assumed that ˆn is conditionally independent of n given t.
+Strikingly, (97) does not depend on r1.
+5.1.6
+Case 2: The initial state is Poisson-distributed, and the phase lengths and am-
+plification efficiencies are given
+• General form of the pdf
+Let t be the Ct value of the PCR process described in Theorem 2. The process begins
+with a Poisson-distributed number of input DNA molecules, with mean λ, which
+are replicated across up to p distinct phases with lengths ⃗τ = (τ1,τ2,...,τp) and am-
+plification efficiencies ⃗r = (r1,r2,...,rp). As noted earlier, t is the time at which the
+number of molecules reaches the quantification threshold, which we denote by x.
+Let us denote the pdf of t by P(t|λ,⃗r,⃗τ,x). By Bayes’ theorem, we have
+P(t|λ,⃗r,⃗τ,x)
+=
+P(λ,⃗r,⃗τ,x|t)P(t)
+P(λ,⃗r,⃗τ,x)
+(98)
+However, λ is independent of ⃗r, ⃗τ, and t, while ⃗r is also independent of t and of the
+precise values taken by the entries of ⃗τ. Therefore, by following the same steps we
+used earlier to derive (65), we find that
+P(t|λ,⃗r,⃗τ,x)
+=
+P(x|λ,⃗r,t,⃗τ)P(t|⃗τ)
+� ∞
+�k−1
+i=1 τi P(x|λ,⃗r,t,⃗τ)P(t|⃗τ)dt
+.
+(99)
+We will derive the pdf, mean, variance, and cdf by assuming, without loss of gener-
+ality, that t ∈ Ik, and then we will specify the functional forms taken by the results
+in the instructive case when t ∈ I1. As before, for simplicity, we will use a uniform
+prior density for t.
+• pdf
+We derive the pdf of the Ct value t by using the general expression given in Equation
+(99), with the probability distribution of the number of molecules given in Theorem
+37
+
+2 serving as the likelihood. Specifically,
+P(t|λ,⃗r,⃗τ,x)
+∝
+(1 − e−z)x
+x
+�
+j=1
+�x−1
+j−1
+�
+j!
+� λe−z
+1 − e−z
+�j
+(100)
+=
+(1 − e−z)x
+x−1
+�
+j=0
+�x−1
+j
+�
+(j + 1)!
+� λe−z
+1 − e−z
+�j+1
+(101)
+since (x
+j)=0 for j>x
+=
+(1 − e−z)x
+∞
+�
+j=0
+�x−1
+j
+�
+(j + 1)!
+� λe−z
+1 − e−z
+�j+1
+(102)
+=
+λe−z(1 − e−z)x−1
+∞
+�
+j=0
+(x − 1)(x − 2)···(x − j)
+(j + 1)! j!
+� λe−z
+1 − e−z
+�j
+(103)
+=
+λe−z(1 − e−z)x−1
+∞
+�
+j=0
+(1 − x)(2 − x)···(j − x)
+(j + 1)! j!
+� −λe−z
+1 − e−z
+�j
+(104)
+=
+λe−z(1 − e−z)x−1
+∞
+�
+j=0
+(1 − x)j
+(2)j
+����λe−z
+1−e−z
+�j
+j!
+(105)
+=
+λe−z(1 − e−z)x−1
+1F1
+�
+1 − x;2; −λe−z
+1 − e−z
+�
+,
+(106)
+where z is given by (67), 1F1 is the hypergeometric function (also called the Kummer
+confluent hypergeometric function of the first kind), defined as
+1F1
+�
+1 − x;2; −λe−r1t
+1 − e−r1t
+�
+=
+∞
+�
+j=0
+(1 − x)j
+(2)j
+� −λe−r1t
+1−e−r1t
+�j
+j!
+,
+and (α)j denotes the rising factorial, i.e. (α)j = α(α+1)(α+2)...(α+j−1) with (α)0 = 1.
+The normalizing constant is given by
+C
+=
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+� ∞
+�k−1
+i=1 τi
+e−jz(1 − e−z)x−jdt.
+(107)
+Let w = e−z. Then,
+C
+=
+1
+rk
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+� θ
+0
+wj−1(1 − w)x−jdw
+=
+1
+rk
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+Bθ(j,x − j + 1),
+(108)
+where θ is given by (70).
+38
+
+Therefore, the pdf is given by
+P(t|λ,⃗r,⃗τ,x)
+=
+rk(1 − e−z)x �x
+j=1
+(x−1
+j−1)
+j!
+� λe−z
+1−e−z
+�j
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+=
+rkλe−z(1 − e−z)x−1 1F1
+�
+1 − x,2, −λe−z
+1−e−z
+�
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+,
+(109)
+Recall that for the single-phase process, θ = 1, so we have
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+B(j,x − j + 1)
+=
+∞
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+B(j,x − j + 1)
+=
+∞
+�
+j=0
+�x−1
+j
+�λj+1
+(j + 1)! B(j + 1,x − j)
+=
+∞
+�
+j=0
+λj+1
+(j + 1)!
+=
+eλ − 1
+x
+,
+(110)
+implying that the pdf is given by
+P(t|λ,r1,x)
+=
+r1xλe−r1t(1 − e−r1t)x−1 1F1
+�
+1 − x,2, −λe−r1t
+1−e−r1t
+�
+eλ − 1
+.
+(111)
+Note that in some cases (eg. when knowledge of the lengths of individual PCR ampli-
+fication phases is not of interest), it may be useful to marginalize ⃗τ out of P(t|λ,⃗r,⃗τ,x).
+This can be achieved by using the fact that
+P(t|λ,⃗r,x)
+=
+�
+Ω
+P(t,⃗τ|λ,⃗r,x)d⃗τ
+=
+�
+Ω
+P(t|λ,⃗r,⃗τ,x)P(⃗τ|λ,⃗r,x)d⃗τ,
+(112)
+where Ω is the domain of ⃗τ.
+• Mean
+39
+
+The mean Ct value is given by
+E(t)
+=
+rk
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+(D)
+��������������������������������������������������������
+� ∞
+�k−1
+i=1 τi
+te−jz(1 − e−z)x−jdt,
+(113)
+where z is given by (67).
+Let w = e−z. Then,
+D
+see (81)
+=
+Bθ(j,x − j + 1)�k−1
+i=1 τi
+rk
++ Γ(j)2θj 3 ˜F2(j,j,j − x;j + 1,j + 1;θ)
+r2
+k
+.
+(114)
+Therefore
+E(t)
+=
+�x
+j=1
+(x−1
+j−1)λj
+j!
+�
+rkBθ(j,x − j + 1)�k−1
+i=1 τi + Γ(j)2θj 3 ˜F2(j,j,j − x;j + 1,j + 1;θ)
+�
+rk
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+.
+(115)
+Recall that for the single-phase process, θ = θ′ = 1, so
+E(t)
+=
+ψ(x + 1)
+r1
+−
+�x
+j=1
+λj
+j! ψ(j)
+r1
+�
+eλ − 1
+� .
+(116)
+• Variance
+The variance is given by E(t2) − E(t)2, where
+E(t2)
+=
+rk
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+(D)
+������������������������������������������������������������
+� ∞
+�k−1
+i=1 τi
+t2e−jz(1 − e−z)x−jdt,
+(117)
+where z is given by (67).
+40
+
+Let w = e−z. Then,
+D
+see (85)
+=
+�
+∂2
+∂n2 + 2 ∂2
+∂n∂x + ∂2
+∂x2
+�
+Bθ(n,x − n + 1)
+rk3
++ 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ)
+r3
+k
++
+Bθ(j,x − j + 1)
+�
+��������
+��k−1
+i=1 τi
+�2 −
+��k−1
+i=1
+riτi
+rk
+�2
+rk
+�
+��������
+(118)
+where θ′ is given by (82).
+Plugging (118) into (117), we obtain
+E(t2)
+=
+1
+r2
+k
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+x
+�
+j=1
+�x−1
+j−1
+�λj
+j!
+�
+�����
+� ∂2
+∂n2 + 2 ∂2
+∂n∂x + ∂2
+∂x2
+�
+Bθ(n,x − n + 1) +
+2lnθ′Γ(n)2θn
+3 ˜F2(n,n,n − x;n + 1,n + 1;θ) + r2
+k Bθ(j,x − j + 1)
+�
+��������
+�
+������
+k−1
+�
+i=1
+τi
+�
+������
+2
+−
+�
+������
+k−1
+�
+i=1
+riτi
+rk
+�
+������
+2�
+��������
+�
+�����.
+.
+(119)
+For the single-phase process, the variance is given by
+Var(t)
+=
+(eλ − 1)�x
+j=1
+λj
+j!
+�
+ψ1(j) + ψ(j)2
+�
+−
+�
+�����
+�x
+j=1
+λj
+j! ψ(j)
+�
+�����
+2
+�
+r1(eλ − 1)
+�2
+− ψ1(x + 1)
+r2
+1
+.
+(120)
+• cdf
+The cdf of the Ct value is given by
+F(t|λ,⃗r,⃗τ,x)
+=
+rk
+�x
+j=1
+(x−1
+j−1)λj
+j!
+� t�k−1
+i=1 τi e−jz′(1 − e−z′)x−jds
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+,
+(121)
+where z′ = rks + �k−1
+i=1(ri − rk)τi.
+41
+
+Let w = e−z′. Then,
+F(t|λ,⃗r,⃗τ,x)
+=
+�x
+j=1
+(x−1
+j−1)λj
+j!
+� θ
+e−z wj−1(1 − w)x���jdw
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+=
+�x
+j=1
+(x−1
+j−1)λj
+j!
+�� θ
+0 wj−1(1 − w)x−jdw −
+� e−z
+0
+wj−1(1 − w)x−jdw
+�
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+=
+�x
+j=1
+(x−1
+j−1)λj
+j!
+�
+Bθ(j,x − j + 1) − Be−z(j,x − j + 1)
+�
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+=
+1 −
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Be−z(j,x − j + 1)
+�x
+j=1
+(x−1
+j−1)λj
+j!
+Bθ(j,x − j + 1)
+,
+(122)
+where θ is given by (70).
+For the single-phase process, using (110), we simplify the cdf to obtain
+F(t|λ,r1,x)
+=
+1 −
+x�x
+j=1
+(x−1
+j−1)λj
+j!
+Be−r1t(j,x − j + 1)
+eλ − 1
+.
+(123)
+• Probability density of λ
+We conclude by deriving the probability density of λ, P(λ|r1,t,x), for the single-phase
+process. This density can be used to estimate λ from measured Ct values, and for
+calculating both the LoD and the LoQ of a PCR process. The steps described below
+can also be used to derive P(λ|⃗r,t,⃗τ), for a PCR process with an arbitrary number of
+phases.
+By Bayes’ Theorem, we have
+P(λ|r1,t,x)
+∝
+�x
+j=1
+(x−1
+j−1)
+j!
+� λw
+1−w
+�j
+eλ − 1
+,
+(124)
+where w = e−r1t.
+42
+
+The normalizing constant is given by
+C
+=
+� ∞
+0
+�x
+j=1
+(x−1
+j−1)
+j!
+� λw
+1−w
+�j
+eλ − 1
+dλ
+=
+x
+�
+j=1
+�x−1
+j−1
+�
+j!
+�
+w
+1 − w
+�j � ∞
+0
+λj
+eλ − 1dλ
+=
+x
+�
+j=1
+�x−1
+j−1
+�
+j!
+�
+w
+1 − w
+�j
+Γ(j + 1)ζ(j + 1)
+=
+x
+�
+j=1
+�x − 1
+j − 1
+��
+w
+1 − w
+�j
+ζ(j + 1),
+(125)
+where ζ(j) is the Riemann zeta function.
+Therefore, the probability density of λ is given by
+P(λ|r1,t,x)
+=
+�x
+j=1
+(x−1
+j−1)
+j!
+� λw
+1−w
+�j
+(eλ − 1)�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+=
+λw 1F1(1 − x,2, −λw
+1−w )
+(eλ − 1)(1 − w)�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+.
+(126)
+The probability that λ takes values between a and b is given by
+P(a ≤ λ ≤ b | r1,t,x)
+=
+�x
+j=1
+(x−1
+j−1)
+j!
+� w
+1−w
+�j � b
+a
+sj
+es−1ds
+�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+=
+�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j [ζb(j + 1) − ζa(j + 1)]
+�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+,
+(127)
+where ζλ(·) is the incomplete Riemann zeta function.
+It follows that the cumulative density function of λ is given by
+F(λ | r1,t,x)
+=
+�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζλ(j + 1)
+�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+.
+(128)
+As discussed earlier in relation to P(n|⃗r,t,⃗τ,x), Equation (126) can be interpreted
+as follows: Suppose that a Ct value t is produced by a PCR process with expected
+number of input molecules λ. If we replace λ by an estimate ˆλ, then (126) gives the
+43
+
+likelihood of ˆλ. For practical purposes (eg. to determine the LoQ), it is useful to
+calculate the probability that ˆλ will be obtained as the estimate of λ from any data t
+that can be produced by a PCR process with expected number of input molecules λ.
+This probability is given by
+P
+� ˆλ|λ,r1,x
+�
+=
+� ∞
+�k−1
+i=1 τi
+P
+� ˆλ,t|λ,r1,x
+�
+dt
+=
+� ∞
+�k−1
+i=1 τi
+P
+� ˆλ|λ,r1,t,x
+�
+P (t|λ,r1,x)dt
+=
+� ∞
+�k−1
+i=1 τi
+P
+� ˆλ|r1,t,x
+�
+P (t|λ,r1,x)dt
+=
+r1x
+�
+eλ − 1
+��
+e ˆλ − 1
+�
+� ∞
+�k−1
+i=1 τi
+(1 − w)x
+��x
+j=1
+(x−1
+j−1)
+j!
+� λw
+1−w
+�j ���x
+j=1
+(x−1
+j−1)
+j!
+� ˆλw
+1−w
+�j �
+�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+dt
+=
+x
+�
+eλ − 1
+��
+e ˆλ − 1
+�
+� θ
+0
+(1 − w)x
+��x
+j=1
+(x−1
+j−1)
+j!
+� λw
+1−w
+�j ���x
+j=1
+(x−1
+j−1)
+j!
+� ˆλw
+1−w
+�j �
+w�x
+j=1
+�x−1
+j−1
+�� w
+1−w
+�j ζ(j + 1)
+dw,
+(129)
+where θ is given by (70) and we have assumed that ˆλ is conditionally independent
+of λ given t.
+It follows that the t-independent probability that ˆλ will take values between a and b
+is given by
+P(a ≤ ˆλ ≤ b | λ,r1,x) =
+� b
+a
+P
+� ˆλ|λ,r1,x
+�
+d ˆλ.
+(130)
+44
+
+5.2
+Supplementary Figures
+Figure 5.1: Limit of detection of the single-phase process. The LoD was determined
+while accounting for either sampling noise alone (solid green line), amplification noise
+alone (solid red line), or both sampling noise and amplification noise (solid blue line).
+It was then plotted versus amplification efficiency, which is expressed on a base-2 scale
+as a percentage. The LoD based on sampling noise alone equals 3, whereas the LoD is
+highest when accounting for both types of noise. In the latter case, it ranges from 157,
+when the efficiency is only 80%, to 6, when the efficiency is 100%. The plot shows a strong
+dependence of the LoD on efficiency.
+45
+
+150 -
+100
+Noise type
+LoD
+Sampling noise only
+Amplification noise only
+Sampling + amplification noise
+50 
+0 -
+80
+85
+90
+95
+100
+Amplification efficiency (%)Figure 5.2: Ratio of expected versus estimated fraction of positive partitions in digital
+PCR. For different expected numbers of input molecules λ, the ratio of the fraction of
+digital PCR partitions expected to test positive was calculated using Equation (24) and
+divided by the standard estimate based on the Poisson distribution (i.e. 1−e−λ). The result
+was plotted versus the amplification efficiency. While the efficiency used in calculations
+is always expressed on a base-e scale, for ease of comprehension it was converted into a
+base-2 scale and displayed as a percentage.
+46
+
+1.00 
+0.75 -
+1.61
+10
+0.25
+0.00 
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+49
+

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+arXiv:2301.00283v1  [quant-ph]  31 Dec 2022
+Scaling limit of the time averaged distribution for
+continuous time quantum walk and Szegedy’s walk on the path
+Yusuke Ide
+Department of Mathematics, College of Humanities and Sciences, Nihon University
+3-25-40 Sakura-josui, Setagaya-ku, Tokyo 156-8550, Japan
+e-mail: ide.yusuke@nihon-u.ac.jp
+Abstract
+In this paper, we consider Szegedy’s walk, a type of discrete time quantum walk, and corresponding continuous time
+quantum walk related to the birth and death chain. We show that the scaling limit of time averaged distribution for
+the continuous time quantum walk induces that of Szegedy’s walk if there exists the spectral gap on so-called the
+corresponding Jacobi matrix .
+1
+Introduction
+Quantum walks, a quantum counterpart of random walks have been extensively developed in various fields
+during the last two decades. Since quantum walks are very simple models therefore they play fundamental
+and important roles in both theoretical fields and applications. There are good review articles for these
+developments such as Kempe [6], Kendon [7], Venegas-Andraca [14,15], Konno [8], Manouchehri and Wang
+[9], and Portugal [11].
+We investigate the time averaged distribution of a variant of discrete time quantum walk (DTQW)
+so-called Szegedy’s walk [13]. On the path graph, the spectral properties of Szegedy’s walk are directly
+connected to the theory of (finite type) orthogonal polynomials. There are studies of the distribution of
+Szegedy’s walk on the path graph for example [1–3,5,10,12].
+In this paper, we focus on scaling limit of the time averaged distributions of both Szegedy’s walk and
+corresponding continuous time quantum walk on the path graph related to the random walk with reflecting
+walls. In order to our main theorem (Theorem 4.1), if there exists the spectral gap, i.e., the limit superior in
+the size of the path graph tends to infinity of the second largest eigenvalue of the Jacobi matrix is less than
+one (the largest eigenvalue), then the scaling limit of Szegedy’s walk is the same as that of corresponding
+continuous time quantum walk. We should note that existence of the spectral gap of the Jacobi matrix is
+equivalent to that of the transition matrix of corresponding random walk. A typical example of this case
+is space homogeneous random walk with pR
+j = p case (the second largest eigenvalue is 2
+�
+p(1 − p) cos π/n)
+treated in [5] except for the symmetric random walk with pR
+j = 1/2. Unfortunately we have not been covered
+with non-spectral gap cases including symmetric random walk and the Ehrenfest model (the second largest
+eigenvalue is 1 − 2/n) treated in [3]. To reveal non-spectral gap case is one of interesting future problems.
+The rest of this paper is organized as follows. In Sec. 2, we define our setting of discrete time random
+walk, continuous time quantum walk and discrete time quantum walk on the path graph. Sec. 3 is devoted to
+show relationships between the time averaged distribution of Szegedy’s walk and continuous time quantum
+walk. In the last section, we state our main theorem (Theorem 4.1) and prove it.
+2
+Definition of the models
+In this paper, we consider the path graph Pn+1 = (V (Pn+1), E(Pn+1)) with the vertex set V (Pn+1) =
+{0, 1, . . ., n} and the (undirected) edge set E(Pn+1) = {(j, j + 1) : j = 0, 1, . . . , n − 1}. On the path graph
+Keywords:
+birth and death chain, Szegedy’s walk, continuous time quantum walk, scaling limit, time averaged distribution
+1
+
+Pn+1, we define a discrete time random walk (DTRW) with reflecting walls as follows:
+Let pL
+j be the transition probability of the random walker at the vertex j ∈ V (Pn+1) to the left (j − 1 ∈
+V (Pn+1)). Also let pR
+j = 1−pL
+j be the transition probability of the random walker at the vertex j ∈ V (Pn+1)
+to the right (j + 1 ∈ V (Pn+1)). For the sake of simplicity, we assume 0 < pL
+j , pR
+j < 1 except for j = 0, n. We
+put the reflecting walls at the vertex 0 ∈ V (Pn+1) and the vertex n ∈ V (Pn+1), i.e., we set pR
+0 = pL
+n = 1.
+We also call this type of DTRW as the birth and death chain.
+Let a positive constant Cπ be
+Cπ := 1 +
+n
+�
+j=1
+pR
+0 · pR
+1 · · · pR
+j−1
+pL
+1 · pL
+2 · · · pL
+j
+then we can define the stationary distribution {π(0), π(1), . . . , π(n)} as
+π(j) =
+
+
+
+1
+Cπ
+if j = 0,
+1
+Cπ ·
+pR
+0 ·pR
+1 ···pR
+j−1
+pL
+1 ·pL
+2 ···pL
+j
+if j = 1, 2, . . ., n.
+Note that π(j) > 0 for all j ∈ V (Pn+1) and the stationary distribution is satisfied with so-called the detailed
+balance condition,
+π(j) · pR
+j = pL
+j+1 · π(j + 1),
+for j = 0, 1, . . .n − 1.
+In order to define a continuous time quantum walk (CTQW) corresponding to the DTRW, we intro-
+duce the normalized Laplacian matrix L.
+Let P be the transition matrix of the DTRW. Also we de-
+fine diagonal matrices D1/2
+π
+:= diag
+��
+π(0),
+�
+π(1), . . . ,
+�
+π(n)
+�
+and D−1/2
+π
+=
+�
+D1/2
+π
+�−1
+.
+Note that
+D−1/2
+��
+= diag
+�
+1/
+�
+π(0), 1/
+�
+π(1), . . . , 1/
+�
+π(n)
+�
+by the definition. The normalized Laplacian matrix L
+is given by
+L := D1/2
+π
+(In+1 − P) D−1/2
+π
+= In+1 − D1/2
+π
+PD−1/2
+π
+,
+where In+1 be the (n + 1) × (n + 1) identity matrix. We should remark that the matrix
+J := D1/2
+π
+PD−1/2
+π
+,
+is referred as the Jacobi matrix. So we can rewrite L as L = In+1 − J.
+By using the detailed balance condition, we obtain
+Jj,k = Jk,j =
+��
+pR
+j pL
+j+1,
+if k = j + 1,
+0,
+otherwise.
+Thus L = In+1 − J is an Hermitian matrix (real symmetric matrix). The CTQW which is discussed in this
+paper is driven by the time evolution operator (unitary matrix)
+UCT QW (t) := exp (itL) :=
+∞
+�
+k=0
+(it)k
+k! Lk,
+where i is the imaginary unit. Let XC
+t
+(t ≥ 0) be the random variable representing the position of the
+CTQWer at time t. The distribution of XC
+t is determined by
+P
+�
+XC
+t = k|XC
+0 = j
+�
+:= |⟨k|UCT QW (t)|j⟩|2 =
+���(UCT QW (t))k,j
+���
+2
+,
+where |j⟩ is the (n + 1)-dimensional unit vector (column vector) which j-th component equals 1 and the
+other components are 0 and ⟨v| is the transpose of |v⟩, i.e., ⟨v| = T |v⟩.
+2
+
+Hereafter we only consider XC
+0 = 0 , i.e., the CTQWer starts from the left most vertex 0 ∈ V (Pn+1),
+cases. The time averaged distribution ¯pC of the CTQW is defined by
+¯pC(j) := lim
+T →∞
+1
+T
+� T
+0
+P
+�
+XC
+t = j|XC
+0 = 0
+�
+dt,
+for each vertex j ∈ V (Pn+1). We define a random variable ¯XC
+n as P
+� ¯XC
+n = j
+�
+= ¯pC(j).
+In this paper, we also deal with a type of discrete time quantum walk (DTQW) corresponding to the
+DTRW so-called Szegedy’s walk. The time evolution operator for the DTQW is defined by U = SC with
+the coin operator C and the shift operator (flip-flop type shift) S. The coin operator C is defined by
+C = |0⟩⟨0| ⊗ I2 +
+n−1
+�
+j=1
+|j⟩⟨j| ⊗ Cj + |n⟩⟨n| ⊗ I2,
+where I2 is the 2 × 2 identity matrix and ⊗ is the tensor product. The local coin operator Cj is defined by
+Cj = 2|φj⟩⟨φj| − I2,
+|φj⟩ =
+�
+pL
+j |L⟩ +
+�
+pR
+j |R⟩,
+where |L⟩ = T [1 0] and |R⟩ = T [0 1]. The shift operator S is given by
+S (|j⟩ ⊗ |L⟩) = |j − 1⟩ ⊗ |R⟩,
+S (|j⟩ ⊗ |R⟩) = |j + 1⟩ ⊗ |L⟩.
+Let XD
+t (t = 0, 1, . . .) be the random variable representing the position of the DTQWer at time t. In this
+paper, we only consider XD
+0 = 0 cases. The distribution of XD
+t
+is defined by
+P
+�
+XD
+t = j|XD
+0 = 0
+�
+: = ∥(⟨j| ⊗ I2) UDT QW (t) (|0⟩ ⊗ |R⟩)∥2
+= |(⟨j| ⊗ ⟨L|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 + |(⟨j| ⊗ ⟨R|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 .
+We also consider the time averaged distribution ¯pD of the DTQW defined by
+¯pD(j) := lim
+T →∞
+1
+T
+T −1
+�
+t=0
+P
+�
+XD
+t = j|XD
+0 = 0
+�
+,
+for each vertex j ∈ V (Pn+1). We define a random variable ¯XD
+n as P
+� ¯XD
+n = j
+�
+= ¯pD(j).
+3
+Relations between ¯XC
+n and ¯XD
+n
+Since the Jacobi matrix J is a real symmetric matrix with simple [4] and symmetric [3] eigenvalues, we
+obtain eigenvalues 1 = λ0 > λ1 > · · · > λn−1 > λn = −1 and corresponding eigenvectors {|vℓ⟩}n
+ℓ=0 as an
+orthonormal basis of n-dimensional complex vector space Cn. Thus we have the spectral decomposition
+J =
+n
+�
+ℓ=0
+λℓ|vℓ⟩⟨vℓ|.
+Noting that L = In+1 − J, the spectral decomposition of UCT QW (t) is given by
+UCT QW (t) =
+n
+�
+ℓ=0
+exp [it (1 − λℓ)] |vℓ⟩⟨vℓ| = eit
+n
+�
+ℓ=0
+e−itλℓ|vℓ⟩⟨vℓ|.
+Because of simple eigenvalues of the Jacobi matrix J, the time averaged distribution ¯pC is expressed by
+¯pC(j) =
+n
+�
+ℓ=0
+|⟨j|vℓ⟩|2 |⟨vℓ|0⟩|2 =
+n
+�
+ℓ=0
+|vℓ(j)|2 |vℓ(0)|2 ,
+3
+
+where vℓ(j) is the jth component of |vℓ⟩.
+On the other hand, the spectral decomposition of UDT QW (t) is given (see e.g. [3,5,12,13]) by
+UDT QW (t) = µ0|u0⟩⟨u0| +
+n−1
+�
+ℓ=1
+�
+1
+2(1 − λ2
+ℓ)
+�
+±
+µ±ℓ|u±ℓ⟩⟨u±ℓ|
+�
++ µn|un⟩⟨un|,
+where
+
+
+
+
+
+µ0 = λ0 = 1,
+|u0⟩ = |v0⟩,
+µ±ℓ = exp
+�
+±i cos−1 λℓ
+�
+,
+|u±ℓ⟩ = |vℓ⟩ − µ±ℓ S|vℓ⟩,
+µn = λn = −1,
+|un−1⟩ = |vn−1⟩,
+with
+|vℓ⟩ = vℓ(0)|0⟩ ⊗ |R⟩ +
+n−1
+�
+j=1
+vℓ(j)|j⟩ ⊗ |φj⟩ + vℓ(n)|n⟩ ⊗ |L⟩.
+All the eigenvalues of UDT QW (t) are also simple, the time averaged distribution ¯pD is expressed by
+¯pD(j) =
+�
+|(⟨j| ⊗ ⟨L|) |u0⟩|2 + |(⟨j| ⊗ ⟨R|) |u0⟩|2�
+|⟨u0| (|0⟩ ⊗ |R⟩)|2
++
+n−1
+�
+ℓ=1
+�
+1
+2(1 − λ2
+ℓ)
+�
+±
+�
+|(⟨j| ⊗ ⟨L|) |u±ℓ⟩|2 + |(⟨j| ⊗ ⟨R|) |u±ℓ⟩|2�
+|⟨u±ℓ| (|0⟩ ⊗ |R⟩)|2
+�
++
+�
+|(⟨j| ⊗ ⟨L|) |un⟩|2 + |(⟨j| ⊗ ⟨R|) |un⟩|2�
+|⟨un| (|0⟩ ⊗ |R⟩)|2 .
+More concrete expression of ¯pD in terms of eigenvalues and eigenvectors of the Jacobi matrix J is given as
+follows (rearrangement of Eq.(10) in [3]):
+¯pD(j) = 1
+2 |v0(j)|2 |v0(0)|2 + 1
+2 |vn(j)|2 |vn(0)|2
++ 1
+2
+n
+�
+ℓ=0
+|vℓ(j)|2 |vℓ(0)|2
++ 1
+2
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+�
+pR
+j−1 |vℓ(j − 1)|2 − λ2
+ℓ |vℓ(j)|2 + pL
+j+1 |vℓ(j + 1)|2�
+|vℓ(0)|2 ,
+with conventions pR
+−1 = vℓ(−1) = pL
+n+1 = vℓ(n + 1) = 0.
+Now we consider the distribution functions ¯F C
+n (x) := P
+� ¯XC
+n ≤ x
+�
+= �
+j≤x ¯pC(j) of ¯XC
+n and ¯F D
+n (x) :=
+P
+� ¯XD
+n ≤ x
+�
+= �
+j≤x ¯pD(j) of ¯XD
+n . For each integer 0 ≤ k ≤ n − 1, we have
+¯F C
+n (k) =
+k
+�
+j=0
+¯pC(j) =
+k
+�
+j=0
+� n
+�
+ℓ=0
+|vℓ(j)|2 |vℓ(0)|2
+�
+.
+4
+
+We also obtain the following expression by using pL
+j + pR
+j = 1, pR
+0 = 1 and pL
+1 |vℓ(1)|2 = λ2
+ℓ |vℓ(0)|2:
+¯F D
+n (k) =
+k
+�
+j=0
+¯pD(j)
+= 1
+2
+k
+�
+j=0
+|v0(j)|2 |v0(0)|2 + 1
+2
+k
+�
+j=0
+|vn(j)|2 |vn(0)|2
++ 1
+2
+k
+�
+j=0
+� n
+�
+ℓ=0
+|vℓ(j)|2 |vℓ(0)|2
+�
++ 1
+2
+k
+�
+j=1
+�n−1
+�
+ℓ=1
+|vℓ(j)|2 |vℓ(0)|2
+�
++ 1
+2
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+�
+pR
+0 |vℓ(0)|2 − pL
+1 |vℓ(1)|2 − pR
+k |vℓ(k)|2 + pL
+k+1 |vℓ(k + 1)|2�
+|vℓ(0)|2
+=
+k
+�
+j=0
+� n
+�
+ℓ=0
+|vℓ(j)|2 |vℓ(0)|2
+�
++ 1
+2
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+�
+−pR
+k |vℓ(k)|2 + pL
+k+1 |vℓ(k + 1)|2�
+|vℓ(0)|2
+= ¯F C
+n (k) + 1
+2
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+�
+−pR
+k |vℓ(k)|2 + pL
+k+1 |vℓ(k + 1)|2�
+|vℓ(0)|2 .
+4
+Scaling limit
+In this section, we state our main result and prove it.
+Theorem 4.1 Assume that there exists the spectral gap, i.e., lim supn→∞ λ1 < 1 = λ0. If
+¯
+XC
+n
+n
+converges
+weakly to the random variable ¯X as n → ∞ then
+¯
+XD
+n
+n
+also converges weakly to the same random variable ¯X.
+Proof of Theorem 4.1
+Let ¯F be the distribution function of the random variable ¯X. We assume that
+lim
+n→∞ P
+� ¯XC
+n
+n
+≤ x
+�
+= ¯F(x)
+(4.1)
+for all points x at which ¯F is continuous. Hereafter we assume ¯F is continuous at x (0 ≤ x ≤ 1). Remark
+that from the definition, Eq. (4.1) means that
+lim
+n→∞
+¯F C
+n (nx) = lim
+n→∞
+¯F C
+n (⌊nx⌋) = lim
+n→∞
+⌊nx⌋
+�
+j=0
+� n
+�
+ℓ=0
+|vℓ(j)|2 |vℓ(0)|2
+�
+= ¯F(x),
+(4.2)
+where ⌊a⌋ denotes the biggest integer which is not greater than a.
+From Eq. (4.2) and the relation
+P
+� ¯XD
+n
+n
+≤ x
+�
+= ¯F D
+n (nx) = ¯F D
+n (⌊nx⌋)
+= ¯F C
+n (⌊nx⌋) + 1
+2
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+�
+− pR
+⌊nx⌋ |vℓ(⌊nx⌋)|2 + pL
+⌊nx⌋+1 |vℓ(⌊nx⌋ + 1)|2
+�
+|vℓ(0)|2 ,
+if we can prove
+lim
+n→∞
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+|vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim
+n→∞
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0,
+(4.3)
+5
+
+then we can conclude
+lim
+n→∞ P
+� ¯XD
+n
+n
+≤ x
+�
+= ¯F(x),
+for all points at which ¯F is continuous.
+From Eq.(4.2), we obtain
+0 ≤
+⌊nx⌋
+�
+j=0
+�n−1
+�
+ℓ=1
+|vℓ(j)|2 |vℓ(0)|2
+�
+≤ ¯F C
+n (⌊nx⌋)
+n→∞
+−−−−→ ¯F(x).
+Also we have
+0 ≤
+⌊nx⌋+1
+�
+j=0
+�n−1
+�
+ℓ=1
+|vℓ(j)|2 |vℓ(0)|2
+�
+≤ ¯F C
+n
+��
+n
+�
+x + 1
+n
+���
+n→∞
+−−−−→ ¯F(x),
+from continuity of ¯F at x. These mean that
+lim
+n→∞
+n−1
+�
+ℓ=1
+|vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim
+n→∞
+n−1
+�
+ℓ=1
+|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0.
+(4.4)
+Therefore combining with Eq. (4.4), we obtain Eq. (4.3) as follows:
+lim sup
+n→∞
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+|vℓ(⌊nx⌋)|2 |vℓ(0)|2 ≤ lim sup
+n→∞
+1
+1 − λ2
+1
+n−1
+�
+ℓ=1
+|vℓ(⌊nx⌋)|2 |vℓ(0)|2
+≤
+1
+1 − lim supn→∞ λ2
+1
+× lim
+n→∞
+n−1
+�
+ℓ=1
+|vℓ(⌊nx⌋)|2 |vℓ(0)|2
+= 0,
+lim sup
+n→∞
+n−1
+�
+ℓ=1
+1
+1 − λ2
+ℓ
+|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 ≤ lim sup
+n→∞
+1
+1 − λ2
+1
+n−1
+�
+ℓ=1
+|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2
+≤
+1
+1 − lim supn→∞ λ2
+1
+× lim
+n→∞
+n−1
+�
+ℓ=1
+|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2
+= 0.
+This completes the proof.
+✷
+References
+[1] Anahara, Y., Konno, N., Morioka, H., Segawa, E.: Comfortable place for quantum walker on finite path.
+Quantum Inf. Process. 21, 242 (2022).
+[2] Higuchi, K., Komatsu, T., Konno, N., Morioka, H., Segawa, E.: A discontinuity of the energy of quantum walk
+in impurities. Symmetry 13, 1134 (2022).
+[3] Ho, C.-L., Ide, Y., Konno, N., Segawa, E., Takumi, K.: A spectral analysis of discrete-time quantum walks
+related to the birth and death chains. J. Stat. Phys. 171, 207–219 (2018).
+[4] Hora, A., Obata, N.: Quantum Probability and Spectral Analysis of Graphs. Springer (2007).
+[5] Ide, Y., Konno, N., Segawa, E.: Time averaged distribution of a discrete-time quantum walk on the path.
+Quantum Inf. Process. 11 (5), 1207–1218 (2012).
+[6] Kempe, J.: Quantum random walks - an introductory overview. Contemporary Physics 44, 307–327 (2003).
+6
+
+[7] Kendon, V.: Decoherence in quantum walks - a review. Math. Struct. in Comp. Sci. 17, 1169–1220 (2007).
+[8] Konno, N.: Quantum Walks. In: Quantum Potential Theory, Franz, U., and Sch¨urmann, M., Eds., Lecture
+Notes in Mathematics: Vol. 1954, pp. 309–452, Springer-Verlag, Heidelberg (2008).
+[9] Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks, Springer (2013).
+[10] Marquezino, F. L., Portugal, R., Abal, G., Donangelo, R.: Mixing times in quantum walks on the hypercube.
+Phys. Rev. A 77, 042312 (2008).
+[11] Portugal, R.: Quantum Walks and Search Algorithms, Springer (2013).
+[12] Segawa, E.: Localization of quantum walks induced by recurrence properties of random walks. J. Comput.
+Nanosci. 10, 1583–1590 (2013).
+[13] Szegedy, M.: Quantum speed-up of Markov chain based algorithms. Proc. of the 45th Annual IEEE Symposium
+on Foundations of Computer Science (FOCS’04), 32–41 (2004).
+[14] Venegas-Andraca, S. E.: Quantum Walks for Computer Scientists, Morgan and Claypool (2008).
+[15] Venegas-Andraca, S. E.: Quantum walks: a comprehensive review, Quantum Inf. Process. 11, 1015–1106
+(2012).
+7
+

5dAyT4oBgHgl3EQfcfda/content/tmp_files/load_file.txt

@@ -0,0 +1,317 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf,len=316
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='00283v1  [quant-ph]  31 Dec 2022  Scaling limit of the time averaged distribution for  continuous time quantum walk and Szegedy’s walk on the path  Yusuke Ide  Department of Mathematics, College of Humanities and Sciences, Nihon University  3-25-40 Sakura-josui, Setagaya-ku, Tokyo 156-8550, Japan  e-mail: ide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='yusuke@nihon-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='jp  Abstract  In this paper, we consider Szegedy’s walk, a type of discrete time quantum walk, and corresponding continuous time  quantum walk related to the birth and death chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We show that the scaling limit of time averaged distribution for  the continuous time quantum walk induces that of Szegedy’s walk if there exists the spectral gap on so-called the  corresponding Jacobi matrix .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  1  Introduction  Quantum walks, a quantum counterpart of random walks have been extensively developed in various fields  during the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Since quantum walks are very simple models therefore they play fundamental  and important roles in both theoretical fields and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' There are good review articles for these  developments such as Kempe [6], Kendon [7], Venegas-Andraca [14,15], Konno [8], Manouchehri and Wang  [9], and Portugal [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  We investigate the time averaged distribution of a variant of discrete time quantum walk (DTQW)  so-called Szegedy’s walk [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' On the path graph, the spectral properties of Szegedy’s walk are directly  connected to the theory of (finite type) orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' There are studies of the distribution of  Szegedy’s walk on the path graph for example [1–3,5,10,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  In this paper, we focus on scaling limit of the time averaged distributions of both Szegedy’s walk and  corresponding continuous time quantum walk on the path graph related to the random walk with reflecting  walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' In order to our main theorem (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1), if there exists the spectral gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', the limit superior in  the size of the path graph tends to infinity of the second largest eigenvalue of the Jacobi matrix is less than  one (the largest eigenvalue), then the scaling limit of Szegedy’s walk is the same as that of corresponding  continuous time quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We should note that existence of the spectral gap of the Jacobi matrix is  equivalent to that of the transition matrix of corresponding random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' A typical example of this case  is space homogeneous random walk with pR  j = p case (the second largest eigenvalue is 2  �  p(1 − p) cos π/n)  treated in [5] except for the symmetric random walk with pR  j = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Unfortunately we have not been covered  with non-spectral gap cases including symmetric random walk and the Ehrenfest model (the second largest  eigenvalue is 1 − 2/n) treated in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' To reveal non-spectral gap case is one of interesting future problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' 2, we define our setting of discrete time random  walk, continuous time quantum walk and discrete time quantum walk on the path graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' 3 is devoted to  show relationships between the time averaged distribution of Szegedy’s walk and continuous time quantum  walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' In the last section, we state our main theorem (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1) and prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  2  Definition of the models  In this paper, we consider the path graph Pn+1 = (V (Pn+1), E(Pn+1)) with the vertex set V (Pn+1) =  {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', n} and the (undirected) edge set E(Pn+1) = {(j, j + 1) : j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' On the path graph  Keywords:  birth and death chain, Szegedy’s walk, continuous time quantum walk, scaling limit, time averaged distribution  1  Pn+1, we define a discrete time random walk (DTRW) with reflecting walls as follows:  Let pL  j be the transition probability of the random walker at the vertex j ∈ V (Pn+1) to the left (j − 1 ∈  V (Pn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Also let pR  j = 1−pL  j be the transition probability of the random walker at the vertex j ∈ V (Pn+1)  to the right (j + 1 ∈ V (Pn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' For the sake of simplicity, we assume 0 < pL  j , pR  j < 1 except for j = 0, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We  put the reflecting walls at the vertex 0 ∈ V (Pn+1) and the vertex n ∈ V (Pn+1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', we set pR  0 = pL  n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  We also call this type of DTRW as the birth and death chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Let a positive constant Cπ be  Cπ := 1 +  n  �  j=1  pR  0 · pR  1 · · · pR  j−1  pL  1 · pL  2 · · · pL  j  then we can define the stationary distribution {π(0), π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' , π(n)} as  π(j) =  \uf8f1  \uf8f2  \uf8f3  1  Cπ  if j = 0,  1  Cπ ·  pR  0 ·pR  1 ···pR  j−1  pL  1 ·pL  2 ···pL  j  if j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Note that π(j) > 0 for all j ∈ V (Pn+1) and the stationary distribution is satisfied with so-called the detailed  balance condition,  π(j) · pR  j = pL  j+1 · π(j + 1),  for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  In order to define a continuous time quantum walk (CTQW) corresponding to the DTRW, we intro-  duce the normalized Laplacian matrix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Let P be the transition matrix of the DTRW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Also we de-  fine diagonal matrices D1/2  π  := diag  ��  π(0),  �  π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' ,  �  π(n)  �  and D−1/2  π  =  �  D1/2  π  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Note that  D−1/2  π  = diag  �  1/  �  π(0), 1/  �  π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' , 1/  �  π(n)  �  by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The normalized Laplacian matrix L  is given by  L := D1/2  π  (In+1 − P) D−1/2  π  = In+1 − D1/2  π  PD−1/2  π  ,  where In+1 be the (n + 1) × (n + 1) identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We should remark that the matrix  J := D1/2  π  PD−1/2  π  ,  is referred as the Jacobi matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' So we can rewrite L as L = In+1 − J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  By using the detailed balance condition, we obtain  Jj,k = Jk,j =  ��  pR  j pL  j+1,  if k = j + 1,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Thus L = In+1 − J is an Hermitian matrix (real symmetric matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The CTQW which is discussed in this  paper is driven by the time evolution operator (unitary matrix)  UCT QW (t) := exp (itL) :=  ∞  �  k=0  (it)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Lk,  where i is the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Let XC  t  (t ≥ 0) be the random variable representing the position of the  CTQWer at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The distribution of XC  t is determined by  P  �  XC  t = k|XC  0 = j  �  := |⟨k|UCT QW (t)|j⟩|2 =  ���(UCT QW (t))k,j  ���  2  ,  where |j⟩ is the (n + 1)-dimensional unit vector (column vector) which j-th component equals 1 and the  other components are 0 and ⟨v| is the transpose of |v⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', ⟨v| = T |v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  2  Hereafter we only consider XC  0 = 0 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', the CTQWer starts from the left most vertex 0 ∈ V (Pn+1),  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The time averaged distribution ¯pC of the CTQW is defined by  ¯pC(j) := lim  T →∞  1  T  � T  0  P  �  XC  t = j|XC  0 = 0  �  dt,  for each vertex j ∈ V (Pn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We define a random variable ¯XC  n as P  � ¯XC  n = j  �  = ¯pC(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  In this paper, we also deal with a type of discrete time quantum walk (DTQW) corresponding to the  DTRW so-called Szegedy’s walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The time evolution operator for the DTQW is defined by U = SC with  the coin operator C and the shift operator (flip-flop type shift) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The coin operator C is defined by  C = |0⟩⟨0| ⊗ I2 +  n−1  �  j=1  |j⟩⟨j| ⊗ Cj + |n⟩⟨n| ⊗ I2,  where I2 is the 2 × 2 identity matrix and ⊗ is the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The local coin operator Cj is defined by  Cj = 2|φj⟩⟨φj| − I2,  |φj⟩ =  �  pL  j |L⟩ +  �  pR  j |R⟩,  where |L⟩ = T [1 0] and |R⟩ = T [0 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The shift operator S is given by  S (|j⟩ ⊗ |L⟩) = |j − 1⟩ ⊗ |R⟩,  S (|j⟩ ⊗ |R⟩) = |j + 1⟩ ⊗ |L⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Let XD  t (t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=') be the random variable representing the position of the DTQWer at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' In this  paper, we only consider XD  0 = 0 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' The distribution of XD  t  is defined by  P  �  XD  t = j|XD  0 = 0  �  : = ∥(⟨j| ⊗ I2) UDT QW (t) (|0⟩ ⊗ |R⟩)∥2  = |(⟨j| ⊗ ⟨L|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 + |(⟨j| ⊗ ⟨R|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  We also consider the time averaged distribution ¯pD of the DTQW defined by  ¯pD(j) := lim  T →∞  1  T  T −1  �  t=0  P  �  XD  t = j|XD  0 = 0  �  ,  for each vertex j ∈ V (Pn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We define a random variable ¯XD  n as P  � ¯XD  n = j  �  = ¯pD(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  3  Relations between ¯XC  n and ¯XD  n  Since the Jacobi matrix J is a real symmetric matrix with simple [4] and symmetric [3] eigenvalues, we  obtain eigenvalues 1 = λ0 > λ1 > · · · > λn−1 > λn = −1 and corresponding eigenvectors {|vℓ⟩}n  ℓ=0 as an  orthonormal basis of n-dimensional complex vector space Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Thus we have the spectral decomposition  J =  n  �  ℓ=0  λℓ|vℓ⟩⟨vℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Noting that L = In+1 − J, the spectral decomposition of UCT QW (t) is given by  UCT QW (t) =  n  �  ℓ=0  exp [it (1 − λℓ)] |vℓ⟩⟨vℓ| = eit  n  �  ℓ=0  e−itλℓ|vℓ⟩⟨vℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Because of simple eigenvalues of the Jacobi matrix J, the time averaged distribution ¯pC is expressed by  ¯pC(j) =  n  �  ℓ=0  |⟨j|vℓ⟩|2 |⟨vℓ|0⟩|2 =  n  �  ℓ=0  |vℓ(j)|2 |vℓ(0)|2 ,  3  where vℓ(j) is the jth component of |vℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  On the other hand, the spectral decomposition of UDT QW (t) is given (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' [3,5,12,13]) by  UDT QW (t) = µ0|u0⟩⟨u0| +  n−1  �  ℓ=1  �  1  2(1 − λ2  ℓ)  �  ±  µ±ℓ|u±ℓ⟩⟨u±ℓ|  �  + µn|un⟩⟨un|,  where  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  µ0 = λ0 = 1,  |u0⟩ = |v0⟩,  µ±ℓ = exp  �  ±i cos−1 λℓ  �  ,  |u±ℓ⟩ = |vℓ⟩ − µ±ℓ S|vℓ⟩,  µn = λn = −1,  |un−1⟩ = |vn−1⟩,  with  |vℓ⟩ = vℓ(0)|0⟩ ⊗ |R⟩ +  n−1  �  j=1  vℓ(j)|j⟩ ⊗ |φj⟩ + vℓ(n)|n⟩ ⊗ |L⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  All the eigenvalues of UDT QW (t) are also simple, the time averaged distribution ¯pD is expressed by  ¯pD(j) =  �  |(⟨j| ⊗ ⟨L|) |u0⟩|2 + |(⟨j| ⊗ ⟨R|) |u0⟩|2�  |⟨u0| (|0⟩ ⊗ |R⟩)|2  +  n−1  �  ℓ=1  �  1  2(1 − λ2  ℓ)  �  ±  �  |(⟨j| ⊗ ⟨L|) |u±ℓ⟩|2 + |(⟨j| ⊗ ⟨R|) |u±ℓ⟩|2�  |⟨u±ℓ| (|0⟩ ⊗ |R⟩)|2  �  +  �  |(⟨j| ⊗ ⟨L|) |un⟩|2 + |(⟨j| ⊗ ⟨R|) |un⟩|2�  |⟨un| (|0⟩ ⊗ |R⟩)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  More concrete expression of ¯pD in terms of eigenvalues and eigenvectors of the Jacobi matrix J is given as  follows (rearrangement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' (10) in [3]):  ¯pD(j) = 1  2 |v0(j)|2 |v0(0)|2 + 1  2 |vn(j)|2 |vn(0)|2  + 1  2  n  �  ℓ=0  |vℓ(j)|2 |vℓ(0)|2  + 1  2  n−1  �  ℓ=1  1  1 − λ2  ℓ  �  pR  j−1 |vℓ(j − 1)|2 − λ2  ℓ |vℓ(j)|2 + pL  j+1 |vℓ(j + 1)|2�  |vℓ(0)|2 ,  with conventions pR  −1 = vℓ(−1) = pL  n+1 = vℓ(n + 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Now we consider the distribution functions ¯F C  n (x) := P  � ¯XC  n ≤ x  �  = �  j≤x ¯pC(j) of ¯XC  n and ¯F D  n (x) :=  P  � ¯XD  n ≤ x  �  = �  j≤x ¯pD(j) of ¯XD  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' For each integer 0 ≤ k ≤ n − 1, we have  ¯F C  n (k) =  k  �  j=0  ¯pC(j) =  k  �  j=0  � n  �  ℓ=0  |vℓ(j)|2 |vℓ(0)|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  4  We also obtain the following expression by using pL  j + pR  j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' pR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='0 = 1 and pL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1 |vℓ(1)|2 = λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ |vℓ(0)|2:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='¯F D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='n (k) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='j=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='¯pD(j)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
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+page_content='|vn(j)|2 |vn(0)|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='|vℓ(j)|2 |vℓ(0)|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='|vℓ(j)|2 |vℓ(0)|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1 − λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='pR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='0 |vℓ(0)|2 − pL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1 |vℓ(1)|2 − pR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k |vℓ(k)|2 + pL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k+1 |vℓ(k + 1)|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='|vℓ(0)|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='j=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='� n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='|vℓ(j)|2 |vℓ(0)|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1 − λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='−pR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k |vℓ(k)|2 + pL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k+1 |vℓ(k + 1)|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='|vℓ(0)|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='= ¯F C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='n (k) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1 − λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='−pR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k |vℓ(k)|2 + pL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='k+1 |vℓ(k + 1)|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='|vℓ(0)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  4  Scaling limit  In this section, we state our main result and prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1 Assume that there exists the spectral gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=', lim supn→∞ λ1 < 1 = λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' If  ¯  XC  n  n  converges  weakly to the random variable ¯X as n → ∞ then  ¯  XD  n  n  also converges weakly to the same random variable ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1  Let ¯F be the distribution function of the random variable ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' We assume that  lim  n→∞ P  � ¯XC  n  n  ≤ x  �  = ¯F(x)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1)  for all points x at which ¯F is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Hereafter we assume ¯F is continuous at x (0 ≤ x ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' Remark  that from the definition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='1) means that  lim  n→∞  ¯F C  n (nx) = lim  n→∞  ¯F C  n (⌊nx⌋) = lim  n→∞  ⌊nx⌋  �  j=0  � n  �  ℓ=0  |vℓ(j)|2 |vℓ(0)|2  �  = ¯F(x),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2)  where ⌊a�� denotes the biggest integer which is not greater than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2) and the relation  P  � ¯XD  n  n  ≤ x  �  = ¯F D  n (nx) = ¯F D  n (⌊nx⌋)  = ¯F C  n (⌊nx⌋) + 1  2  n−1  �  ℓ=1  1  1 − λ2  ℓ  �  − pR  ⌊nx⌋ |vℓ(⌊nx⌋)|2 + pL  ⌊nx⌋+1 |vℓ(⌊nx⌋ + 1)|2  �  |vℓ(0)|2 ,  if we can prove  lim  n→∞  n−1  �  ℓ=1  1  1 − λ2  ℓ  |vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim  n→∞  n−1  �  ℓ=1  1  1 − λ2  ℓ  |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='3)  5  then we can conclude  lim  n→∞ P  � ¯XD  n  n  ≤ x  �  = ¯F(x),  for all points at which ¯F is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='2), we obtain  0 ≤  ⌊nx⌋  �  j=0  �n−1  �  ℓ=1  |vℓ(j)|2 |vℓ(0)|2  �  ≤ ¯F C  n (⌊nx⌋)  n→∞  −−−−→ ¯F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  Also we have  0 ≤  ⌊nx⌋+1  �  j=0  �n−1  �  ℓ=1  |vℓ(j)|2 |vℓ(0)|2  �  ≤ ¯F C  n  ��  n  �  x + 1  n  ���  n→∞  −−−−→ ¯F(x),  from continuity of ¯F at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' These mean that  lim  n→∞  n−1  �  ℓ=1  |vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim  n→∞  n−1  �  ℓ=1  |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='4)  Therefore combining with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='4), we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='3) as follows:  lim sup  n→∞  n−1  �  ℓ=1  1  1 − λ2  ℓ  |vℓ(⌊nx⌋)|2 |vℓ(0)|2 ≤ lim sup  n→∞  1  1 − λ2  1  n−1  �  ℓ=1  |vℓ(⌊nx⌋)|2 |vℓ(0)|2  ≤  1  1 − lim supn→∞ λ2  1  × lim  n→∞  n−1  �  ℓ=1  |vℓ(⌊nx⌋)|2 |vℓ(0)|2  = 0,  lim sup  n→∞  n−1  �  ℓ=1  1  1 − λ2  ℓ  |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 ≤ lim sup  n→∞  1  1 − λ2  1  n−1  �  ℓ=1  |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2  ≤  1  1 − lim supn→∞ λ2  1  × lim  n→∞  n−1  �  ℓ=1  |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
+page_content='  ✷  References  [1] Anahara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
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+page_content='  [15] Venegas-Andraca, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'}
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+arXiv:2301.03302v1  [eess.SY]  9 Jan 2023
+A Rolling Horizon Game Considering Network Effect
+in Cluster Forming for Dynamic Resilient
+Multiagent Systems
+Yurid Nugraha a, Ahmet Cetinkaya b, Tomohisa Hayakawa a, Hideaki Ishii c,
+Quanyan Zhu d
+aDepartment of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
+bDepartment of Functional Control Systems, Shibaura Institute of Technology, Tokyo, 135-8548, Japan
+cDepartment of Computer Science, Tokyo Insitute of Technology, Yokohama 226-8502, Japan
+dDepartment of Electrical and Computer Engineering, New York University, Brooklyn NY, 11201, USA
+Abstract
+A two-player game-theoretic problem on resilient graphs in a multiagent consensus setting is formulated. An attacker is
+capable to disable some of the edges of the network with the objective to divide the agents into clusters by emitting jamming
+signals while, in response, the defender recovers some of the edges by increasing the transmission power for the communication
+signals. Specifically, we consider repeated games between the attacker and the defender where the optimal strategies for the
+two players are derived in a rolling horizon fashion based on utility functions that take both the agents’ states and the sizes of
+clusters (known as network effect) into account. The players’ actions at each discrete-time step are constrained by their energy
+for transmissions of the signals, with a less strict constraint for the attacker. Necessary conditions and sufficient conditions
+of agent consensus are derived, which are influenced by the energy constraints. The number of clusters of agents at infinite
+time in the face of attacks and recoveries are also characterized. Simulation results are provided to demonstrate the effects of
+players’ actions on the cluster forming and to illustrate the players’ performance for different horizon parameters.
+Key words: Multiagent Systems, Cybersecurity, Game Theory, Consensus, Cluster Forming, Network Effect/Network
+Externality
+1
+Introduction
+Applications of large-scale networked systems have
+rapidly grown in various areas of critical infrastructures
+including power grids and transportation systems. Such
+systems can be considered as multiagent systems where
+a number of agents capable of making local decisions
+interact over a network and exchange information to
+reach a common goal [2]. While wireless communica-
+tion plays an important role for the functionality of the
+network, it is also prone to cyber attacks initiated by
+malicious adversaries [11,25].
+Email addresses: yurid@dsl.sc.e.titech.ac.jp (Yurid
+Nugraha), ahmet@shibaura-it.ac.jp (Ahmet Cetinkaya),
+hayakawa@sc.e.titech.ac.jp (Tomohisa Hayakawa),
+ishii@c.titech.ac.jp (Hideaki Ishii),
+quanyan.zhu@nyu.edu (Quanyan Zhu).
+Jamming attacks in consensus problems of multiagent
+systems have been studied in [3, 5, 28]. Noncooperative
+games between attackers and other players protecting
+the network are widely used to analyze security prob-
+lems, including jamming attacks [12, 17] and injection
+attacks [18,24,26].
+In a jamming attack formulation, it is natural to consider
+that the jammer/the attacker has an energy constraint
+such that, if it is not connected to energy sources, it is
+impossible to attack all communication links of the net-
+work at all times [4,5]. In the context of game-theoretical
+approaches, this constraint becomes important to char-
+acterize the strategic behaviors of the players [17].
+When the links in the network are attacked, the agents
+may become disconnected from other agents, resulting
+in several groups of connected agents, or clusters. The
+
+work [13] proposed the notion of network effect/network
+externality, which refers to the utility of an agent in
+a certain cluster depending on how many other agents
+belong to that particular cluster. Such a concept has
+been used to analyze grouping of agents on, e.g., social
+networks and computer networks, as discussed in [10,16].
+Rolling horizon control has been used to handle sys-
+tems with uncertainties. It is also studied in the context
+of networked control [15,30], where there may be addi-
+tional uncertainties related to communications among
+agents in the networks. Rolling horizon approaches are
+also discussed in noncooperative security game settings
+in [34,35], where horizon lengths affect the resilience of
+the system. Rolling horizon approaches have also been
+used to handle the constraints in the system, e.g., in an
+agent with obstacle avoidance constraints [14,27].
+In this paper, we consider a security problem in a two-
+player game setting between an attacker, who is moti-
+vated to disrupt the communication among agents by
+attacking communication links, and a defender, who at-
+tempts to recover some of the attacked links. We for-
+mulate the problem based on [6, 20], which use graph
+connectivity to characterize the game and the players’
+strategies. The game in this paper is played repeatedly
+over discrete time in the context of multiagent consen-
+sus.
+As a results of these persistent attacks and recover-
+ies, under consensus protocol cluster forming emerges
+among the agents of the networks with different clus-
+ters having different agents’ states. Cluster forming in
+multiagent systems has been studied in, e.g., [1, 7, 29],
+where the relations among certain agents may be hos-
+tile. In this paper, we approach clustering from a dif-
+ferent viewpoint based on a game-theoretic formulation.
+Specifically, the players of the game consider network
+effect/network externality [13] to form clusters among
+agents. Their utilities are determined by how the net-
+work is disconnected into groups of agents as well as how
+the players’ actions affect the states of the agents at each
+time. Under this setting, the number and the size of the
+clusters are influenced by how strong the attacks are;
+the stronger attacker is supposed to be able to separate
+agents into more smaller clusters, and vice versa.
+In the resilient network setting, it is common that there
+exists a network manager who is aware of the incoming
+attack, since the agents try to communicate with their
+neighbor agents at all time and thus quickly know if some
+of their neighbors do not send any signal. The network
+manager then tries to prepare a defense plan to quickly
+recover from such attacks and to repel the subsequent
+attacks.
+From the attacker’s viewpoint, it is also common that the
+attacker knows which edges of the network are the most
+vulnerable as well as how powerful the network manager
+is, e.g., the manager’s remaining resources. Therefore,
+we believe that this sequential model can be applied to
+several real-world settings.
+The main contribution of this paper is that we introduce
+a repeated game played repeatedly over time to model
+the decision making process between the attacker and
+the defender in the context of network security. It is then
+natural to explore how these games affect the networks
+and state evolution of the agents. Consensus protocol is
+considered due to its simple characterization, where all
+agents should converge in the case of no attack. More
+specifically, in comparison to [6, 20], our contribution
+is threefold: (i) We introduce more options for the at-
+tacker’s jamming signal strengths; (ii) the game consists
+of multiple attack-recovery actions, resulting in more
+complicated strategies; and (iii) we consider a rolling
+horizon approach for the players so that their strategies
+may be modified as they obtain new knowledge of the
+status of the system.
+More specifically, it is now possible for the attacker to
+disable links with stronger intensity of attack signals
+so that the defender is unable to recover those links
+(the decision on which edges are to be attacked with
+stronger attack signals is made at the same time as the
+decision on which edges are to be attacked with nor-
+mal attack signals); this feature is motivated by [32,33].
+In practice, this is possible when the attacker emits
+stronger jamming signals that takes more resource that
+results in much lower signal-to-interference-plus-noise
+ratio (SINR) so that it is not possible for the defender to
+recover the communication on those links with its lim-
+ited recovery strength. On the other hand, we consider
+games consisting of multiple parts, where the players
+need to consider their future utilities and energy con-
+straints when deciding their strategies at any point in
+time. This setting enables the the players to think fur-
+ther ahead and prioritize their long-term payoffs, com-
+pared to in a single-step case. The players recalculate and
+may override their strategies as time goes on, according
+to the rolling horizon approach. A related formulation
+without rolling horizon is discussed in [19], where the
+players are not able to change their strategies decided at
+earlier times.
+The paper is organized as follows. In Section 2, we in-
+troduce the framework for the attack-recovery sequence,
+cluster forming among agents, and energy consumption
+models of the players. The utility functions of the games
+in rolling horizon approach of the repeated games is dis-
+cussed in Section 3, whereas the game structure is char-
+acterized in Section 4. In Section 5, we analyze some con-
+ditions of consensus among agents, which are related to
+the parameters of the underlying graph and the players’
+energy constraints. We continue by discussing the clus-
+ter forming of agents when consensus is not achieved in
+Section 6. The equilibrium characterization of the game
+under certain conditions is discussed in Section 7. We
+2
+
+then provide numerical examples on consensus and clus-
+ter forming in Section 8 and conclude the paper in Sec-
+tion 9. The conference version of this paper appeared
+in [21], where we consider a more restricted situation on
+how often players update their strategies.
+The notations used in this paper are fairly standard. We
+denote by |·| the cardinality of a set. The floor function
+and the ceiling function are denoted by ⌊·⌋ and ⌈·⌉, re-
+spectively. The sets of positive and nonnegative integers
+are denoted by N and N0, respectively.
+2
+Attack/Recovery Characterization for Multi-
+agent Systems Under Consensus Dynamics
+We consider a multiagent system of n agents communi-
+cating to each other in discrete time in the face of jam-
+ming attacks. The agents are aiming to converge to a
+consensus state by interacting with each other over the
+communication network. The network topology for the
+normal operation is given by an undirected and con-
+nected graph G = (V, E). The graph consists of the set V
+of vertices representing the agents and the set E ⊆ V ×V
+of edges representing the communication links. The edge
+connectivity [2] of the connected graph G is denoted by
+λ.
+Each agent i has the scalar state xi[k] following the
+discrete-time update rule at time k ∈ N0 given by
+xi[k + 1] = xi[k] + ui[k],
+x[0] = x0,
+(1)
+where ui[k] denotes the control input applied to agent i.
+We assume that ui[k] is constructed as the weighted sum
+of the state differences between agent i and its neighbor
+agents, commonly used in, e.g., [8], which is given by
+ui[k] =
+�
+j∈Ni[k]
+aij(xj[k] − xi[k]),
+(2)
+where Ni[k] denotes the set of agents that can communi-
+cate with agent i at time k, and aij represents the weight
+of edge (i, j) ∈ E such that Σn
+j=1,j̸=iaij < 1, i ∈ V to
+ensure that the agents achieve consensus without any
+attack.
+We assume that the jamming attacks on an edge affect
+the communication between the two agents connected
+by that attacked edge. As a result, the set Ni[k] may
+change, and the resulting communication topology can
+be disconnected at time k. Such jamming attacks are
+represented by the removal of edges in G. On the other
+hand, within the system there is a defender that may be
+capable of maintaining the communication among the
+agents, e.g., by asking agents to send stronger commu-
+nication signals to overcome the jamming signals. This
+action is represented as rebuilding some of the attacked
+edges.
+From this sequence of attacks and recoveries, we charac-
+terize the attack-recovery process as a two-player game
+between the attacker and the defender in terms of the
+communication links in the network. In other words, the
+graph characterizing the networked system is resilient if
+the group of agents is able to recover from the damages
+caused by the attacker. However, there may be cases
+where the resiliency level of the graph is reduced if the
+jamming signals are sufficiently strong such that the de-
+fender cannot recover. Note that to achieve consensus,
+the agents need not be connected for all time.
+In this paper, we consider the case where the attacker has
+two types of jamming signals in terms of their strength,
+strong and normal. The defender is able to recover only
+the edges that are attacked with normal strength. In the
+following subsections, we first describe the sequence of
+attacks and recoveries and characterize some constraints
+on the players’ energy and computational ability that
+we need to impose as well as how the objective of the
+problem is formulated.
+2.1
+Attack-Recovery Sequence
+In our setting, at each discrete time k, the players (the
+attacker and the defender) decide to attack/recover cer-
+tain edges in two stages, with the attacker acting first
+and then the defender. Specifically, at time k the at-
+tacker attacks G by deleting the edges EA
+k
+⊆ E with
+normal jamming signals and E
+A
+k ⊆ E with strong jam-
+ming signals with EA
+k ∩ E
+A
+k = ∅, whereas the defender
+recovers ED
+k ⊆ EA
+k . As mentioned earlier, the defender
+is not able to recover the edges attacked with strong
+jamming signals, i.e., ED
+k ∩ E
+A
+k
+= ∅. Due to the at-
+tacks and then the recoveries, the network changes from
+G to GA
+k := (V, E \ (EA
+k ∪ E
+A
+k )) and further to GD
+k :=
+(V, (E \ (EA
+k ∪ E
+A
+k )) ∪ ED
+k ) at time k. The agents then
+communicate to their neighbors Ni[k] based on this re-
+sulting graph GD
+k .
+In this game, the players attempt to choose the best
+strategies in terms of edges attacked/recovered (E
+A
+k , EA
+k )
+and ED
+k to maximize their own utility functions. Here,
+the games are played every game period T time steps
+and the lth game is defined over the horizon of h steps
+from time (l − 1)T to (l − 1)T + h − 1, with l ∈ N and
+1 ≤ T ≤ h. The players make decisions in a rolling hori-
+zon fashion; the optimal strategies obtained at (l − 1)T
+for the future time may be overridden when the play-
+ers recalculate their strategies at time lT when the next
+game starts. Fig. 1 illustrates the discussed sequence over
+time with h = 8 and T = 4, where the filled circles in-
+dicate the implemented strategies and the empty circles
+3
+
+PSfrag replacements
+k
+Edge
+0
+1T
+2T
+l = 1
+l = 2
+l = 3
+horizon length h
+2nd game
+horizon length h
+game period T
+Fig. 1. Illustration of the games played over discrete time k with rolling
+horizon approaches by the players.
+PSfrag replacements
+0
+1
+2
+3
+k
+4
+Energy
+κA
+– Time
+Fig. 2. Energy constraint of the attacker considered
+in the formulation. The dashed line represents the
+total supplied energy to spend. The filled circles
+representing the actual energy consumed by the
+player should be below the dashed line.
+indicate the strategies of the game that are discarded.
+In this setting, the horizon length h indicates the com-
+putational ability, i.e., how long in the future the play-
+ers can plan their strategies, whereas the game period
+T ≤ h indicates the players’ adaptability, i.e., how long
+the players apply the obtained strategies without updat-
+ing (shorter T means that a player is more adaptable).
+The rolling horizon game structure will be discussed in
+Section 4 in more detail.
+2.2
+Energy Constraints
+The actions of the attacker and the defender are af-
+fected by the constraints on their energy resources. It
+is assumed that the total supplied energy for the play-
+ers increases linearly in time; furthermore, the energy
+consumed by the players is proportional to the number
+of attacked/recovered edges. Here we suppose that the
+players initially possess certain amount of energy κA and
+κD for the attacker and the defender, respectively. More-
+over, the players are assumed to be able to supply en-
+ergy wirelessly to devices that obstruct/retain commu-
+nication signals between the agents so that the energy
+supply rates to these devices are limited by the constant
+values of ρA and ρD every discrete time step. These de-
+vices are supposed to have unlimited battery capacity
+and thus can be supplied constantly by the players with
+a linear rate ρA or ρD.
+For the attacker, the strong attacks on E
+A
+k take β
+A >
+0 energy per edge per unit time whereas the normal
+attacks on EA
+k take βA > 0 cost per edge, with β
+A > βA.
+The total energy used by the attacker is constrained as
+k
+�
+m=0
+(β
+A|E
+A
+m|+βA|EA
+m|) ≤ κA + ρAk
+(3)
+for any time k, where κA ≥ ρA > 0. This implies that
+the total energy spent by the attacker cannot exceed the
+available energy characterized as the sum of the initial
+energy κA and the supplied energy ρAk by time k. This
+energy constraint restricts the number of edges that the
+attacker can attack. Note that the attacker’s available
+energy increases by ρA at each k. The condition κA ≥
+ρA allows the attacker to have at least the same attack
+ability at time k = 0.
+Fig. 2 illustrates the energy constraint of the attacker,
+where the dashed line with slope ρA represents the total
+supplied energy and the filled circles indicate the total
+energy spent. A critical case is when βA < ρA, since it is
+possible for the attacker to attack at least one edge for all
+times. This will have implications on the consensus and
+cluster forming of the agents, as we will discuss later.
+The energy constraint for the defender is similar to (3):
+k
+�
+m=0
+βD|ED
+m|≤ κD + ρDk,
+(4)
+with κD ≥ ρD > 0 and βD > 0. Note that there is a
+single term on the left-hand side because there is only
+one type of recovery signals for the agents.
+3
+Utility Functions with Cluster Forming and
+Agent-group Index Considerations
+In our game setting, the attacker tries to make the
+graph disconnected to separate the agents into clusters.
+Here, we introduce a few notions related to group-
+ing/clustering of agents. In a given subgraph G′ = (V, E′)
+of G, the agents may be divided into n(G′) number
+of groups, with the groups V′
+1, V′
+2, . . . , V′
+n(G′) being a
+partition of V with ∪n(G′)
+p=1 V′
+p = V and V′
+p ∩ V′
+q = ∅, if
+p ̸= q. There is no edge connecting different groups, i.e.,
+ei′,j′ /∈ E′, ∀i′ ∈ V′
+p, j′ ∈ V′
+q. We also call each subset of
+agents taking the same state at infinite time as a clus-
+4
+
+ter, i.e., limk→∞(xi[k] − xj[k]) = 0 implies that agents
+i and j belong to the same cluster.
+In the considered game, the attacker and the defender
+are concerned about the number of agents in each
+group. Specifically, we follow the notion of network ef-
+fect/network externality [13], where the utility of an
+agent in a certain group depends on how many other
+agents belong to that particular group. In the context
+of this game, the attacker wants to isolate agents so
+that fewer agents are in each group, while the defender
+wants as many agents as possible in the same group. We
+then represent the level of grouping in the graph G′ by
+the function c(·), which we call the agent-group index,
+given by
+c(G′) :=
+n(G′)
+�
+p=1
+|V′
+p|2−|V|2
+(≤ 0).
+(5)
+The value of c(G′) is 0 if G′ is connected, since there is
+only one group (i.e., n(G′) = 1). A larger value (closer to
+0) of c(G′) implies that there are fewer groups in graph
+G′, and/or each group has more agents. The agent-group
+indices of some graphs are shown in Fig. 3. Here, it is
+interesting that c(GD) is smaller than c(GC), even though
+GC has more groups. It is because the largest cluster is
+constituted by more agents in GC than the case of GD.
+Thus, for an attacker who tries to reduce the number of
+agents in one cluster, GD is preferable to GC.
+In our problem setting, the players also consider the ef-
+fects of their actions on the agent states when attack-
+ing/recovering. For example, the attacker may want to
+separate agents having state values with more differences
+in different groups. We specify the agents’ state differ-
+ence zk as
+zk(E
+A
+k , EA
+k , ED
+k ) := xT[k + 1]Lcx[k + 1],
+(6)
+with Lc, for simplicity, being the Laplacian matrix of the
+complete graph with n agents. That is, (6) represents the
+sum of squares of the state differences of all the agent
+pairs. This implies that all state differences between any
+pair of agents are worth the same and thus the players
+do not prioritize any connection between agents.
+The attacked and recovered edges (E
+A
+k , EA
+k , ED
+k ) will af-
+fect x[k + 1] in accordance with (1) and (2), and in turn
+the value of zk. Note that the value of zk is nonincreas-
+ing over time [2] even if some agents are left discon-
+nected from other agents under attacks. This sum-of-
+square characterization of the agents’ state difference is
+commonly used and essentially the same to our previous
+work [19] for the continuous-time setting; here, we ex-
+tend the formulation to comply with the discrete-time
+1
+2
+3
+4
+5
+6
+7
+1
+2
+3
+4
+5
+6
+7
+1
+2
+3
+4
+5
+6
+7
+1
+2
+3
+4
+5
+6
+7
+PSfrag replacements
+(a) GA
+(b) GB
+(c) GC
+(d) GD
+Fig. 3. Graphs and their agent-group indices: (a) c(GA) = 0,
+(b) c(GB) = −12, (c) c(GC) = −22, and (d) c(GD) = −24.
+Note that c(GC) is larger than c(GD), even with more number
+of groups.
+setting by considering the states at one time step ahead
+k + 1.
+Now, we combine the two measures in (5) and (6) to
+construct the utility functions for the game in a zero-
+sum manner. Specifically, for the lth game starting at
+time k = (l−1)T , the attacker and the defender’s utility
+functions take account of the agent-group index c(·) and
+the difference zk of agents’ states over h horizon length
+from time (l − 1)T to (l − 1)T + h − 1. With weights
+a, b ≥ 0, the utilities for the lth game U A
+l for the attacker
+and U D
+l
+for the defender are, respectively, defined by
+U A
+l :=
+(l−1)T +h−1
+�
+k=(l−1)T
+(azk − bc(GD
+k )),
+(7)
+U D
+l := −U A
+l .
+(8)
+In our setting both players attempt to maximize their
+utilities at the start of each game l. The values of a
+and b represent the preference of the players towards
+either a long-term agent clustering or a short-term agent-
+grouping. A higher value of a implies that the players
+prefer to focus on the agent states and the subsequent
+cluster forming, whereas a higher value of b implies that
+they focus on the agent-grouping more. We suppose that
+both players know the underlying topology G as well as
+the states of all agents xi[k].
+4
+Rolling Horizon Game Structure
+We are interested in finding the subgame perfect equi-
+librium [9] of this game outlined in Section 3. To this
+end, the game is divided into some subgames/decision-
+making points. The subgame perfect equilibrium must
+be an equilibrium in every subgame. The optimal strat-
+egy of each player is obtained by using a backward in-
+duction approach, i.e., by finding the equilibrium from
+the smallest subgames. The tie-break condition happens
+when the players’ strategies result in the same utility. In
+this case, we suppose that the players choose to save their
+energy by attacking/recovering less edges unless they
+have enough energy to attack/recover all edges in ev-
+ery subsequent steps, in which case they attack/recover
+more edges.
+Due to the nature of the rolling horizon approach, the
+5
+
+strategies obtained from the lth game, i.e., attacked and
+recovered edges, are applied only from time (l − 1)T to
+lT − 1. Specifically, in the lth game for time (l − 1)T
+to (l − 1)T + h − 1, the strategies of both players
+are denoted by ((E
+A
+l,1, EA
+l,1, ED
+l,1), . . . , (E
+A
+l,h, EA
+l,h, ED
+l,h)),
+with (E
+A
+l,α, EA
+l,α, ED
+l,α) indicating the strategies at the
+αth step of the lth game with α ∈ {1, . . ., h}. Note
+that here we show the strategies with two subscripts
+representing the game and the step indices along
+the time axis. From the above set of strategies, only
+((E
+A
+l,1, EA
+l,1, ED
+l,1), . . . , (E
+A
+l,T , EA
+l,T , ED
+l,T ))
+is
+applied.
+Re-
+call that h is taken to be greater than or equal to
+T . Therefore, for the lth game from time (l − 1)T
+to lT − 1, the strategy applied will be written as
+((E
+A
+(l−1)T , EA
+(l−1)T , ED
+(l−1)T ), . . . , (E
+A
+lT −1, EA
+lT −1, ED
+lT −1)) :=
+((E
+A
+l,1, EA
+l,1, ED
+l,1), . . . , (E
+A
+l,T , EA
+l,T , ED
+l,T )).
+We look at how the optimal edges can be found by an
+example with h = 2 and T = 1 or 2. In this case, for the
+lth game over time (l−1)T and (l−1)T +1, the optimal
+strategies of the players are given by
+ED∗
+l,2 (E
+A
+l,2, EA
+l,2) ∈ arg max
+ED
+l,2
+U D
+l,2,
+(9)
+(E
+A∗
+l,2 (ED
+l,1), EA∗
+l,2 (ED
+l,1)) ∈ arg
+max
+(E
+A
+l,2,EA
+l,2)
+U A
+l,2,
+(10)
+ED∗
+l,1 (E
+A
+l,1, EA
+l,1) ∈ arg max
+ED
+l,1
+U D
+l ,
+(11)
+(E
+A∗
+l,1 , EA∗
+l,1 ) ∈ arg
+max
+(E
+A
+l,1,EA
+l,1)
+U A
+l ,
+(12)
+where U A
+l,α and U D
+l,α are defined as parts of U A
+l
+and
+U D
+l , respectively, calculated from the αth step to the
+last (hth) step of the lth game, i.e., U A
+l,α = −U D
+l,α :=
+�(l−1)T +h−1
+(l−1)T +α−1(azk − bc(GD
+k )). In this case with h = 2,
+the functions U A
+l,2 and U D
+l,2 are based on the values of
+azk and bGD
+k at k = (l − 1)T + 1 only. Note that to
+find (E
+A∗
+l,1 , EA∗
+l,1 ), one needs to obtain ED∗
+l,1 (E
+A
+l,1, EA
+l,1) be-
+forehand. Likewise, to find ED∗
+l,1 , one needs to obtain
+(E
+A∗
+l,2 (ED
+l,1), EA∗
+l,2 (ED
+l,1)). Similarly, to find (E
+A∗
+l,2 , EA∗
+l,2 ), the
+edges ED∗
+l,2 (E
+A
+l,2, EA
+l,2) must be obtained beforehand. Note
+that deriving the optimal strategies above is subject to
+the energy constraints (3) and (4).
+For h > 2, the players’ optimal strategies consist of 2h
+parts similar to those in (9)–(12), with one time step
+consisting of two parts of strategies corresponding to the
+number of players. They are solved by the players at
+every time k = (l − 1)T of the lth game, l ∈ N. With
+T = h, the players do not have chance to override their
+strategies, which removes the rolling horizon aspect of
+the game.
+We will find the optimal strategies of the players by
+computing all possible combinations, since the choices of
+edges are finite. From the optimization problems speci-
+fied above, the players examine at most 3|E|2|E|h number
+of combinations of attacked and recovered edges for util-
+ity evaluations, since they have to foresee the opponent’s
+response as well. Note that the attacker has three pos-
+sible actions on an edge: no attack, attack with normal
+signals, and attack with strong signals, whereas the de-
+fender has only two actions: recover or not recover. Here
+we can see that the number of computation increases ex-
+ponentially with respect to the number of edges in the
+underlying graph. To address scalability issues, we may
+find edges that are easier to attack first, i.e., edges that
+result in the formation of new groups if attacked, and
+limit the strategy choices over those edges only.
+Our previous works [19, 20] considered related games
+in continuous time, where the timings for launching at-
+tack/defense actions are also part of the decision vari-
+ables. This aspect complicated the formulation, making
+it difficult to study games over a time horizon. In this pa-
+per, we simplify the timing issue and instead introduce
+the rolling horizon feature. This enables the players to
+consider the cluster forming in a longer time range, which
+is especially important when consensus among agents is
+obstructed by adversaries.
+With this rolling horizon setting, it is important for a
+player to know what the opponent’s previous action at
+the previous step of the game is in order to know its
+position at the game tree, i.e., which subgame is the
+player’s playing. For example, if the defender does not
+know which edges are previously attacked, then it cannot
+properly calculate the value of the utility function (8).
+5
+Consensus Analysis
+In this section, we examine the effect of the game struc-
+ture and players’ energy constraints on consensus.
+We will begin the analysis by looking at the case of
+certain energy conditions of the players. Specifically, if
+a player has enough energy to attack/recover all edges
+from a certain step of the game, then it will use all of
+their energy to attack/recover as many edges as they
+can in the subsequent steps. We will confirm this point
+formally in the following. For simplicity, we denote the
+total energy that the defender consumed before the lth
+game as ˜βD
+l := �(l−1)T −1
+k=0
+βD|ED
+k | and the total energy
+that the defender may consume from the 1st to the αth
+step of the lth game as ˆβD
+α := �α
+m=1 βD|ED
+l,m|, where
+we omit the index l from the left-hand side, with a
+slight abuse of notation. Similarly, for the attacker we
+denote ˜βA
+l
+:= �(l−1)T −1
+m=0
+(βA|EA
+m|+β
+A|E
+A
+m|) and ˆβA
+α :=
+�α
+m=1(βA|EA
+l,m|+β
+A|E
+A
+l,m|).
+6
+
+We discuss in Lemma 1 (resp., Lemma 2) the optimal
+strategy of the defender (resp., attacker) at the αth step
+of the game given certain energy conditions mentioned
+in Section 2. This characterization of optimal strategy
+of the defender (resp., attacker) will be useful to obtain
+the necessary (resp., sufficient) conditions for consensus
+not to happen.
+5.1
+Necessary Conditions for not Reaching Consensus
+This subsection discusses necessary conditions for the
+agents to be separated into different clusters for infinitely
+long duration without achieving overall consensus. We
+first discuss the defender’s optimal strategy on some
+games with specific conditions in Lemmas 1 and 2. In
+Lemma 1, we state the defender’s optimal strategy at
+any step of the lth game given a certain energy condition.
+Lemma 1 If the defender’s total energy ˜βD
+l + ˆβD
+ˆα−1 con-
+sumed before the ˆαth step of the lth game satisfies
+˜βD
+l + ˆβD
+ˆα−1
+≤ κD + ρD((l − 1)T + ˆα − 1) − (h − ˆα + 1)|E|βD,
+(13)
+then ED∗
+l,α = EA∗
+l,α for all α ≥ ˆα, i.e., the defender will
+recover all normally attacked edges from the ˆαth step.
+PROOF. We first look at the last (hth) step of the lth
+game. Since the game consists of a horizon of h steps, the
+last step of the game corresponds to the last decision-
+making point, in which the players’ strategies cannot in-
+fluence the decision already made in the previous steps of
+the same game. Hence, in the last step of the lth game the
+players do not save their energy by attacking/recovering
+less edges.
+From the defender’s energy constraint (4), it is clear that
+at any time k, the set of edges that the defender recov-
+ers is bounded as |ED
+k |≤
+κD+ρDk−�k−1
+m=0 βD|ED
+m|
+βD
+. Thus, at
+the hth step, recovered edges satisfy |ED
+l,h|≤ |ED′
+l,h| with
+|ED′
+l,h|:= min{⌊
+κD+ρD((l−1)T +h−1)−( ˜βD
+l + ˆβD
+h−1)
+βD
+⌋, |EA∗
+l,h |}.
+Depending on which edges are normally attacked,
+the defender may not recover the maximum num-
+ber |ED′
+l,h| of edges. If the defender’s optimal strat-
+egy given normally attacked edges EA
+l,h is not to re-
+cover |ED′
+l,h| number of edges, i.e., recover less, then
+the defender will be able to obtain more utility
+U D
+l,h(E
+A
+l,h, EA
+l,h, ED
+l,h) > U D
+l,h(E
+A
+l,h, EA
+l,h, ED′
+l,h). However, un-
+der (13) with α = h the defender has sufficiently high en-
+ergy, and thus the utility becomes U D
+l,h(E
+A
+l,h, EA
+l,h, ED
+l,h) >
+U D
+l,h(E
+A
+l,h, EA
+l,h, EA
+l,h) = U D
+l,h(E
+A
+l,h, ∅, ∅). It then follows
+that as long as the defender has enough energy, it will
+recover all optimal edges attacked normally at the hth
+step, i.e., ED∗
+l,h = EA∗
+l,h .
+Next, we investigate the effect of this property on the
+earlier steps of the lth game. Since the defender’s strat-
+egy at the hth step is not affected by its strategy at the
+previous (i.e., (h−1)th) step when κD +ρD((l −1)T +h
+−1) − (˜βD
+l + ˆβD
+h−1) ≥ βD|E|, here the defender does not
+need to recover fewer edges at the (h − 1)th step to save
+energy; this is because it already has enough energy to
+recover EA∗
+l,h at the hth step.
+Now, we derive that if κD + ρD((l − 1)T + h − 2)−
+(˜βD
+l + ˆβD
+h−2) ≥ 2βD|E| at the (h − 1)th step, then the
+defender will also recover ED∗
+l,h−1 = EA∗
+l,h−1. To recover all
+attacked edges at steps α ≥ ˆα, it is then sufficient that
+the defender’s energy satisfies (13) so that κD + ρD((l −
+1)T + α − 1) ≥ ˜βD
+l + ˆβD
+α−1 + βD|E|, i.e., the worst-case
+scenario of the energy constraint (4) when the defender
+recovers all edges, is always satisfied when α ≥ ˆα.
+□
+From the proof above, note that if the defender’s strat-
+egy is not to recover all normally attacked edges given
+even if (13) is satisfied, i.e., EA
+l,α = ˆEA ̸= ED
+l,α, then
+the attacker will not attack ˆEA set of edges in the first
+place. This is because by attacking ˆEA (and consid-
+ering ED
+l,α ̸= ˆEA) the attacker’s utility for step α ≥
+ˆα becomes U A
+l,α(·, ˆEA, ED
+l,α ̸= ˆEA) < U A
+l,α(·, ∅, ∅), since
+U D
+l,α(·, ˆEA, ED
+l,α ̸= ˆEA) > U D
+l,α(·, ∅, ∅) = U D
+l,α(·, ˆEA, ˆEA)
+and U D
+l = −U A
+l .
+We also remark that in order to derive the same optimal
+strategy for the defender the quantity (h − α + 1)|E|
+in the right-hand side of inequality (13) can be relaxed
+to the maximum number of edges that the attacker can
+attack from step ˆα to step h given its energy condition.
+However, this number of edges may change every game,
+making the inequality complicated to express.
+Lemma 2 gives an interval over which, at least once,
+either not attacking with normal signals or recovering
+nonzero edges is optimal.
+Lemma 2 There is at least one occurrence of either
+ED
+k ̸= ∅ or EA
+k = ∅ every ⌈ h|E|βD−ρD
+ρDT
++ 1⌉ time steps.
+PROOF. It follows from Lemma 1 that in a game with
+index l′ where (13) is satisfied for α = 1, the defender
+always recovers edges that are attacked normally in the
+1st step, i.e., ED
+l′,1 ̸= ∅ if EA
+l′,1 ̸= ∅. We then investigate
+in which game inequality (13) is satisfied for α = 1.
+Since the defender gains ρD every time k, if ED
+k = ∅ for
+7
+
+any k ∈ {0, . . ., (l′ − 1)T − 1}, then (13) at the first
+step of the l′th game can be written as κD+ρD(l′−1)T
+βD
+≤
+h|E|. With κD = ρD as a worst-case scenario, the left-
+hand side becomes
+ρD(1+(l′−1)T )
+βD
+, and we then obtain
+l′ ≥ ⌈ h|E|βD−ρD
+ρDT
++ 1⌉.
+Note that the above fact holds when the defender
+does not recover any edge for any k ∈ {(j − 1)(l′ ���
+1)T, . . . , j(l′ − 1)T − 1}, j
+∈
+N. If the defender
+recovers one or more attacked edges at any k
+∈
+{0, . . ., (l′ − 1)T − 1}, then the above result may
+not hold, i.e., the defender may not be able to re-
+cover all EA
+l′ . However, it follows that during time
+k ∈ {(j − 1)(l′ − 1)T, . . . , j(l′ − 1)T − 1}, either 1) the
+defender recovers nonzero edges (ED
+k
+̸= ∅), or 2) the
+attacker attacks no edges with normal signals (EA
+k = ∅)
+at least once.
+□
+Lemmas 1 and 2 above imply that the defender is guar-
+anteed to make recoveries from normal attacks every cer-
+tain interval. Hence, the attacker needs to attack some
+edges strongly to prevent the recovery in order to sepa-
+rate agents into different clusters, as we discuss next.
+The following two results provide necessary conditions
+for consensus not to take place. We consider a more gen-
+eral condition in Proposition 3, whereas in Theorem 4
+we consider a more specific situation for the utility func-
+tions that leads to a tighter condition. Recall that λ rep-
+resents the connectivity of G.
+Proposition 3 A necessary condition for consensus not
+to happen is ⌊ρA/βA⌋ ≥ λ.
+PROOF. In deriving this necessary condition, we sup-
+pose that there is no recovery by the defender at any time
+k. Without any recovery from the defender (ED
+k = ∅),
+the attacker must attack at least λ number of edges with
+normal signals (which take less energy) at any time k to
+make GD
+k disconnected at all times. Otherwise, there will
+be time steps where the graph GD
+k is connected, which
+implies that consensus will still be reached.
+If the attacker attacks λ edges with normal jamming
+signals at all times, the energy constraint (3) becomes
+(βAλ − ρA)k ≤ κA. Thus, the condition ρA/βA ≥ λ has
+to be satisfied to ensure that the attacker can make GD
+k
+disconnected for all k. Note that, if the attacker does not
+have enough energy to disconnect GD
+k given no recovery,
+then it definitely cannot disconnect GD
+k in the face of
+recovery by the defender.
+□
+We now limit the class of utility functions in (7), (8) to
+the case of b = 0 in the weights. This means that the
+players do not take account of the agent-group index in
+the graph, but only the states in consensus. In this case,
+the attacker may need more energy to prevent consensus
+as shown in the next theorem.
+Theorem 4 Suppose that b = 0. A necessary condition
+for consensus not to happen is ρA/β
+A ≥ λ.
+PROOF. We prove by contrapositive; especially, we
+prove that consensus always happens if ρA/β
+A < λ.
+We first suppose that the attacker attempts to attack
+λ edges strongly at all times to disconnect the graph
+GD
+k . From (3), the energy constraint of the attacker at
+time k becomes (β
+Aλ − ρA)k ≤ κA. This inequality is
+not satisfied for sufficiently large k if ρA/β
+A < λ, since
+β
+Aλ − ρA becomes positive and κA is finite. Therefore,
+the attacker cannot attack λ edges strongly at all times
+if ρA/β
+A < λ, and is forced to disconnect the graph by
+attacking with normal jamming signals instead.
+Next, by Lemma 2 above, we show that there exists an
+interval of time where the defender always recovers if
+there are edges attacked normally, i.e., ED
+l′ ̸= ∅ is optimal
+given that EA
+l′ ̸= ∅.
+From the definitions in (7), (8), given that b = 0, we can
+see that the defender obtains a higher utility if the agents
+are closer. This means that given a nonzero number
+of edges to recover (at time jl′T described above), the
+defender recovers the edges connecting further agents.
+Specifically, for some i ∈ N, for interval [jl′T, (j +i)l′T ],
+there is a time step where U D
+l (ED
+k
+= E1) ≥ U D
+l (E2),
+with edges E1 connecting agents with further states than
+agents connected by E2. This fact implies that when re-
+covering, the defender always chooses the further dis-
+connected agents. Since by communicating with the con-
+sensus protocol as in (1) the agents’ states are getting
+closer, the defender will choose different edges to re-
+cover if the states of agents connected by recovered edges
+ED
+k become close enough. Consequently, if ρA/β
+A < λ,
+then there exists i ∈ N where the union of graphs,
+i.e., the graph having the union of the edges of each
+graph (V, �((E \(E
+A
+k ∪EA
+k ))∪ED
+k )) over the time interval
+[j(l′ − 1)T, (j + i)(l′ − 1)T ], becomes a connected graph,
+where l′ = ⌈ h|E|βD−ρD
+ρDT
++ 1⌉ as in Lemma 2 above. These
+intervals [j(l′−1)T, (j+i)(l′−1)T ] occur infinitely many
+times, since the defender’s energy bound keeps increas-
+ing over time.
+It is shown in [31] that with protocol (1), the agents
+achieve consensus in the time-varying graph as long as
+the union of the graphs over bounded time intervals
+is a connected graph. This implies that consensus is
+8
+
+achieved if (V, �((E \(E
+A
+k ∪EA
+k ))∪ED
+k )) is connected over
+[l′
+i, l′
+i + 1, . . . , l′
+i+j]. Thus, if ρA/β
+A < λ then consensus
+is achieved.
+□
+The result in Theorem 4 only holds for b = 0, since with
+b > 0 the defender may choose to recover the edges con-
+necting agents that already have similar states to maxi-
+mize c(GD
+k ) (instead of those connecting further agents).
+In such a case, the network may remain disconnected and
+thus the agents may converge to different states. As we
+see from these results, the weight values affect the nec-
+essary conditions to prevent consensus, whereas the ef-
+fect of the weights on the sufficient condition (discussed
+later) is less straightforward. The effect of the values of
+a and b on consensus is illustrated in Section 8.
+5.2
+Sufficient Condition to Prevent Consensus
+The next result provides a sufficient condition for pre-
+venting consensus. It shows that the attacker can pre-
+vent consensus if it has sufficiently large recharge rate ρA
+given the network topology G. We first state Lemma 5
+about the attacker’s optimal strategy under some energy
+conditions, similar to the discussion on the defender’s
+case above.
+Lemma 5 The attacker’s optimal strategy is E
+A∗
+l,α = E if
+• the attacker’s recharge rate satisfies ρA/β
+A ≥ |E|,
+or
+• the attacker’s total energy ˜βA
+l + ˆβA
+α−1 that it con-
+sumes before αth step of the lth game satisfies
+˜βA
+l + ˆβA
+α−1
+≤ κA + ρA((l − 1)T + α − 1) − (h − α + 1)β
+A|E|.
+(14)
+PROOF. We first observe that in the hth step of the lth
+game the attacker does not save their energy by attack-
+ing fewer edges. Since zl,h(E, ∅, ∅) > zl,h(E
+A
+l,h, EA
+l,h, ED
+l,h)
+and c((V, ∅)) ≥ c((V, (E \ (E
+A
+l,h ∪ EA
+l,h) ∪ ED
+l,h))) are al-
+ways satisfied for any edges E
+A
+l,h, EA
+l,h, ED
+l,h, the function
+U A
+h always has the highest value if the attacker strongly
+attacks all edges E. It then follows that the attacker with
+enough energy, i.e., κA + ρA((l − 1)T + h − 1) − (˜βA
+l +
+ˆβA
+h−1) ≥ β
+A|E| is satisfied, will choose to attack all edges
+with strong signals.
+Similar to the proof in Lemma 1, inequalities zl,α(E, ∅,
+∅) > zl,α(E
+A
+l,α, EA
+l,α, ED
+l,α) and c((V, ∅)) ≥ c((V, (E\(E
+A
+l,α∪
+EA
+l,α) ∪ ED
+l,α))) are always satisfied for any step α. Hence,
+the attacker will choose to attack all edges with strong
+signals in any step α given enough energy. This can be
+achieved if the attacker has high enough stored energy,
+i.e., (14) is satisfied, or if the attacker has high enough
+recharge rate, i.e., ρA ≥ β
+A|E|. These conditions enable
+the attacker to attack all edges strongly while still sat-
+isfying the energy constraint (3) above for all steps.
+□
+Proposition 6 A sufficient condition for all agents not
+to achieve consensus at infinite time is that the attacker’s
+parameters satisfy ρA/β
+A ≥ |E|.
+PROOF. By Lemma 5, the attacker always strongly at-
+tacks all edges with strong signals in a game at any step
+α given either sufficient recharge rate or sufficient stored
+energy at the beginning of the game. Consequently, if
+the attacker’s recharge rate satisfies ρA/β
+A ≥ |E|, the
+attacker will attack E with stronger jamming signals at
+all steps of all games, separating every agent at all times.
+As a result, there are n clusters formed, and hence, ob-
+viously, consensus is not reached.
+□
+Remark 7 Note that the necessary conditions and the
+sufficient condition above consider zk = xTLcx in (6)
+which is a nonincreasing function. It is possible to con-
+sider other Laplacian matrices, e.g., Laplacian of the un-
+derlying graph G, however the function zk may not be non-
+increasing anymore. For example, we consider a simple
+path graph 1-2-3 with initial states x0 = [10, 0, −5]T and
+Laplacian of graph G considered in state difference func-
+tion zk. With weights of the utility functions (7) and (8)
+a = 1 and b = 0 and under consensus protocol (1) and (2)
+with weights a12 = 0.1 and a23 = 0.8, the players’ utili-
+ties in the first game with h = 1 are U A
+1 = −U D
+1 = 148
+without any attacks, and U A
+1 = U A
+0 = −U D
+1 = 125 if both
+edges are attacked. This implies that not attacking any
+edge may actually be optimal for the attacker even with
+large enough energy. As a consequence, with Laplacian
+of graph G considered in state difference function zk, the
+analysis becomes more complicated and some of the the-
+oretical results do not hold anymore, e.g., the sufficient
+condition in Proposition 6.
+5.3
+Example on a Gap Between Necessary Condition
+and Sufficient Condition
+In this subsection we provide an example that illustrates
+the gap between the necessary condition for preventing
+consensus in Theorem 4 and the sufficient condition in
+Proposition 6. Here we suppose that the defender has a
+very high recharge rate (i.e., ρD is much larger than βD)
+such that it can recover any normally-attacked edges at
+any k (note that the condition in Theorem 4 only consists
+of the attacker’s parameters). This will force the attacker
+9
+
+1
+2
+3
+4
+Fig. 4. Graph G used in the case study.
+Table 1
+Agent state difference for various values of ρA/β
+A
+ρA/β
+A
+z20
+Consensus
+1
+0.113
+Yes
+1.1
+0.115
+Yes
+1.2
+1.3405
+No
+1.4
+235.345
+No
+1.8
+706.8
+No
+2
+1354
+No
+to attack with strong jamming signals to disconnect any
+agent.
+We consider a graph G as in Fig. 4, with x[0] =
+[−5, 0, −20, 10], h = 2, and κA = ρA. The weight of the
+utility functions are set to be a = 1 and b = 0. We test
+various values of 1 ≤ ρA/β
+A ≤ 2, implying that the
+attacker can attack one edge with strong signals at all
+time without running out of energy. Thus, the attacker
+needs to attack e12 (min-cut edge of G) at all times in
+order to prevent consensus, since it is the only edge
+which, if attacked, will make the graph disconnected.
+Note that this ratio 1 ≤ ρA/β
+A ≤ 2 satisfies the neces-
+sary condition for preventing consensus in Theorem 4,
+but not the sufficient condition in Proposition 6.
+Specifically in this example we test whether consensus
+is prevented or not for various value of ρA/β
+A based on
+agent states at time k = 20. It is interesting to note
+from Table 1 that even with a relatively small value of
+ρA/β
+A < |E|, consensus can still be prevented by the
+attacker.
+From this example, we observe that there is a gap be-
+tween the necessary condition and the sufficient condi-
+tion. Note that this gap may be larger for a more con-
+nected G as well as for network consisting of more agents,
+where typically |E|>> λ. Later in Section 8, we pro-
+vide more detailed examples which illustrate the effect
+of these parameters’ values on consensus.
+As the last result of the section, we state that for a special
+case with the complete graph under b = 0 and h =
+1, i.e., a single-step game without rolling horizon, the
+condition in Theorem 4 is also sufficient, i.e., there is no
+gap between the necessary condition and the sufficient
+condition.
+Proposition 8 Suppose that b = 0 and h = 1. In the
+complete graph G, a sufficient condition for consensus
+not to happen is ρA/β
+A ≥ n − 1.
+PROOF. With h = 1, the attacker will spend all of its
+energy at the only step of the game. With ρA/β
+A ≥ n−1,
+the attacker is always able to disconnect the complete
+graph G.
+In the complete graph G, every agent is connected to
+all other agents regardless of their states, implying that
+there is no agent that can be prioritized to be isolated by
+the attacker (different from the example above). Then,
+with b = 0, the attacker is ensured to separate the fur-
+thest agent. This implies that, at each game (and at each
+k), the attacker will always attack the same edges, re-
+sulting in disconnected GD
+k at each time.
+□
+We note that in different class of graphs (including in
+other symmetric graphs such as cycle graphs or star
+graphs), it is more challenging to derive a tighter suffi-
+cient condition. This is because agents have direct access
+only to some other agents which makes cluster forming
+based on the agent states more difficult.
+6
+Clustering Analysis
+In this section, we derive some results on the number of
+formed clusters of agents at infinite time. From Propo-
+sition 6, the result implies the simple case where if the
+attacker has enough energy such that ρA/β
+A ≥ |E|, then
+the attacker can attack all the edges of the underlying
+topology G so that the number of clusters is n (i.e., all
+the agents are separated).
+The next result discusses a relation between the at-
+tacker’s cost and energy recharge rate with the maxi-
+mum number of clusters that the attacker may create
+through jamming. In the subsequent results of this sec-
+tion, we suppose that b = 0.
+We first define a vector which characterizes the maxi-
+mum number of clusters of G, given the parameters ρA
+and β
+A. Specifically, we define a vector Θ ∈ R|E| with el-
+ements Θj := max|EA|=j n(V, E \ EA), with n(V, E \ EA)
+being the number of agent groups of (V, E \ EA).
+Proposition 9 An upper bound on the number of
+formed clusters at infinite time is Θ⌊ρA/β
+A⌋.
+PROOF. The vector Θ consists of the maximum num-
+ber of formed groups n(V, E \ EA) given the number of
+10
+
+attacked edges as the element index. Since some edges
+need to be attacked consistently in order to divide the
+agents into different clusters, the number of formed clus-
+ters at infinite time is never more than the maximum
+number of groups at any time k given the same number
+of strongly attacked edges.
+Recall that ⌊ρA/β
+A⌋ is the maximum achievable num-
+ber of edges that can be strongly attacked at all times.
+Given the known graph topology G, we then can imply
+that Θ⌊ρA/β
+A⌋ gives the maximum number of clusters at
+infinite time.
+□
+We continue by addressing a special case where all the
+agents in the network are connected with each other.
+Corollary 10 In the complete graph G, the attacker can-
+not divide the agents into more than
+1 +
+(n−1)
+�
+j=1
+min
+�
+1,
+�
+2ρA
+jβ
+A(2n − j − 1)
+��
+(15)
+number of clusters.
+PROOF. In the complete graph, every agent is con-
+nected to all other n − 1 agents. From Proposition 9, we
+can derive the vector Θ of the complete graph G as
+Θ =[1, . . . , 1, 2, . . ., 2, 3, . . . , n − 1, n]T,
+where the value of the (n−1)th entry is 2, the value of the
+((n−1)+(n−2))th entry is 3, and so on. This is because
+in the complete graph G the attacker needs to attack
+(n−1) number of edges to disconnect the graph, further
+(n − 2) number of edges to make three groups of agents,
+further (n − 3) number of edges to make four groups of
+agents, and so on, until (n − 1) + (n − 2) + · · · + 1 =
+n(n − 1)/2 agents to make n groups. The value of the
+⌊ρA/β
+A⌋th entry of this Θ matrix for the complete graph
+can be written as in (15). This value determines the
+upper bound of the number of clusters.
+□
+In Proposition 9, we use the information of the graph
+structure to obtain the vector Θ. We remark that if the graph
+structure G is not known, then the number of clusters at
+infinite time is in general upper bounded by ⌊ρA/β
+A⌋ + 1.
+This is because the attacker can attack continuously at all
+time at most ⌊ρA/β
+A⌋ number of edges, and in the most
+vulnerable graph with λ = 1, i.e., tree graphs, any attacked
+edge will result in a new group.
+To illustrate the relationship between Θ and ρA/β
+A, we look
+Table 2
+Possible cases of attack and recovery actions
+Case
+c(GA
+l,α)
+c(GD
+l,α)
+1
+c(GA
+l,α) = c(G)
+c(GD
+l,α) = c(GA
+l,α)
+2
+c(GA
+l,α) < c(G)
+c(GD
+l,α) = c(GA
+l,α)
+3
+c(GA
+l,α) < c(G)
+c(GD
+l,α) > c(GA
+l,α)
+7
+Equilibrium Characterization
+In this game the strategy choices are all finite in form of
+edges attacked and recovered. Here, we characterize the
+equilibrium/optimal strategies of the players in certain
+situations for the case where the players’ horizon length
+is 1 so that they myopically update their strategies every
+time step.
+In this section, we state some results when a = 0, i.e.,
+when the players do not consider the agents’ states but
+agent-group index in determining their strategies so that
+the defender (resp., attacker) has higher (resp., lower)
+utility when more agents belong to the same group. Sim-
+ilar to the analysis in [20], here we explore some possible
+optimal strategy candidates for the players in a game.
+However, since a game consists of several steps in this
+formulation, the subgame perfect equilibrium is more in-
+volved to characterize, compared to the case of a game
+consisting of one step as in [20].
+In the αth step of each game, there are three possibilities
+in function c(·) as shown in Table 2 (Cases 1, 2, and 3).
+From this table, we characterize the optimal strategies
+of both players in each case:
+• Case 1: When c(G) = c(GD
+l,α), the attacker’s utility
+in one time step is c(G), which implies that the at-
+tacker should not attack any edge either with nor-
+mal signals or strong signals, with the utilities of
+both players equal to zero. The players’ strategies
+in this case are called Combined Strategy 1.
+• Case 2: When c(GD
+l,α) = c(GA
+l,α), the defender does
+not recover any attacked edge, whereas the attacker
+should attack some edges either with strong or nor-
+mal signals. The players’ strategies in this case are
+classified as Combined Strategy 2.
+• Case 3: Here both players will attack/recover
+nonzero number of edges. In particular, the at-
+tacker will attack with normal signals and poten-
+tially with strong signals. The players’ strategies
+here are called Combined Strategy 3.
+at the graph in Fig. 4 from the last section. Here, Θ =
+[2, 2, 3, 4]T, whereas the values of ⌊ρA/β
+A⌋ + 1 are 2, 3, 4,
+5 for ρA/β
+A = 1, 2, 3, and 4, respectively. Note that for
+any value of ρA/β
+A, inequality Θ⌊ρA/βA⌋ ≤ ⌊ρA/β
+A⌋ + 1 is
+always satisfied, indicating that knowing the graph structure
+helps to better estimate the upper bound of the number of
+clusters.
+11
+
+We will then discuss the equilibrium for this game in
+Proposition 11 below. For simplicity, we only consider
+the case when h = 1. The case of h > 1 can be examined
+based on the characterization here for h = 1.
+Proposition 11 The optimal strategies for the players
+with h = 1 satisfy the following:
+(1) Combined Strategy 1 if ˜βA
+l +βA > κA +ρA(l −1)T ,
+(2) Otherwise,
+(a) Combined Strategy 2 if
+(i) ˜βD
+l + βD > κD + ρD(l − 1)T , or
+(ii) ˜βD
+l
++ βD
+≤
+κD + ρD(l − 1)T
+and
+U A
+l (⌊(κA + ρA(l − 1)T − ˜βA
+l )/β
+A⌋, ∅, ∅) =
+maxE
+A
+k ,EA
+k ,ED
+k U A
+l (E
+A
+k , EA
+k , ED
+k ),
+(b) Combined Strategy 3 if ˜βD
+l
++ βD ≤ κD +
+ρD(l − 1)T and U A
+l (⌊(κA + ρA(l − 1)T −
+˜βA
+l )/β
+A⌋, ∅, ∅) ̸= maxE
+A
+k ,EA
+k ,ED
+k U A
+l (E
+A
+k , EA
+k , ED
+k ).
+PROOF. With a = 0, we observe that the defender al-
+ways recovers from the optimal attack at the last step
+given sufficient energy, which implies that it always re-
+covers for h = 1 if ˜βD
+l + βD ≤ κD + ρD((l − 1)T ) is sat-
+isfied. Similar to the defender, the attacker obtains the
+least utility, i.e., zero, by not attacking for the case of
+h = 1. Therefore, the attacker will attack at least one
+edge as long as it has enough energy to do so. We prove
+each point of the proposition statement as below.
+(1): We now suppose that ˜βA
+l + βA > κA + ρA((l − 1)T )
+(point (1) in the statement) is satisfied, i.e., the attacker
+does not have enough energy to even attack one edge
+normally. In this case, Combined Strategy 1 becomes
+optimal since there is no other choice, i.e., the attacker
+cannot attack even one edge with normal signals. In the
+rest of the proof, we assume that ˜βA
+l +βA ≤ κA+ρA((l−
+1)T ) is satisfied.
+(2a(i)): We now continue by providing the conditions
+for Combined Strategy 2. Similarly to the attacker
+above, we observe that the defender cannot recover any
+edge if ˜βD
+l + βD > κD + ρD((l − 1)T ), implying that
+c(GA
+l,α) < c(G) and c(GD
+l,α) = c(GA
+l,α) (corresponds to
+point (2a(i))).
+(2a(ii)): We then suppose that ˜βD
+l
++ βD ≤ κD +
+ρD((l − 1)T ) is satisfied. It then follows that given
+enough energy for the defender, the attacker needs to
+attack nonzero number of edges with strong signals to
+satisfy c(GA
+l,α) < c(G) and c(GD
+l,α) = c(GA
+l,α). In order for
+Combined Strategy 2 to be optimal, the attacker then
+needs to attack edges strongly without attacking with
+normal signals at all, i.e., EA
+k = ∅. Thus, β
+A needs to be
+sufficiently low to make strong attack feasible. Specif-
+ically, U A
+l (E
+A′
+k , ∅, ∅)
+=
+maxE
+A
+k ,EA
+k ,ED
+k U A
+l (E
+A
+k , EA
+k , ED
+k ),
+with |E
+A′
+k |= ⌊(κA + ρA((l − 1)T ) − ˜βA
+l )/β
+A⌋ indicating
+the maximum number of edges the attacker attacks
+strongly. This corresponds to point (2a(ii)).
+(2b): Consequently, if ˜βD
+l + βD ≤ κD + ρD((l − 1)T )
+and U A
+l (E
+A′
+k , ∅, ∅) ̸= maxE
+A
+k ,EA
+k ,ED
+k U A
+l (E
+A
+k , EA
+k , ED
+k ) are
+true, then the attacker normally attacks nonzero num-
+ber of edges and the defender recovers nonzero number
+of edges, which imply that Combined Strategy 3 is op-
+timal (point 2b).
+□
+Remark 12 The characterization of optimal strategies
+in Proposition 11 also holds for a more general class of
+agent-group indices other than c(G′) defined in (5), as
+long as the utility function structure (7) and (8) does not
+change. Specifically, it holds for those indices that belong
+to the class given by
+C := {˜c : 2V × 2E → R : ˜c((V, E ∪ E′)) ≥ ˜c((V, E)),
+E, E′ ⊆ E}.
+(16)
+The condition ˜c((V, E ∪ E′)) ≥ ˜c((V, E)) implies that not
+attacking results in the maximum value of ˜c(GA
+l,α) of the
+attacker. Similarly, for the defender, this condition im-
+plies that not recovering given the attacks results in the
+minimum value of ˜c(GD
+l,α). This condition is necessary
+for ensuring the equilibrium as in Proposition 11, since it
+guarantees that attacking/recovering nonzero number of
+edges (corresponding to Combined Strategy 3) is always
+optimal for the players as long as they have the energy to
+do so.
+In general, since the cases discussed above are for one
+step only, for longer h > 1 the optimal strategies will
+take form of a set of combined strategies. For exam-
+ple, if h = 3, the sequence of optimal strategies may be
+{Combined Strategy 1, Combined Strategy 2, Combined
+Strategy 2}. On the other hand, for a > 0, the condition
+in Proposition 11 becomes more complicated to charac-
+terize since attacking more edges does not necessarily
+result in the highest possible utility.
+8
+Simulation Results
+8.1
+Consensus and Clustering across Parameters
+Here we show how the consensus varies across different
+weights of the utility functions and the initial states.
+8.1.1
+Varying Weights a and b
+We consider the 4-agents line/path graph 1–2–3–4 with
+initial states x0 = [1, 0.75, 0.75, −1]T. The parameters
+12
+
+0
+5
+10
+15
+20
+25
+30
+Time
+-1
+-0.5
+0
+0.5
+1
+State
+agent 1
+agent 2
+agent 3
+agent 4
+Fig. 5. Agent states with a = 0.1 and b = 0.9
+0
+5
+10
+15
+20
+25
+30
+Time
+-1
+-0.5
+0
+0.5
+1
+State
+agent 1
+agent 2
+agent 3
+agent 4
+Fig. 6. Agent states with a = 0.9 and b = 0.1
+0
+5
+10
+15
+20
+25
+30
+Time
+Edges
+Fig. 7. Attacked and recovered edges with a = 0.1 and b = 0.9
+0
+5
+10
+15
+20
+25
+30
+Time
+Edges
+Fig. 8. Attacked and recovered edges with a = 0.9 and b = 0.1
+are βA = βD = 1, h = β
+A = 2, κA = ρA = 2.6, ρD =
+0.3, and κD = 0.8, which satisfy the necessary condition
+for preventing consensus in Proposition 3, but not the
+sufficient condition in Proposition 6. With b = 1 − a,
+Figs. 5 and 6 show the agent states with small a (at a =
+0.1) and large a (at a = 0.9), respectively. Figs. 7 and 8
+illustrate the status of the edges in GD
+k over discrete time
+k. There, no line in the corresponding edge implies that
+the edge is strongly attacked; likewise, dashed red lines:
+normally attacked, dashed black lines: recovered, and
+solid black lines: not attacked.
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+0
+2
+4
+6
+8
+10
+12
+170
+180
+190
+200
+210
+220
+230
+Fig. 9. Comparison of zk and − � c(GD
+k ) (k = 20) versus a
+1
+2
+3
+4
+6
+5
+7
+8
+9
+10
+Fig. 10. Graph used for simulation in Section 8.1.2
+We observe that for small a, the attacker more often
+divides the agents into more groups, indicated by more
+dashed red lines in Fig. 7. As a result, the attacker fails
+to prevent consensus among the agents (Fig. 5), despite
+the condition in Proposition 3 being satisfied. On the
+other hand, with large a, the attacker is more focused
+to make the difference among agents’ states larger while
+separating the agents into fewer groups compared to the
+case with small a. These features can be seen in Fig. 8,
+where there are no black lines in the edge e34, and thus
+no consensus among the agents in Fig. 6.
+We next present a comparison in the optimal state dif-
+ference zk(E
+A∗
+k , EA∗
+k , ED∗
+k ) and agent-group index c(GD
+k )
+across different a and b = 1 − a in Fig. 9. We observe
+that with larger a, the attacker successfully prevents con-
+sensus among agents (shown with larger value of zk) at
+time k = 20. On the other hand, with smaller a (corre-
+sponding to larger b), the attacker obtains higher c(GD
+k )
+at the cost of low zk, implying that the attacker fails to
+prevent consensus. It is interesting that the values of zk
+and � c(GD
+k ) remain almost constant for some different
+a, implying that there is a critical value of weights a and
+b that determine the consensus and the number of clus-
+ters at infinite time; in this case, the critical value of a
+is located in 0.4 < a < 0.5.
+8.1.2
+Varying Initial States x0
+We also observe how the initial states x0 affect the agent-
+group index of the agents. We consider the graph shown
+in Fig. 10, which consists of 10 agents. All parameters
+other than the initial states are set to be the same and
+satisfy the conditions in Proposition 3. Specifically, we
+set βA = βD = 1, β
+A = 2, κA = ρA = 2.1, κD = ρD =
+0.7, and a = 1 − b = 0.9. The state trajectories of the
+agents with varying x0 are shown in Figs. 11–13. Here
+we consider three cases of initial states x0:
+13
+
+(1) x0 = [1, 0.9, 0.8, 0.4, 0.44, 0.35, 0.48, 0.2, 0.19, 0.28]T,
+(2) x0 = [1, 0.9, 0.8, 0.4, 0.44, 0.35, 0.48, −0.5, −0.1, −0.2]T,
+(3) x0 = [0.6, 0.5, 0.8, 0.4, 0.44, 0.35, 0.48, 0.58, 0.8, 0.75]T.
+Note that in Case (1), agents 1–3 have closer initial states
+and are far from the other agents. Similarly, in Case (2),
+agents 8–10 have initial states that are different from
+the other agents. However, in Case (3), agent states are
+distributed approximately evenly in the range [0.35, 0.8]
+so that it is hard for the attacker to divide them into
+clusters.
+From Fig. 11, we can see that in Case (1), agents 1–3,
+which have weak connection to other agents (only con-
+nected by one edge), are grouped together and converge
+to the same state. This occurs by attacking the edge con-
+necting agents 3 and 5. On the other hand, in Fig. 12
+for Case (2), agents 8–10 are separated from the others
+because the edge connecting agents 5 and 8 is attacked
+continuously. Clearly, in Cases (1) and (2) it is easier for
+the attacker to separate agents since their initial states
+form clusters matching the network topology.
+In Case (3), however, the initial state values do not ex-
+hibit such properties and as a result, the states converge
+towards the same value as shown in Fig. 13. In this sim-
+ulation, the attacker is not able to effectively attack cer-
+tain edges at all times; as a consequence, the agents are
+not divided into clusters and thus consensus happens.
+The attacker may be able to prevent consensus with
+higher weight a, as discussed in Section 8.1.1 above.
+For obtaining Figs. 11–13, we solve combinatorial opti-
+mization problems to find optimal strategies of the play-
+ers. We remark that the computational complexity of
+this problem depends on the number of edges E of G. We
+have reduced the complexity by disregarding some com-
+binations of edges that are clearly not optimal; for ex-
+ample, attacking only the edge connecting agents 4 and
+7 does not disconnect the graph, and thus cannot be the
+best move for the attacker.
+8.1.3
+Varying Energy and Cost Parameters
+We continue by discussing the effect of the attacker’s
+recharge rate ρA and unit costs of attacks βA and β
+A
+on the consensus and cluster forming. Recall that in the
+theoretical results in Sections 5 and 6, the ratios of ρA
+to β
+A and ρA to βA are used to derive the necessary con-
+ditions and sufficient conditions for preventing consen-
+sus as well as the upper bound of the number of clusters
+formed at infinite time.
+Assuming that b = 0, the number of clusters is dictated
+by ρA/β
+A as discussed in Proposition 9. We show the
+number of clusters over different topologies of the un-
+derlying graph G in Fig. 14. We consider networks with
+0
+5
+10
+15
+20
+25
+30
+Time
+0
+0.2
+0.4
+0.6
+0.8
+1
+State
+Fig. 11. Agent states in Case 1
+0
+5
+10
+15
+20
+25
+30
+Time
+-0.5
+0
+0.5
+1
+State
+Fig. 12. Agent states in Case 2
+0
+5
+10
+15
+20
+25
+30
+Time
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+State
+Fig. 13. Agent states in Case 3
+n = 5, with the edges positioned to yield the most con-
+nected topology, i.e., maximum λ, given the same num-
+ber of edges |E|. Note that, with n = 5, there are at
+most n(n−1)/2 = 10 number of edges in the underlying
+graph G (which happens for the complete graph G). We
+observe that with ρA/β
+A ≥ |E|, the agents are divided
+into 5 clusters (all agents are separated) as shown in the
+upper left area of the figure indicated by “5” as derived
+in Proposition 6 whereas in the lower right area indi-
+cated by “1” the agents converge to the same cluster. It
+is clear that in a more connected graph, the agents are
+more likely to converge to a fewer number of clusters.
+8.2
+Players’ Performance Under Varying Horizon
+Length and Game Period
+In this subsection, we evaluate the players’ performance
+under varying horizon length h and game period T . To
+evaluate the performance of the players, we introduce
+the applied utilities ˆU A
+k := azk(E
+A∗
+k , EA∗
+k , ED∗
+k )−bc(GD∗
+k )
+and
+ˆU D
+k
+:=
+−azk(E
+A∗
+k , EA∗
+k , ED∗
+k ) + bc(GD∗
+k ), with
+GD∗
+k
+= (V, ((E \ (E
+A∗
+k
+∪ EA∗
+k )) ∪ ED∗
+k ). These are el-
+14
+
+PSfrag replacements
+|E|
+ρA
+β
+A
+Fig. 14. Number of clusters at k = 50 with b = 0. The un-
+derlying graphs used are those with 5 agents with maximum
+10 edges.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Time
+0
+5
+10
+15
+20
+25
+30
+Fig. 15. �
+k ˆU A
+k in the path graph (solid lines) and the com-
+plete graph (dashed lines) for varying value of h. The ap-
+plied utility for h = 2 and h = 3 in the path graph is almost
+identical.
+ements of utility functions U A
+l
+and U D
+l
+correspond-
+ing to the αth step, α
+=
+k mod T + 1, of the
+game with index l = ⌊k/T ⌋ + 1, where the obtained
+strategies (E
+A∗
+(l−1)T +α−1, EA∗
+(l−1)T +α−1, ED∗
+(l−1)T +α−1)
+=
+(E
+A∗
+l,α, EA∗
+l,α, ED∗
+l,α) are applied. Since U A
+l
+= −U D
+l , having
+higher applied utility for the attacker implies lower ap-
+plied utility for the defender. Note that the values of h
+and T are uniform among the players.
+In this subsection, we consider the weight aij = ˆa, ˆa <
+1/n in (2) which implies that different agents have dif-
+ferent convergence speeds depending on the number of
+their neighbors. Furthermore, we consider various ini-
+tial states x0 for the agents in order to more accurately
+evaluate the attacker’s performance and the pattern of
+applied utilities ˆU A
+k . We use up to 1000 randomly gener-
+ated initial states in this simulation for each agent rang-
+ing from −1 to 1. Throughout this subsection, we use
+parameters n = 3, ρA = 1.1, κA = 7, β
+A = 2βA = 1.
+8.2.1
+Players’ Performance Under Varying Horizon
+Length
+We now consider the case of varying value of horizon
+length h when the network is a path graph and a com-
+plete graph. Note that the value of h is still uniform
+among the attacker and the defender. The evolutions of
+the attacker’s applied utility ˆU A
+k with varying h (with
+T = 1 for every h) are shown in Fig. 15.
+Table 3
+Difference in the optimal actions and the resulting utilities
+in the path graph G between h = 2 and h = 3
+Initial states
+|E
+A∗
+0 |
+�19
+k=0 ˆU A
+k
+h = 2
+h = 3
+h = 2
+h = 3
+[0.824, −0.798, −0.413]T
+2
+2
+37.74
+[−0.983, 0.649, 0.535]T
+2
+2
+39.89
+[−0.787, −0.786, −0.265]T
+2
+1
+28.41
+30.00
+[0.624, 0.629, −0.821]T
+2
+1
+37.92
+43.45
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Time
+0
+5
+10
+15
+20
+25
+30
+Fig. 16. �
+k ˆU A
+k in the path graph (solid lines) and the com-
+plete graph (dashed lines) for varying T . The applied utility
+for T = 1 and T = 2 in the path graph is almost identical.
+Since the path graph is the least connected graph, the
+attacker will be able to make multiple groups of agents
+relatively easily compared to more connected graphs. As
+a result, the attacker may not need to have a very long
+horizon length h to improve its utility since it does not
+need to save energy as much compared to the case of
+the complete graph. This is shown with the overlapping
+red and yellow solid lines in the Fig. 15, implying that
+the horizon length h = 3 is already as good as the case
+of h = 2. On the other hand, the blue solid line is far
+below the red and the yellow ones, implying that having
+h being too short can result in a worse utility for the
+attacker over time.
+The differences of the attacker’s strategies for some no-
+table cases in the path graph G between h = 2 and
+h = 3 are shown in Table 3. Here, we see the difference
+in the optimal actions between the attacker with h = 2
+and h = 3 in the path graph G even though the plots
+of applied utilities in Fig. 15 are very similar. We ob-
+serve that when the initial states of some agents are suf-
+ficiently close, the attacker with h = 2 keeps attacking
+both edges at k = 0, whereas the attacker with h = 3
+chooses to save its energy by attacking fewer edges. At
+k = 19 the attacker with h = 3 obtains higher applied
+utility, indicating that it is able to better use its energy
+than the attacker with h = 2 by attacking later.
+On the other hand, since the complete graph is the most
+connected graph, here the attacker will need more energy
+to disconnect the graph and obtain some utility. Conse-
+quently, even with longer h, the difference of � ˆU A
+k is
+smaller compared to the path graph case. The difference
+15
+
+5
+5
+5
+5
+5
+4
+5
+4
+3
+4
+3
+310
+5
+5
+5
+5
+5
+5
+5
+9
+5
+5
+5
+5
+5
+5
+5
+8
+5
+5
+5
+5
+5
+5
+5
+7
+5
+5
+5
+5
+5
+5
+53
+3
+2
+3
+2
+2
+2
+2
+2
+2
+2
+1
+1
+1
+1
+1
+1
+1
+8
+6
+109
+5
+5
+5
+5
+5
+5
+4
+a
+5
+5
+5
+5
+5
+5
+4
+3
+4
+5
+5
+5
+5
+4
+3
+3
+3
+5
+5
+5
+4
+3
+3
+2
+2
+5
+5
+4
+3
+2
+2
+2
+1
+5
+4
+3
+2
+1
+1
+1
+1
+2
+3
+4
+5
+9
+7
+ed6between the red and the yellow dashed lines is clearer
+however, suggesting that the attacker still benefits by
+having h = 3 (compared to the very little difference in
+the path graph case). The attacker’s different behavior
+for the path graph and the complete graph G suggests
+that in a less connected graph, the effectiveness of longer
+h may saturate from a lower value compared to the one
+in a more connected graph G, given the attacker’s energy
+parameters.
+In general, we observe that having a longer h may re-
+sult in a better applied utility for the attacker over time
+due to its role as a leader of the game, i.e., the attacker
+moves first and is able to choose its strategy that min-
+imizes the defender’s best response. Additionally, there
+is also a clear pattern on when � ˆU A
+k increases; this im-
+plies that the variation of initial states may not affect
+the attacker’s optimal strategy, except in some cases as
+explained above.
+We also remark that the effect of different values of h is
+also influenced by the underlying graph G. Specifically,
+in a less connected graph G, having a very short horizon
+may even be more harmful compared to the case with a
+more connected G. For example, in Fig. 16, the difference
+of � ˆU A
+k in the path graph between h = 1 and h = 2 is
+much more apparent than in the complete graph. The
+possible reason is that in the path graph, it is easier for
+the attacker to disconnect all agents and make n groups
+at some time steps. Thus, with large enough h, the at-
+tacker can save enough energy to make n groups more
+often. On the other hand, we also observe that increasing
+horizon length from h = 2 to h = 3 has minimal effect
+on the attacker’s utility for the path graph, indicating
+that increasing horizon length past a certain value may
+not be beneficial anymore. As we see later, the similar
+phenomenon also happens for varying values of T .
+8.2.2
+Players’ Performance Under Varying Game Pe-
+riod
+We then continue by simulating the case of varying value
+of game period T (value of h is set to be h = 3 for both
+players so that the assumption T ≤ h is always satisfied).
+The average value of � ˆU A
+k over time is shown in Fig. 16,
+where in general, the attacker with shorter game period
+T has higher applied utility especially at later time for
+both the path graph and the complete graph G.
+The attacker with shorter T will be more adaptive to
+the changes of the agents’ and players’ conditions. In the
+context of this game, the attacker with shorter T may
+delay the attack further to maximize its utility later.
+This in turn increases the attacker’s utility at later time,
+similar to the case of longer h discussed above. Note that
+the yellow dashed and solid lines are the same as the
+yellow lines in Fig. 15, and we observe that the green
+and the purple lines do not differ as much as the red and
+Table 4
+Average total number of edges attacked in the path graph G
+h
+T
+�k
+m=0|E A∗
+m | (Normal)
+�k
+m=0|E
+A∗
+m | (Strong)
+k = 9
+k = 19
+k = 9
+k = 19
+1
+1
+7
+16
+5
+6
+2
+0
+0
+8
+13.959
+3
+0
+0
+7.993
+13.971
+2
+0
+0
+8
+13.970
+3
+2.970
+4.970
+7.003
+11.015
+the blue lines in Fig. 15, indicating that for the attacker,
+having a large value of T may not be as disadvantageous
+as having short h.
+Table 4 shows the average number of edges attacked by
+normal and strong jamming signals given different values
+of h and T . It is interesting to note that for h > T ,
+the attacker never attacks any edge with normal signals,
+indicating that it prefers to save its energy to use it later
+for more powerful attacks. Consequently, the number of
+edges attacked strongly with h > T becomes more than
+those in the case of h = T , which results in the larger
+applied utilities as described above. We can also observe
+that in the case of h = 3 and T = 1, the attacker is
+able to strongly attack more edges than the other cases
+in Table 4 in average at k = 19, even though at k = 9
+it attacks slightly fewer edges than the case of closer
+values of h and T . This suggests that the attacker tends
+to save its energy more in the case of larger value of h
+and smaller T .
+9
+Conclusion
+We have formulated a two-player game in a cluster form-
+ing of resilient multiagent systems played over time. The
+players consider the impact of their actions on future
+communication topology and agent states, and adjust
+their strategies according to a rolling horizon approach.
+Necessary conditions and sufficient conditions for form-
+ing clusters among agents have been derived. We have
+discussed the effect of the weights of the utility func-
+tions and different initial states on cluster forming, and
+evaluated the effects of varying horizon length and game
+period on the players’ performance.
+Possible future extensions include the case where the
+players’ utility functions are not zero-sum, the case
+where the players do not have perfect knowledge, and
+the setting where each agent is capable to decide its own
+strategies in a decentralized way. We have also consid-
+ered in [22] the case where the players’ horizon lengths
+and game periods are not uniform. This case can be fur-
+ther generalized to decentralized settings where agents
+decide their own strategies in an asynchronous way.
+16
+
+Furthermore, it is also interesting to consider a case
+where the players may not have a complete knowledge of
+the other players. This incomplete version of the game
+is considered in [23].
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+17
+

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@@ -0,0 +1,4133 @@
+The non-intrusive reduced basis two-grid method applied to
+sensitivity analysis
+January 3, 2023
+Elise Grosjean 1, Bernd Simeon 1
+Abstract
+This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity anal-
+ysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these
+two approaches may become too costly.
+To reduce computational times, Proper Orthogonal Decomposition
+(POD) and Reduced Basis Methods (RBMs) have already been investigated. The majority of these algorithms
+are however intrusive in the sense that the High-Fidelity (HF) code must be modified. To address this issue,
+non-intrusive strategies are employed. The NIRB two-grid method uses the HF code solely as a “black-box”,
+requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline
+stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive
+than an HF evaluation.
+In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods.
+On a classical model problem, the heat equation, we prove that HF evaluations of sensitivities reach an optimal
+convergence rate in L∞(0, T; H1(Ω)), and then establish that these rates are recovered by the proposed NIRB
+approximations. These results are supported by numerical simulations. We then numerically demonstrate that
+a further deterministic post-treatment can be applied to the direct method. This further reduces computational
+costs of the online step while only computing a coarse solution of the initial problem. All numerical results are
+run with the model problem as well as a more complex problem, namely the Brusselator system.
+1
+Introduction.
+Sensitivity analysis is a critical step in optimizing the parameters of a parametric model. The goal is to see how
+sensitive its results are to small changes of its input parameters. It is especially useful in the biomedical field
+when experiments are extremely complex or prohibitively expensive. Indeed, conducting several experiments to
+determine the impact of all parameters involved in biological processes may be difficult, if not impossible.
+Several methods have been developed for computing sensitivities, see [4] for an overview. We focus here
+on two differential-based sensitivity analysis approaches in connection with models given as reaction-diffusion
+equations.
+• “The direct method”, also known as the ”forward method”, which may be used when dealing with dis-
+cretized solutions of parametric Partial Differential Equations (PDEs). The sensitivities (of the solution or
+other outputs of interest) are computed directly from the original problem. One drawback is that it necessi-
+tates solving a new system for each parameter of interest, i.e., for P parameters of interest, P + 1 problems
+have to be solved.
+• “The adjoint state method”, also known as the ”backward method”. It may be a viable option [44] when the
+direct method becomes prohibitively expensive. In this setting, the goal is to compute the sensitivities of an
+objective function that one aims at minimizing. The associated Lagrangian is formulated, and by choosing
+appropriate multipliers, a new system known as ”the adjoint” is derived. This approach is preferred in
+many situations since it avoids calculating the sensitivities with respect to the solutions. For example,
+in the framework of inverse problems, one can determine the ”true” parameter from several measures
+1Felix-Klein-Institut f¨ur Mathematik, Kaiserslautern TU, 67657, Deutschland
+1
+arXiv:2301.00761v1  [math.NA]  2 Jan 2023
+
+(which are frequently provided by multiple sensors) while combining it with a gradient-type optimization
+algorithm. As a result, we get the ”integrated effects” on the outputs over a time interval. The advantage is
+that it only requires two systems to solve regardless of the number of parameters of interest.
+Thus, the direct method is appealing when there are relatively few parameters or a large number of objective
+functions, whereas the adjoint state method is preferred when there are many parameters and few objective
+functions.
+Earlier works.
+For extremely complex simulations, both methods may still be impractical. Several reduction
+techniques have thus been investigated in order to reduce the complexity of the sensitivity computation. Among
+them, Reduced Basis Methods (RBMs) are a well-developed field [36, 40, 3]. They use an offline-online decompo-
+sition, in which the offline step is time-consuming but is only performed once, and the online step is significantly
+less expensive than a High-Fidelity (HF) evaluation. In the context of sensitivity analysis, the majority of these
+studies rely on a Galerkin projection onto the adjoint state system in the online part. In what follows, we present
+a brief review of previous works on RBMs combined with both sensitivity methods.
+• Let us begin with the direct method. It has been employed and studied with RB spaces in various applica-
+tions, e. g., [39, 47, 13]). The sensitivities may also be useful to enhance the reduced state approximation.
+Unlike the other studies cited below, the sensitivities in [25] are computed to improve RB methods (see
+also [24] with a Lagrangian formulation or [23] with a finite difference approach [23]). Still to improve an
+approximation, in [31], a combined method is proposed (based on local and global approximations with
+series expansion and a RB expression), which was first developed in [30].
+Note also that variance-based sensitivity analysis has been investigated using RBMs [28] and non-intrusive
+RB [34].
+• The adjoint state formulation can be thought of as a PDE-constrained optimization. The first applications
+of this method in conjunction with computational reduction approaches can be found in [27] in the context
+of RBMs, where several RB sub-spaces are compared or in [42] with the POD method, with an affine
+parameter dependence. Currently, particular emphasis is being placed on developing accurate a-posteriori
+error estimates in order to improve basis generation [41, 46, 11, 12] with Proper Orthogonal Decomposition
+(POD) and/or RBMs.
+RBMs and POD have also been investigated in the context of optimal control under uncertainty [9]. In
+recent studies, the case of infinite-dimensional control function is considered with RB approximations on
+the state, adjoint, and control variables [29, 1]. Even if the adjoint state method is frequently preferred,
+writing its associated reduced problem can be difficult when the adjoint formulation is not straightforward.
+It may also be reformulated to take advantage of previously developed RB theory. For example, in [38], it
+is rewritten as a saddle-point problem for Stokes-type problems.
+To conclude this brief overview of RBMs applied to sensitivity analysis, we add that non-intrusive methods have
+been developed, in the framework of the inverse problem, without computing the sensitivities (see the PBDW
+method [37, 22, 10] with a direct formulation).
+Motivation.
+Even though the Galerkin projection is prevalent in the literature, its main disadvantage lies in its
+intrusiveness. Indeed, in order to approximate the solution of a PDE, the matrices computed from its variational
+formulation must be changed in the HF code. This may be difficult if the HF is very complex or even impossible
+if it has been purchased, as is often the case in an industrial context. From an engineering standpoint, Non-
+Intrusive Reduced Basis (NIRB) methods are more practical to implement than intrusive RBMs because they only
+require the execution of the HF code as a ”black-box” solver. Apparently, NIRB methods have not yet been used
+to approximate sensitivities except for statistical approaches such as variance-based sensitivity analysis.
+In this paper, we aim at computing the sensitivities with respect to some parameters of interest µ ∈ G,
+with the direct and adjoint methods combined with NIRB techniques. We focus on the NIRB two-grid method
+[7, 17, 8, 43] (see also different NIRB methods [6, 2, 15] from the two-grid method). Like most RBMs, the NIRB
+two-grid method relies on the assumption that the manifold of all solutions S = {u(µ), µ ∈ G} has a small
+Kolmogorov width [33] (in what follows, uh(µ) will refer to the HF solution for the parameter µ).
+The two-grid algorithm can be employed for a variety of PDEs and is simple to implement. It has been
+studied with FEM in the context of elliptic equations [7] and parabolic equations [19] (see also [17] for finite
+volume schemes). Furthermore, because it is non-intrusive, it is suitable for a wide range of problems. The
+2
+
+effectiveness of this method relies on its offline/online decomposition (as most RBMs). The offline part is time-
+consuming but it is only performed once. On the contrary, the specific feature of the NIRB approach is to solve
+the parametric problem on a coarse mesh only during the online step, and then to rapidly improve the precision
+of the coarse solution. It makes this portion of the algorithm much cheaper than a HF evaluation.
+In this paper, we combine the two-grids framework with both sensitivity analysis methods. Then, drawing in-
+spiration from recent works [18], we efficiently apply a deterministic process to further reduce the computational
+cost of its online stage with the direct method, in the context of parabolic equations. During the online stage,
+this additional step allows us to solve only the initial problem on the coarse mesh, regardless of the number of
+parameters of interest, making this novel approach very appealing. We highlight the fact that because the direct
+approach requires a new system to be solved for each parameter, the adjoint method is preferred in many studies
+(as cited above), despite the fact that its formulation is more complex and yields integrated sensitivities over
+time.
+Outline of the paper.
+This article is about extending the NIRB two-grid method to the computation of sensitiv-
+ities and performing the associated numerical analysis. We present and illustrate the NIRB algorithms applied
+to both sensitivity analysis methods with several numerical results. With the direct method, we have carried out
+a thorough theoretical analysis of the heat equation as model problem. In this setting, we have optimal conver-
+gence rates in L∞(0, T; H1(Ω)) for the spatial HF semi-discretized sensitivity solution and for its fully-discretized
+form. It turns out that we obtain theoretically and numerically these optimal rates also for the NIRB sensitivity
+approximations. Our main theoretical result is given by Theorem 4.1.
+The rest paper is organized as follows. Section 2 describes both sensitivity methods along with established
+convergence results and the NIRB two-grid algorithm for parabolic equations. In Section 3, we present the al-
+gorithms for the direct and adjoint methods with the NIRB two-grid approach, as well as the new version of
+the algorithm for the direct method. Section 4 is devoted to the theoretical results on the rate of convergence
+for the NIRB sensitivity approximation. In the last section 5, several numerical results are presented and illus-
+trate the theoretical ones. The implementation and the use of Automatic Differentiation (AD) is discussed as well.
+2
+Mathematical Background.
+Let Ω be a bounded domain in Rd, with d ≤ 3 and a smooth enough boundary ∂Ω, and consider a parametric
+problem P on Ω. For the NIRB two-grid method, we consider two spatial ”grids” of Ω:
+• one fine mesh, denoted Th, where its size h is defined as
+h = max
+K∈Mh
+hK,
+(1)
+• and on coarse mesh, denoted TH, with its size defined as
+H = max
+K∈MH
+HK >> h,
+(2)
+where the diameter hK (or HK) of any element K in a mesh is equal to sup
+x,y∈K
+|x − y|.
+In this section, we first introduce our model problem, that of the heat equation, in a continuous setting, and then
+its spatial (over the two meshes) and time discretizations. Then, we recall the NIRB algorithm in the context of
+parabolic equations, and finally, we detail the sensitivity problems for this model problem.
+In the next sections, C will denote various positive constants independent of the size of the meshes h and
+H and of the parameter µ, and C(µ) will denote constants independent of the sizes of the meshes h and H but
+dependent of µ.
+2.1
+A model problem: The heat equation.
+2.1.1
+The continuous problem.
+We consider the following heat equation on the domain Ω with homogeneous Dirichlet conditions, which takes
+the form
+3
+
+�
+�
+�
+�
+�
+ut − ∇ · (A(µ)∇u) = f, in Ω×]0, T],
+u(·, 0) = u0(·), in Ω,
+(3)
+u(·, t) = 0, on ∂Ω×]0, T],
+where
+f ∈ L2(Ω × [0, T]), while u0 ∈ H1
+0(Ω) and µ = (µ1, · · · , µP) ∈ G ⊂ RP is the parameter, such that
+A : Ω × G → Md(R) is measurable, bounded, and uniformly elliptic.
+(4)
+For any t > 0, the solution u(·, t) ∈ H1
+0(Ω), and ut(·, t) ∈ L2(Ω) stands for the derivative of u with respect to
+time.
+We use the conventional notations for space-time dependent Sobolev spaces [35]
+Lp(0, T; V) := {u(x, t) | ∥u∥Lp(0,T;V) :=
+� � T
+0
+��u(·, t)
+��p
+V dt
+�1/p
+< ∞}, 1 ≤ p < ∞,
+L∞(0, T; V) := {u(x, t) | ∥u∥L∞(0,T;V) := ess sup
+0≤t≤T
+��u(·, t)
+��
+V < ∞},
+where V is a real Banach space with norm∥·∥V . The variational form of (3) is given by:
+�
+�
+�
+�
+�
+�
+�
+Find u ∈ L2(0, T; H1
+0(Ω)) with ut ∈ L2(0, T; H−1(Ω)) such that
+(ut(t, ·), v) + a(u(t, ·), v; µ) = ( f (t, ·), v), ∀v ∈ H1
+0(Ω) and t ∈ (0, T),
+(5)
+u(·, 0) = u0(·), in Ω,
+where a is given by
+a(w, v; µ) =
+�
+Ω A(x; µ)∇w(x) · ∇v(x) dx,
+∀w, v ∈ H1
+0(Ω).
+(6)
+We remind that (5) is well posed (see [14] for the existence and the uniqueness of solutions to problem (5)) and
+we refer to the notations of [14]. Note that we will use the notation (·, ·) to denote the classical L2-inner product
+on Ω.
+2.1.2
+The various discretizations.
+For the NIRB algorithm, we use the two spatial grids on the variational formulation (5) of our problem (3). We
+employed P1 finite elements to discretize in space. Thus, we introduce Vh and VH, the continuous piecewise
+linear finite element functions (on fine and coarse meshes, respectively) that vanish on the boundary ∂Ω. We
+consider the so-called Ritz projection operator P1
+h : H1
+0(Ω) → Vh (P1
+H on VH is defined similarly) which is given
+by
+(∇P1
+hu, ∇v) = (∇u, ∇v),
+∀v ∈ Vh, for u ∈ H1
+0(Ω).
+(7)
+In the context of time-dependent problems, a time stepping method of finite difference type is used to get a fully
+discrete approximation of the solution of (3). As for the spatial domain, we consider two different time grids:
+• One time grid, denoted F, is associated to fine solutions (for the generation of the snapshots). To avoid
+making notations more cumbersome, we will consider a uniform time step ∆tF. The time levels can be
+written tn = n ∆tF, where n ∈ N∗.
+• Another time grid, denoted G, is used for coarse solutions. By analogy with the fine grid, we consider a
+uniform grid with time step ∆tG. Now, the time levels are written �tm = m ∆tG, where m ∈ N∗.
+As in the elliptic context [7], the NIRB algorithm is designed to recover the optimal estimate in space. Yet,
+since there is no such argument as the Aubin-Nitsche argument for time stepping methods, we must consider
+time discretizations that provide the same precision with larger time steps. Thus, we consider a higher order
+time scheme for the coarse solution. As in [19], we used an Euler scheme (first order approximation) for the
+fine solution and a Crank-Nicolson scheme (second order approximation) for the coarse solution on our model
+problem.
+Thus, we deal with two kind of notations for the discretized solutions:
+4
+
+• uh(x, t) and uH(x, t) that respectively denote the fine and coarse solutions of the spatially semi-discrete
+solution, at time t ≥ 0.
+• un
+h(x) and um
+H(x) that respectively denote the fine and coarse full-discretized solutions at time tn = n × ∆tF
+and �tm = m × ∆tG.
+Remark 2.1. To simplify the notations, we consider that both time grids end at time T here,
+T = NT ∆tF = MT ∆tG.
+The semi-discrete form of the variational problem (5) writes for the fine mesh (similarly for the coarse mesh):
+�
+�
+�
+�
+�
+�
+�
+Find uh(t) = uh(·, t) ∈ Vh for t ∈ [0, T] such that
+(uh,t(t), vh) + a(uh(t), vh; µ) = ( f (t), vh), ∀vh ∈ Vh and t ∈]0, T],
+(8)
+uh(·, 0) = u0
+h(·) = P1
+h(u0)(·).
+From the definition of P1
+h (7), the initial condition u0
+h (and similarly for the coarse mesh) is such that
+(∇u0
+h, ∇vh) = (∇u0, ∇vh), ∀vh ∈ Vh,
+(9)
+and hence, it corresponds to the finite element solution of the corresponding elliptic problem of (3) with A(1) = Id
+(that of the Poisson’s equation) and whose exact solution is u0.
+The full discrete form of the variational problem (5) for the fine mesh with an implicit Euler scheme writes:
+�
+�
+�
+�
+�
+�
+�
+Find un
+h ∈ Vh for n = 0, . . . , NT such that
+(∂un
+h, vh) + a(un
+h, vh; µ) = ( f (tn), vh), ∀vh ∈ Vh and n = 1, . . . , NT,
+(10)
+uh(·, 0) = u0
+h(·),
+where the time derivative in the variational form of the problem (8) has been replaced by a backward difference
+quotient, ∂un
+h =
+un
+h−un−1
+h
+∆tF
+.
+For the coarse mesh with a Crank-Nicolson scheme, and with the notation ∂um
+H = um
+H−um−1
+H
+∆tG
+, it becomes:
+�
+�
+�
+�
+�
+�
+�
+Find um
+H ∈ VH for m = 0, . . . , MT, such that
+(∂um
+H, vH) + a( um
+H+um−1
+H
+2
+, vH; µ) = ( f (�tm− 1
+2 ), vH), ∀vH ∈ VH and m = 1, . . . MT,
+uH(·, 0) = u0
+H(·),
+(11)
+where �tm− 1
+2 = �tm+�tm−1
+2
+.
+For the NIRB approximation, we will need to interpolate in space and in time the coarse solution. So let us
+introduce the quadratic interpolation in time of a coarse solution at time tn ∈ Im = [�tm−1,�tm] defined on [�tm−2,�tm]
+from the coarse approximations at times �tm−2,�tm−1, and �tm, for all m = 2, . . . , MT. To this purpose, we employ
+the following parabola on [�tm−2,�tm]:
+For m ≥ 2, ∀n ∈ Im = [�tm−1,�tm],
+I2
+n[um
+H](µ) :=
+um−2
+H
+(µ)
+(�tm − �tm−2)(�tm−2 − �tm−1)
+�
+− (tn)2 + (�tm−1 + �tm)tn − tm−1tm�
++
+um−1
+H
+(µ)
+(�tm−2 − �tm−1)(�tm−1 − �tm)
+�
+− (tn)2 + (�tm + �tm−2)tn − tmtm−2�
++
+um
+H(µ)
+(�tm−1 − �tm)(�tm − �tm−2)
+�
+− (tn)2 + (�tm−2 + �tm−1)tn − tm−2tm−1�
+.
+(12)
+For tn ∈ I1 = [�t0,�t1], we use the same parabola defined by the coarse approximations at times �t0, �t1, �t2 as the
+one used over [�t1,�t2]. We denote by �
+uH
+n(µ) = I2
+n[um
+H](µ) the quadratic interpolation of um
+H at a time n. Note that
+we choose this interpolation in order to keep an approximation of order 2 in time ∆tG (it works also with other
+quadratic interpolations).
+In the next section, we recall the NIRB algorithm in the context of parabolic equations.
+5
+
+2.2
+Reminders on the Non-Intrusive Reduced Basis method (NIRB) in the context of
+parabolic equations.
+Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G. With the NIRB algorithm, we aim at
+quickly approximating this solution by using a reduced space, denoted XN
+h , constructed from N fully discretized
+solutions of (10), namely the so-called snapshots. Since each snapshot is a HF finite element approximation in
+space at a time tn, n = 0, ..., NT (NT being potentially very high), not all of the time steps may be required for the
+construction of the reduced space. Here, for each parameter µi, i = 1, . . . , Nµ, selected for the basis construction,
+the number of time steps employed (which depends on i) is denoted Ni. Thus, the reduced basis is defined as
+XN
+h := Span{u
+(nj)i
+h
+(µi)| i = 1, . . . , Nµ, j = 1, . . . , Ni, (nj)i ⊂ {1, · · �� , NT}},
+(13)
+with N :=
+Nµ
+∑
+i=1
+Ni.
+We recall the offline/online decomposition of the NIRB procedure with parabolic equations:
+• “Offline step”
+The offline part of the algorithm allows us to construct the reduced space XN
+h .
+1. From training parameters (µi)i∈{1,...,Ntrain}, we define Gtrain =
+∪
+i∈{1,...,Ntrain}µi. Then, we employ a greedy
+procedure to adequately choose the parameters (µi)i=1,...,Nµ within Gtrain to construct the RB. For this
+procedure, we refer to algorithm 1 (described for the setting Nµ = N in order to simplify notations).
+Note that a POD-greedy algorithm may also be employed [19, 21, 20, 32].
+Algorithm 1 Greedy algorithm
+Input: tol, {un
+h(µ1), · · · , un
+h(µNtrain) with µi ∈ Gtrain, n = 0, . . . , NT}.
+Output: Reduced basis {Φh
+1, · · · , Φh
+N}.
+Choose µ1, n1 =
+arg max
+µ∈Gtrain, n∈{0,...,NT}
+���un
+h(µ)
+���
+L2(Ω) ,
+Set Φh
+1 =
+un1
+h (µ1)
+���un1
+h (µ1)
+���
+L2
+Set G1 = {µ1, n1} and X1
+h = span{Φh
+1}.
+for k = 2 to N do:
+µk, nk = arg
+max
+(µ, n)∈(Gtrain×{0,...,NT})\Gk−1
+���un
+h(µ) − Pk−1(un
+h(µ))
+���
+L2, with Pk−1(un
+h(µ)) :=
+k−1
+∑
+i=1
+(un
+h(µ), Φh
+i ) Φk
+i .
+Compute �
+Φh
+k = unk
+h (µk) − Pk−1(unk
+h (µk)) and set Φh
+k =
+�
+Φh
+k
+����
+�
+Φh
+k
+����
+L2(Ω)
+Set Gk = Gk−1 ∪ {µk} and Xk
+h = Xk−1
+h
+⊕ span{Φh
+k}
+Stop when
+���un
+h(µ) − Pk−1(un
+h(µ))
+���
+L2 ≤ tol, ∀µ ∈ Gtrain, ∀n = 0, . . . , NT.
+end for
+The greedy algorithm is usually less expensive than the POD-greedy (thanks to a-posteriori error
+estimates for stationary problems). Although for time dependent problems, the latter is more rea-
+sonable when the snapshots are computed for all time steps, our choice of using a greedy procedure
+is motivated by the fact that it is more efficient with the post-treatment introduced below. The RB
+functions (time-independent), denoted (Φh
+i )i=1,...,N, are generated at the end of this step, from fine
+fully-discretized solutions {un
+h(µi)}i∈{1,...,Nµ}, n={0,...,NT} (solving problem (10) with HF solver). Note
+that even if all the time steps are computed, only Ni are used for each i ∈ {1, . . . , Nµ} in the RB
+construction. Since at each step k, all sets added in the basis are in the orthogonal complement of
+Xk−1
+h
+, it yields an L2 orthogonal basis without further processing.
+Hence, XN
+h
+can be defined as
+XN
+h = Span{Φh
+1, . . . , Φh
+N}.
+6
+
+Remark 2.2. In practice, the algorithm is halted with a stopping criterion such as an error threshold or a
+maximum number of basis functions to generate.
+2. Then, we solve the following eigenvalue problem:
+�
+�
+�
+�
+�
+Find Φh ∈ XN
+h , and λ ∈ R such that:
+∀v ∈ XN
+h ,
+�
+Ω ∇Φh · ∇v dx = λ
+�
+Ω Φh · v dx,
+(14)
+We get an increasing sequence of eigenvalues λi, and orthogonal eigenfunctions (Φh
+i )i=1,··· ,N, which
+do not depend on time, orthonormalized in L2(Ω) and orthogonalized in H1(Ω). Note that with
+Gram-Schmidt procedure, we only obtain an L2-orthonormalized RB.
+3. For any parameter µk, k = 1, . . . , Nµ, the classical NIRB approximation differs from the HF uh(µk)
+computed in the offline stage [19]. Thus, as proposed in [7], to improve NIRB accuracy, we use a
+”rectification post-processing”. To this purpose, we need a rectification matrix for each fine time step,
+denoted Rn, and constructed from coarse snapshots, generated by solving (11) and whose parameters
+are the same as for the fine snapshots.
+Thus, for all n = 1, . . . , NT, we compute the vectors
+Rn
+u,i = ((An)TAn + δIN)−1(An)TBn
+i ,
+i = 1, · · · , N,
+(15)
+where
+∀i = 1, · · · , N,
+and
+∀µk ∈ Gtrain,
+An
+k,i =
+�
+Ω
+�
+uH
+n(µk) · Φh
+i dx,
+(16)
+Bn
+k,i =
+�
+Ω un
+h(µk) · Φh
+i dx,
+(17)
+and where IN refers to the identity matrix and δ is a regularization parameter.
+Remark 2.3. Note that since every time step has its own rectification matrix, the matrix An is a “flat” rectan-
+gular matrix (Ntrain ≤ N), and thus the parameter δ is required for the inversion of (An)TAn.
+We also remark that with the rectification post-treatment, the standard greedy algorithm 1 may leads to more
+accurate approximations, compared to the POD-greedy algorithm. It comes from the fact that the coefficients of
+the matrix are directly derived from the snapshots in that case.
+• “Online step”
+The online part of the algorithm is much faster than a HF evaluation.
+4. We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step
+m = 0, . . . , MT.
+5. We quadratically interpolate in time the coarse solution on the fine time grid with (12).
+6. Then, we linearly interpolate �
+uH
+n(µ) on the fine mesh in order to compute the L2-inner product with
+the RB functions. The approximation used in the two-grid method is
+For n = 0, . . . , NT,
+uN,n
+Hh (µ) :=
+N
+∑
+i=1
+( �
+uH
+n(µ), Φh
+i ) Φh
+i ,
+(18)
+and with the rectification post-treatment step, it becomes
+Rn
+u[uN
+Hh](µ) :=
+N
+∑
+i,j=1
+Rn
+u,ij ( �
+uH
+n(µ), Φh
+j ) Φh
+i ,
+(19)
+where Rn
+u is the rectification matrix at time tn, given by (15).
+7
+
+In [19], we have proven the following estimate on the heat equation
+for n = 0, . . . , NT,
+���u(tn)(µ) − uN,n
+Hh (µ)
+���
+H1(Ω) ≤ ε(N) + C1(µ)h + C2(N)H2 + C3(µ)∆tF + C4(N)∆t2
+G,
+(20)
+where C1, C2, C3 and C4 are constants independent of h and H, ∆tF and ∆tG. The term ε(N) depends on a proper
+choice of the RB space as a surrogate for the best approximation space associated to the Kolmogorov N-width.
+It decreases when N increases and it is linked to the error between the fine solution and its projection on the
+reduced space XN
+h , given by
+�����un
+h(µ) −
+N
+∑
+i=1
+(un
+h(µ), Φh
+i ) Φh
+i
+�����
+H1(Ω)
+.
+(21)
+The constant C2 increases with N and thus, a trade-off needs to be done between increasing N to obtain a more
+accurate manifold, and keeping a constant C2 as low as possible.
+If H is such as H2 ∼ h, ∆t2
+G ∼ ∆tF, and ε(N) is small enough, with C2(N) and C4(N) not too large, the estimate
+(20) entails an error estimate in O(h + ∆tF), and thus, we recover an optimal error estimate in L∞(0, T; H1(Ω)).
+Before adapting NIRB to the sensitivity analysis context, we first recall how to derive the sensitivities functions
+in the next section.
+2.3
+Sensitivity analysis: The direct problem.
+In this section, we recall the sensitivity systems (continuous and discretized versions) for P parameters of interest.
+Then, we prove the numerical results of the direct method on the model problem. To not make the notations too
+cumbersome, we will consider A(µ) = µ Id, with µ ∈ R+∗ for the analysis theorems.
+2.3.1
+The continuous setting.
+In this setting, we consider P parameters of interest, denoted µp = 1, . . . , µP, and we want to approximate the
+exact derivatives
+Ψp(t, x; µ) := ∂u
+∂µp
+(t, x; µ).
+(22)
+In order to seek these sensitivities, we solve P new systems, which can directly be obtained by differentiating the
+initial problem with respect to µp. The continuous initial problem (5) may be rewritten
+�
+�
+�
+�
+�
+Find u(t) ∈ V for t ∈ [0, T] such that
+(ut(t), v) = F(u(t), v; µ) := −a(u(t), v; µ) + ( f (t), v), ∀v ∈ V, t > 0,
+u(·, 0) = u0(·),
+where the bilinear form a is defined by (6). Using the chain rule and since the time and the parameter derivatives
+can commute,
+(Ψp,t(t), v) = ∂F
+∂u (u(t), v; µ) · Ψp(t) + ∂F
+∂µ(u(t), v).
+Since the initial condition here does not depend on µ, we obtain the following problem
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find Ψp(t) ∈ V for t ∈ [0, T] such that
+(Ψp,t(t), v) + a(Ψp(t), v; µ) = −( ∂A
+∂µp (µ)∇u(t), ∇v), for v ∈ V, for t > 0,
+Ψ0
+p = 0,
+(23)
+which is well-posed since u ∈ L2(0, T; H1
+0(Ω)), and under the assumptions (4), the so-called ”parabolic regularity
+estimate” implies that u ∈ L2(0, T; H2(Ω)) ∩ L∞(0, T; H1
+0(Ω)) [14, 45].
+8
+
+2.3.2
+The spatially semi-discretized version.
+As previously for the state solution, we discretize in space and in time the sensitivity problems (23).
+The corresponding spatially semi-discretized formulations (on Th) read
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find Ψp,h(t) ∈ Vh for t ∈ [0, . . . , T] such that
+(Ψp,h,t(t), vh) + a(Ψp,h(t), vh; µ) = −( ∂A
+∂µp (µ)∇uh(t; µ), ∇vh), for vh ∈ Vh, for t ∈]0, T],
+Ψ0
+p,h(·) = P1
+h(Ψ0
+p)(·),
+(24)
+where P1
+h is given by (7). Before proceeding with the proof of Theorem (4.1), we need several results that can
+be deduced from [45], but require some precisions. Indeed, first, in [45], the estimates are proven on the heat
+equation with a non-varying diffusion coefficient. Secondly, the right-hand side function f vanishes when seek-
+ing the error estimates, whereas in our case, the right-hand side function depends on u and necessitates precised
+estimates.
+On the semi-discretized formulation, the following estimate holds.
+Theorem 2.4. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . Consider u ∈ H1(0, T; H2(Ω)) be the
+solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8). Let Ψ and Ψh be the corresponding
+sensitivities , respectively given by (23) and (24). Then
+∀t ∈ [0, T],
+��Ψh(t) − Ψ(t)
+��
+L2(Ω) ≤ Ch2����Ψ0���
+H2(Ω) +
+� T
+0 ∥Ψt∥H2(Ω) ds
+�
++ C(µ)h2� � T
+0 ∥ut∥2
+H2(Ω) ds
+�1/2
+.
+Proof. As in [45], we first decompose the error with two components θ and ρ such that
+∀t ∈ [0, T], e(t) := Ψh(t) − Ψ(t) = (Ψh(t) − P1
+hΨ(t)) + (P1
+hΨ(t) − Ψ(t)),
+= θ(t) + ρ(t).
+(25)
+• For the estimate on ρ(t), a classical FEM estimate [45, 5] is
+���P1
+hv − v
+���
+L2(Ω) + h
+���∇(P1
+hv − v)
+���
+L2(Ω) ≤ Ch2∥v∥H2(Ω) ,
+∀v ��� H2 ∩ H1
+0,
+(26)
+which leads to
+��ρ(t)
+��
+L2(Ω) ≤ Ch2��Ψ(t)
+��
+H2(Ω) , ∀t ∈ [0, T],
+≤ Ch2����Ψ0���
+H2(Ω) +
+� T
+0 ∥Ψt∥H2(Ω) ds
+�
+, ∀t ∈ [0, T].
+(27)
+• For the estimate on θ(t), let us consider v ∈ Vh,
+∀t ∈]0, T], (θt(t), v) + µ(∇θ(t), ∇v) = (Ψh,t(t), v) + µ(∇Ψh(t), ∇v) − (P1
+hΨt(t), v) − µ(∇P1
+hΨ(t), ∇v).
+Since v ∈ H1
+0, by definition of P1
+h (7), the semi-discretized weak formulations (24) implies
+(θt(t), v) + µ(∇θ(t), ∇v) = −(∇uh(t), ∇v) − (P1
+hΨt(t), v) − µ(∇P1
+hΨ(t), ∇v),
+= −(∇uh(t), ∇v) − (P1
+hΨt(t), v) − µ(∇Ψ(t), ∇v).
+Thanks to the continuous weak formulation (23), and since the operator P1
+h and the time derivative com-
+mute, it can be rewritten
+(θt(t), v) + µ(∇θ(t), ∇v) = (∇u(t) − ∇uh(t), ∇v) + (Ψt(t) − (P1
+hΨ)t(t), v),
+= (∇u(t) − ∇uh(t), ∇v) − (ρt(t), v).
+Choosing v = θ(t), it yields
+(θt(t), θ(t)) + µ
+��∇θ(t)
+��2
+L2(Ω) = (∇u(t) − ∇uh(t), ∇θ(t)) − (ρt(t), θ(t)),
+9
+
+and using the continuous and semi-discretized weak formulations on the state variable u(t) ((5) and (8)
+respectively), we obtain
+(θt(t), θ(t)) + µ
+��∇θ(t)
+��2
+L2(Ω) = 1
+µ(uh,t(t) − ut(t), θ(t)) − (ρt(t), θ(t)),
+(28)
+where the first term of the right-hand side is a new contribution (compared to the proof of Theorem 1.2
+[45]). Since
+(θt(t), θ(t)) = 1
+2
+d
+dt(
+��θ(t)
+��2
+L2(Ω)) =
+��θ(t)
+��
+L2(Ω)
+d
+dt
+��θ(t)
+��
+L2(Ω) ,
+(29)
+and, since the second term in (28) is positive, it becomes with Cauchy-Schwarz inequality (the case where
+θ(t) = 0 for some t may easily be handled)
+d
+dt
+��θ(t)
+��
+L2(Ω) ≤ 1
+µ
+��uh,t(t) − ut(t)
+��
+L2(Ω) +
+��ρt(t)
+��
+L2(Ω) .
+Integrating over time, it follows that
+��θ(t)
+��
+L2(Ω) ≤
+��θ(0)
+��
+L2(Ω)
+�
+��
+�
+T1
++ 1
+µ
+� T
+0
+��uh,t − ut
+��
+L2(Ω) ds
+�
+��
+�
+T2
++
+� T
+0
+��ρt
+��
+L2(Ω) ds
+�
+��
+�
+T3
+.
+(30)
+– From the initial conditions, since u0
+h = P1
+hu0, T1 = 0.
+Note that other optimal order choices of
+discretized initial conditions (such as the L2 orthogonal projection onto Vh) lead to an estimate in
+Ch2���Ψ0���
+H2(Ω) for T1.
+– To estimate T2, in analogy with θ and ρ, let us introduce θu and ρu, such that
+∀t ∈ [0, T], uh(t) − u(t) = (uh(t) − P1
+hu(t)) + (P1
+hu(t) − u(t)),
+= θu(t) + ρu(t).
+(31)
+We remark that the term T2 can also be written
+T2 = 1
+µ
+� T
+0
+��θu,t + ρu,t
+��
+L2(Ω) ds ≤ 1
+µ
+� T
+0 ∥θu,t∥L2(Ω) +
+��ρu,t
+��
+L2(Ω) ds.
+Then, by Cauchy-Schwarz inequality,
+T2 ≤
+√
+T
+µ
+�� � T
+0 ∥θu,t∥2
+L2(Ω) ds
+�1/2
++
+� � T
+0
+��ρu,t
+��2
+L2(Ω) ds
+�1/2�
+,
+(32)
+We can bound � T
+0 ∥θu,t∥2
+L2(Ω), using the variational formulations (5) and (8). We first write for t ∈]0, T]:
+(θu,t(t), v) + µ(∇θu(t), ∇v) = (uh,t(t), v) + µ(∇uh(t), ∇v) − (P1
+hut(t), v) − µ(∇P1
+hu(t), ∇v),
+= ( f (t), v) − (P1
+hut(t), v) − µ(∇u(t), ∇v),
+= −(ρu,t(t), v).
+Formally by using v = θu,t(t) and (29), it entails
+��θu,t(t)
+��2
+L2(Ω) + µ
+2
+d
+dt
+��∇θu(t)
+��2
+L2(Ω) = −(ρu,t(t), θu,t(t)),
+such that (with Young’s inequality)
+��θu,t(t)
+��2
+L2(Ω) + µ d
+dt
+��∇θu(t)
+��2
+L2(Ω) ≤
+��ρu,t(t)
+��2
+L2(Ω) .
+Integrating over time, we obtain
+� T
+0 ∥θu,t∥2
+L2(Ω) ds + µ
+��∇θu(t)
+��2
+L2(Ω) ≤ µ
+��∇θu(0)
+��2
+L2(Ω) +
+� T
+0
+��ρu,t
+��2
+L2(Ω) ds,
+10
+
+and since the second term is always positive and that we have chosen u0
+h = P1
+hu0, it yields
+� T
+0 ∥θu,t∥2
+L2(Ω) ≤
+� T
+0
+��ρu,t
+��2
+L2(Ω) .
+(33)
+Remark 2.5. Note that with another choice of discretized initial solution, we would have
+��∇θu(0)
+��2
+L2(Ω) ≤
+���∇u0
+h − ∇u0���
+2
+L2(Ω) + Ch2���u0���
+2
+H2(Ω) ,
+which would have lead to an estimate in O(h) on the L2(Ω) error estimate of Ψ(t). In practice, this is not an
+issue since the effect of the initial data exponentially decreases [45].
+Therefore, from (32), we obtain
+T2 ≤ 2
+√
+T
+µ
+� � T
+0
+��ρu,t
+��2
+L2(Ω) ds
+�1/2
+.
+(34)
+By definition of P1
+h (7), we have
+��ρu,t(t)
+��
+L2(Ω) =
+���P1
+hut(t) − ut(t)
+���
+L2(Ω) ≤ Ch2��ut(t)
+��
+H2(Ω) ,
+(35)
+and thus, (34) yields
+T2 ≤ C2
+√
+T
+µ
+h2� � T
+0 ∥ut∥2
+H2(Ω) ds
+�1/2
+.
+(36)
+– Finally, for T3, we only need to use (35) again, but with Ψ instead of u. Therefore
+T3 =
+� T
+0
+��ρt
+��
+L2(Ω) ds ≤ Ch2
+� T
+0 ∥Ψt∥H2(Ω) ds .
+(37)
+Combining (27), (30), (36), and (37) concludes the proof.
+We can derive a similar result for the H1
+0 norm.
+Theorem 2.6. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . Consider u ∈ H1(0, T; H2(Ω)) be the
+solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8). Let Ψ and Ψh be the corresponding
+sensitivities , respectively given by (23) and (24).
+∀t ∈ [0, T],
+��Ψ(t) − Ψh(t)
+��
+H1(Ω) ≤ Ch
+����Ψ0���
+H2(Ω) +
+� T
+0 ∥Ψt∥H2(Ω) ds
+�
++ C(µ)h2
+�� � T
+0 ∥ut∥2
+H2 ds
+�1/2
++
+� � T
+0 ∥Ψt∥2
+H2(Ω) ds
+�1/2�
+.
+Proof. Using the same notation as before (25), we first decompose the error with the two components θ and ρ
+such that
+∀t ∈ [0, T], ∇Ψh(t) − ∇Ψ(t) = ∇θ(t) + ∇ρ(t).
+(38)
+• For the estimate on ρ(t), we use (26) to obtain
+��∇ρ(t)
+��
+L2(Ω) ≤ Ch
+��Ψ(t)
+��
+H2(Ω) , ∀t ∈ [0, T],
+which leads to
+��∇ρ(t)
+��
+L2(Ω) ≤ Ch
+����Ψ0���
+H2(Ω) +
+� T
+0 ∥Ψt∥H2(Ω)
+ds
+�
+, ∀t ∈ [0, T].
+(39)
+• For the estimate on θ(t), let us consider v ∈ Vh. As in the previous proof, ∀t ∈ [0, T], we write
+(θt(t), v) + µ(∇θ(t), ∇v) = (Ψh,t(t), v) + µ(∇Ψh(t), ∇v) − (P1
+hΨt(t), v) − µ(∇P1
+hΨ(t), ∇v).
+11
+
+Instead of replacing v by θ(t) as in the L2 estimate, here we formally replace v by θt(t), thus
+∀t ∈]0, T],
+��θt(t)
+��2
+L2(Ω) + µ(∇θ(t), ∇θt(t)) = (∇u(t) − ∇uh(t), ∇θt(t)) − (ρt(t), θt(t)).
+Thanks to the variational formulations on the state solution u ((5) and (8) respectively)
+��θt(t)
+��2
+L2(Ω) + µ(∇θ(t), ∇θt(t)) = ( 1
+µ(uh,t(t) − ut(t)), θt(t)) − (ρt(t), θt(t)),
+and thus (with Young’s inequality),
+��θt(t)
+��2
+L2(Ω) + µ(∇θ(t), ∇θt(t)) ≤ 1
+2
+�����
+1
+µ(uh,t(t) − ut(t))
+�����
+2
+L2(Ω)
++ 1
+2
+��θt(t)
+��2
+L2(Ω) + 1
+2
+��ρt(t)
+��2
+L2(Ω) + 1
+2
+��θt(t)
+��2
+L2(Ω) ,
+≤
+1
+2µ2
+��uh,t(t) − ut(t)
+��2
+L2(Ω) + 1
+2
+��ρt(t)
+��2
+L2(Ω) +
+��θt(t)
+��2
+L2(Ω) .
+Thus,
+µ(∇θ(t), ∇θt(t)) ≤
+1
+2µ2
+��uh,t(t) − ut(t)
+��2
+L2(Ω) + 1
+2
+��ρt(t)
+��2
+L2(Ω) ,
+(40)
+and by (29), we have
+d
+dt
+��∇θ(t)
+��2
+L2(Ω) ≤ 1
+µ3
+��uh,t(t) − ut(t)
+��2
+L2(Ω) + 1
+µ
+��ρt(t)
+��2
+L2(Ω) .
+Integrating over time, it entails
+��∇θ(t)
+��2
+L2(Ω) ≤
+��∇θ(0)
+��2
+L2(Ω)
+�
+��
+�
+T′
+1
++ 1
+µ3
+� T
+0
+��uh,t − ut
+��2
+L2(Ω) ds
+�
+��
+�
+T′
+2
++ 1
+µ
+� T
+0
+��ρt
+��2
+L2(Ω) ds
+�
+��
+�
+T′
+3
+.
+(41)
+– From the initial conditions, T′
+1 = 0.
+– We can also write T′
+2 as
+T′
+2 = 1
+µ3
+� T
+0
+��θu,t + ρu,t
+��2
+L2(Ω) ds .
+Therefore using (33),
+T′
+2 ≤ 2
+µ3
+� T
+0 ∥θu,t∥2
+L2(Ω) +
+��ρu,t
+��2
+L2(Ω) ds ≤ 4
+µ3
+� T
+0
+��ρu,t
+��2
+L2(Ω) ds ≤ Ch4
+µ3
+� T
+0 ∥ut∥2
+H2(Ω) ds .
+(42)
+– Similarly,
+T′
+3 ≤ Ch4
+µ
+� T
+0 ∥Ψt∥2
+H2(Ω) ds .
+(43)
+Hence, combining (38) with (39), (41), (42) and (43) concludes the proof.
+2.3.3
+The fully-discretized versions.
+From (24), we can derive the fully-discretized systems for the fine and coarse grids.
+The direct sensitivity problems with respect to the parameter µp on the fine mesh Th with an Euler scheme read
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find Ψn
+p,h ∈ Vh for n ∈ {0, . . . , NT} such that
+(∂Ψn
+p,h, vh) + a(Ψn
+p,h, vh; µ) = −( ∂A
+∂µp (µ)∇un
+h(µ), ∇vh), for vh ∈ Vh, for n = {1, . . . , NT},
+(44)
+Ψ0
+p,h(·) = P1
+hΨ0
+p(·),
+where, as before, the time derivative in the variational form of the problem (23) has been replaced by a backward
+difference quotient, ∂Ψn
+h =
+Ψn
+h−Ψn−1
+h
+∆tF
+.
+With the fully-discretized version (44), the following estimate holds.
+12
+
+Theorem 2.7. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ .
+Consider u ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un
+h be the fully-discretized
+variational form (10). Let Ψ and Ψn
+h be the corresponding sensitivities , respectively given by (23) and (44). Then
+∀n = 0, . . . , NT,
+��Ψn
+h − Ψ(t)
+��
+L2(Ω) ≤ Ch2���Ψ0���
+H2(Ω) + h2�
+C
+� tn
+0 ∥Ψt∥H2(Ω) ds + C(µ)
+� � tn
+0 ∥ut∥2
+H2(Ω) ds
+�1/2�
++ ∆tF
+�
+C
+� tn
+0 ∥Ψtt∥L2(Ω) ds + C(µ)
+� � tn
+0 ∥utt∥2
+L2(Ω) ds
+�1/2�
+.
+Proof. Now, we define θn and ρn on the discretized time grid (tn)n=0,...,NT.
+∀n = 0, . . . , NT, en := Ψn
+h − Ψ(tn) = (Ψn
+h − P1
+hΨ(tn)) + (P1
+hΨ(tn) − Ψ(tn)),
+= θn + ρn.
+(45)
+• In analogy with (26) the estimate on ρn is
+��ρn��
+L2(Ω) ≤ Ch2��Ψ(tn)
+��
+H2(Ω) ≤ Ch2����Ψ0���
+H2(Ω) +
+� tn
+0 ∥Ψt∥H2(Ω) ds
+�
+, ∀n ∈ {0, . . . , NT}.
+(46)
+• For θn, the equation (28) becomes
+(∂θn, θn) + µ
+��∇θn��2
+L2(Ω) = 1
+µ(∂un
+h − ut(tn), θn) − (wn
+1 + wn
+2, θn),
+= 1
+µ(∂un
+h − ut(tn), θn) − (wn, θn),
+(47)
+where wn
+1 and wn
+2 are defined by
+wn
+1 := (P1
+h − I)∂Ψ(tn),
+wn
+2 := ∂Ψ(tn) − Ψt(tn),
+and
+wn := wn
+1 + wn
+2.
+(48)
+By definition of ∂ and by Cauchy-Schwarz inequality (and since the second term of the left-hand side of
+(47) is always positive),
+��θn��2
+L2(Ω) ≤
+����θn−1���
+L2(Ω) + ∆tF
+� 1
+µ
+���∂un
+h − ut(tn)
+���
+L2(Ω) +
+��wn��
+L2(Ω)
+����θn��
+L2(Ω) ,
+and by repeated application, and since
+���θ0���
+L2(Ω) = 0 (again, the case where some θn are equal to 0 can be
+easily handled), it entails
+��θn��
+L2(Ω) ≤ ∆tF
+n
+∑
+j=1
+1
+µ
+���∂uj
+h − ut(tj)
+���
+L2(Ω)
+�
+��
+�
+T2,n
++ ∆tF
+n
+∑
+j=1
+���wj���
+L2(Ω)
+�
+��
+�
+T3,n
+,
+(49)
+– We first decompose T2,n in two contributions
+∆tF
+µ
+n
+∑
+j=1
+���∂uj
+h − ut(tj)
+���
+L2(Ω) ≤ ∆tF
+µ
+n
+∑
+j=1
+����∂θj
+u
+���
+L2(Ω) +
+���wj
+u
+���
+L2(Ω)
+�
+,
+where
+wj
+u := wj
+1,u + wj
+2,u with wj
+1,u := (P1
+h − I)∂u(tj),
+and
+wj
+2,u := ∂u(tj) − ut(tj).
+(50)
+Then by using Cauchy-Schwarz inequality (as in the semi-discretized case (32)),
+∆tF
+µ
+n
+∑
+j=1
+���∂uj
+h − ut(tj)
+���
+L2(Ω) ≤
+√
+tn
+µ
+�� n
+∑
+j=1
+∆tF
+���∂θj
+u
+���
+2
+L2(Ω)
+�
+��
+�
+Tθ
+�1/2
++
+� n
+∑
+j=1
+∆tF
+���wj
+u
+���
+2
+L2(Ω)
+�
+��
+�
+Tw
+�1/2�
+.
+(51)
+13
+
+* Let us begin by the estimate on Tθ. On the state solution u, by choosing v = ∂θn
+u, from (10) (the
+operator ∂ and the spatial derivative can commute), we have
+���∂θn
+u
+���
+2
+L2(Ω) + µ(∇θn
+u, ∂∇θn
+u) = −(wn
+u, ∂θn
+u),
+(52)
+where θn
+u is the discrete version of (31). By definition of ∂ (and with Young’s inequality),
+���∂θn
+u
+���
+2
+L2(Ω) + µ
+∆tF
+��∇θn
+u
+��2
+L2(Ω) ≤
+µ
+2∆tF
+���∇θn
+u
+��2
+L2(Ω) +
+���∇θn−1
+u
+���
+2
+L2(Ω)
+�
++ 1
+2
+���wn
+u
+��2
+L2(Ω) +
+���∂θn���
+2
+L2(Ω)
+�
+,
+which entails
+���∂θn
+u
+���
+2
+L2(Ω) ≤
+µ
+∆tF
+���∇θn−1
+u
+���
+2
+L2(Ω) −
+µ
+∆tF
+��∇θn
+u
+��2
+L2(Ω) +
+��wn
+u
+��2
+L2(Ω) , ∀n = 1, . . . , NT.
+(53)
+Summing over the time steps, we get
+n
+∑
+j=1
+���∂θj
+u
+���
+2
+L2(Ω) ≤
+���
+n
+∑
+j=1
+µ
+∆tF
+����∇θj−1
+u
+���
+2
+L2(Ω) −
+���∇θj
+u
+���
+2
+L2(Ω)
+�
++
+���wj
+u
+���
+2
+L2(Ω)
+���,
+and we obtain
+n
+∑
+j=1
+���∂θn
+u
+���
+2
+L2(Ω) ≤
+��� µ
+∆tF
+����∇θ0
+u
+���
+2
+L2(Ω) −
+��∇θn
+u
+��2
+L2(Ω)
+���� +
+n
+∑
+j=1
+���wj
+u
+���
+2
+L2(Ω) .
+From the initial condition, θ0
+u = 0,
+n
+∑
+j=1
+���∂θn
+u
+���
+2
+L2(Ω) ≤
+µ
+∆tF
+��∇θn
+u
+��2
+L2(Ω) +
+n
+∑
+j=1
+���wj
+u
+���
+2
+L2(Ω) .
+(54)
+From (53) and by repeated application, we find for the first right-hand side term that
+��∇θn
+u
+��2
+L2(Ω) ≤ ∆tF
+µ
+n
+∑
+j=1
+���wj
+u
+���
+2
+L2(Ω) ,
+which gives for (54), multiplying by ∆tF to recover Tθ,
+n
+∑
+j=1
+∆tF
+���∂θj
+u
+���
+2
+L2(Ω) ≤ 2
+n
+∑
+j=1
+∆tF
+���wj
+u
+���
+2
+L2(Ω) .
+(55)
+Now, going back to (51), we obtain
+∆tF
+µ
+n
+∑
+j=1
+���∂uj
+h − ut(tj)
+���
+L2(Ω) ≤ C
+µ
+� n
+∑
+j=1
+∆tF
+���wj
+u
+���
+2
+L2(Ω)
+�
+��
+�
+Tw
+�1/2
+≤ C
+µ
+� n
+∑
+j=1
+∆tF
+����wj
+1,u
+���
+2
+L2(Ω) +
+���wj
+2,u
+���
+2
+L2(Ω)
+��1/2
+.
+(56)
+* It remains to estimate Tw.
+· Let us first consider the construction for w1,u
+wj
+1,u = (P1
+h − I)∂u(tj) =
+1
+∆tF
+(P1
+h − I)
+� tj
+tj−1 ut ds =
+1
+∆tF
+� tj
+tj−1(P1
+h − I)ut ds ,
+14
+
+since P1
+h and the time integral commute. Thus, from Cauchy-Schwarz inequality,
+∆tF
+n
+∑
+j=1
+���wj
+1,u
+���
+2
+L2(Ω) ≤ ∆tF
+n
+∑
+j=1
+�
+Ω
+� 1
+∆t2
+F
+� tj
+tj−1((P1
+h − I)ut)2 ds ∆tF
+�
+≤
+n
+∑
+j=1
+� tj
+tj−1
+���(P1
+h − I)ut
+���
+2
+L2(Ω)
+ds ,
+≤ Ch4
+n
+∑
+j=1
+� tj
+tj−1∥ut∥2
+H2(Ω) , by definition of P1
+h,
+≤ Ch4
+� tn
+0 ∥ut∥2
+H2(Ω)
+ds.
+(57)
+· To estimate the L2 norm of w2,u, we write
+wj
+2,u =
+1
+∆tF
+(u(tj) − u(tj−1)) − ut(tj) = − 1
+∆tF
+� tj
+tj−1(s − tj−1)utt(s) ds,
+such that we end up with
+∆tF
+n
+∑
+j=1
+���wj
+2,u
+���
+2
+L2(Ω) ≤
+n
+∑
+j=1
+�����
+� tj
+tj−1(s − tj−1)utt(s) ds
+�����
+2
+L2(Ω)
+≤ ∆t2
+F
+� tn
+0 ∥utt∥2
+L2(Ω) ds,
+(58)
+– We still have to find a bound for T3,n, defined in (49).
+* For the estimates on wj
+1,
+wj
+1 =
+1
+∆tF
+� tj
+tj−1(P1
+h − I)Ψt ds ,
+and thus,
+∆tF
+n
+∑
+j=1
+���wj
+1
+���
+L2(Ω) ≤ Ch2
+� tn
+0 ∥Ψt∥H2(Ω) ds .
+* For wj
+2, we have
+∆tFwj
+2 = Ψ(tj) − Ψ(tj−1) − ∆tFΨt(tj) = −
+� tj
+tj−1(s − tj−1)Ψtt(s) ds ,
+and therefore
+∆tF
+n
+∑
+j=1
+���wj
+2
+���
+L2(Ω) ≤
+n
+∑
+j=1
+�����
+� tj
+tj−1(s − tj−1)Ψtt(s) ds
+�����
+L2(Ω)
+≤ ∆tF
+� tn
+0 ∥Ψtt∥L2(Ω) ds .
+Altogether,
+T3,n ≤ Ch2
+� tn
+0 ∥Ψt∥H2(Ω) ds + ∆tF
+� tn
+0 ∥Ψtt∥L2(Ω) ds ,
+(59)
+and the proof ends by using (46), (49), (56), (57), (58), and (59).
+With the fully-discretized version (44), the following estimate holds with H1 norm.
+Theorem 2.8. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ .
+Consider u ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un
+h be the fully-discretized
+variational form (10). Let Ψ and Ψn
+h be the corresponding sensitivities , respectively given by (23) and (44). Then
+∀n = 0, . . . , NT,
+��∇Ψn
+h − ∇Ψ(t)
+��
+L2(Ω) ≤ h
+�
+C
+���Ψ0���
+H2(Ω) + C(µ)
+� tn
+0 ∥Ψt∥H2(Ω) ds + C(µ)
+� � tn
+0 ∥ut∥2
+H2(Ω) ds
+�1/2�
++ C(µ)∆tF
+� � tn
+0 ∥Ψtt∥L2(Ω) ds +
+� � tn
+0 ∥utt∥2
+L2(Ω) ds
+�1/2��
+.
+15
+
+Proof. The proof combines the ideas of the two previous ones, since we seek the estimate in the H1 norm (as in
+the semi-discretized problem) but with the fully-discretized version.
+• In analogy with (26), the estimate on ρn is now given by
+��∇ρn��
+L2(Ω) ≤ Ch
+��Ψ(tn)
+��
+H2(Ω) ≤ Ch
+����Ψ0���
+H2(Ω) +
+� tn
+0 ∥Ψt∥H2(Ω) ds
+�
+, ∀n = 0, . . . , NT.
+(60)
+• For θn, instead of choosing v = θn as in (47), we take v = ∂θn
+���∂θn���
+2
+L2(Ω) + µ(∇θn, ∇∂θn) = 1
+µ(∂un
+h − ut(tn), ∂θn) − (wn, ∂θn),
+(61)
+and we obtain (as before with the semi-discretized version (40))
+µ(∇θn, ∇∂θn) ≤
+1
+2µ2
+���∂un
+h − ut(tn)
+���
+2
+L2(Ω) + 1
+2
+��wn��2
+L2(Ω) .
+By definition of ∂
+µ
+��∇θn��2
+L2(Ω) ≤ (√µ∇θn, √µ∇θn−1) + ∆tF
+2µ2
+���∂un
+h − ut(tn)
+���
+2
+L2(Ω) + ∆tF
+2
+��wn��2
+L2(Ω) ,
+which entails (by Young’s inequality)
+µ
+��∇θn��2
+L2(Ω) ≤ µ
+���∇θn−1���
+2
+L2(Ω) + ∆tF
+µ2
+���∂un
+h − ut(tn)
+���
+2
+L2(Ω) + ∆tF
+��wn��2
+L2(Ω) ,
+and, by recursion (as in (49))
+µ
+��∇θn��2
+L2(Ω) ≤ ∆tF
+µ2
+n
+∑
+j=1
+���∂uj
+h − ut(tj)
+���
+2
+L2(Ω)
+�
+��
+�
+T′
+2,n
++ ∆tF
+n
+∑
+j=1
+���wj���
+2
+L2(Ω)
+�
+��
+�
+T′
+3,n
+.
+(62)
+– To estimate T′
+2,n, we write
+T′
+2,n ≤ 2
+µ2
+� n
+∑
+j=1
+∆tF
+���∂θj
+u
+���
+2
+L2(Ω)
+�
+��
+�
+Tθ
++
+n
+∑
+j=1
+∆tF
+���wj
+u
+���
+2
+L2(Ω)
+�
+��
+�
+Tw
+�
+,
+and thanks to the previous estimate on Tθ (55), we find that
+T′
+2,n ≤ 6
+µ2
+� n
+∑
+j=1
+∆tF
+���wj
+u
+���
+2
+L2(Ω)
+�
+��
+�
+Tw
+�
+,
+which, by (57) and (58), yields
+T′
+2,n ≤ C
+µ2
+�
+h4
+� tn
+0 ∥ut∥2
+H2(Ω) ds + ∆t2
+F
+� tn
+0 ∥utt∥2
+L2(Ω) ds
+�
+.
+– To find a bound for T′
+3,n, we simply use (57) and (58) again but with the sensitivity function Ψ instead
+of u.
+Combining the estimates on T′
+2,n and T′
+3,n with (62), and (60) concludes the proof.
+16
+
+With ∂Ψm
+H = Ψm
+H−Ψm−1
+H
+∆tG
+, on the coarse mesh TH with the Crank-Nicolson scheme, the fully-discretized system
+(11) yields
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find Ψm
+p,H ∈ VH for m ∈ {0, . . . , MT} such that
+(∂Ψm
+p,H, vH) + a(
+Ψm
+p,H+Ψm−1
+p,H
+2
+, vH; µ) = −( ∂A
+∂µp (µ) ∇um
+H(µ)+∇um−1
+H
+(µ)
+2
+, ∇vH), for vH ∈ VH, for m = {1, . . . , MT}, (63)
+Ψ0
+p,H(·) = P1
+HΨ0
+p(·).
+We have the following result in the L2 norm with the Crank-Nicolson scheme on the coarse mesh TH.
+Theorem 2.9. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ .
+Consider u ∈ H2(0, T; H2(Ω)) ∩ H3(0, T; L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and um
+H be the fully-discretized
+variational form (11) (on the coarse mesh TH). Let Ψ and Ψm
+H be the corresponding sensitivities , respectively given by (23)
+and (44). Then
+∀m = 0, . . . , MT,
+���Ψm
+h − Ψ(�tm)
+���
+L2(Ω) ≤ CH2����Ψ0���
+H2(Ω) +
+� �tm
+0
+∥Ψt∥H2(Ω) ds + C(µ)
+� � �tm
+0
+∥ut∥2
+H2(Ω) ds
+���1/2�
++ C∆t2
+G
+� � �tm
+0
+∥Ψttt∥L2(Ω) ds +
+� � �tm
+0
+∥∆utt∥2
+L2(Ω) ds
+�1/2
++ C(µ)
+�� � �tm
+0
+∥uttt∥2
+L2(Ω) ds]1/2 +
+� �tm
+0
+∥∆Ψtt∥L2(Ω) ds
+��
+.
+Proof.
+• For ρm we have the same estimate as before (46) (but with the coarse size H).
+• We introduce the following notation
+�
+um
+H = 1
+2(um
+H + um−1
+H
+).
+(64)
+Thanks to the Crank-Nicolson formulation on Ψm
+H (63) and um
+H (11) on the coarse mesh TH (and on the weak
+formulation on u (5) and by definition of P1
+H (7)),
+(∂θm, v) + µ(∇�
+θm, ∇v) = (∂Ψm
+H, v) − (∂P1
+H(Ψ(�tm)), v) + µ(∇�
+Ψm
+H, ∇v) − µ
+2
+�
+(∇P1
+HΨ(�tm), ∇v) + (∇P1
+HΨ(�tm−1), ∇v)
+�
+,
+= −(∇ �
+um
+H, ∇v) − (∂P1
+H(Ψ(�tm)), v) − µ
+2
+�
+(∇Ψ(�tm), ∇v) + (∇Ψ(�tm−1), ∇v)
+�
+,
+= −(∇ �
+um
+H, ∇v) −(∂P1
+H(Ψ(�tm)), v) + (∂Ψ(�tm), v)
+�
+��
+�
+−wm
+I
+−(∂Ψ(�tm), v) + (Ψt(�tm− 1
+2 ), v)
+�
+��
+�
+−wm
+II
+− (Ψt(�tm− 1
+2 ), v) − µ
+2
+�
+(∇Ψ(�tm), ∇v) + (∇Ψ(�tm−1), ∇v)
+�
+= −(∇ �
+um
+H, ∇v) − wm
+I − wm
+II + (∇u(�tm− 1
+2 ), ∇v) + µ(∇Ψ(�tm− 1
+2 ), ∇v)
+− µ
+2
+�
+(∇Ψ(�tm), ∇v) + (∇Ψ(�tm−1), ∇v)
+�
+= (∇u(�tm− 1
+2 ) − �
+∇um
+H, ∇v) − (wm
+I + wm
+II + µwm
+III, v),
+where wm
+I , wm
+II and wm
+III are defined by
+wm
+I := (P1
+H − I)∂Ψ(�tm), wm
+II := ∂Ψ(�tm) − Ψt(�tm− 1
+2 ), and wm
+III := ∆ψ(�tm− 1
+2 ) − 1
+2(∆Ψ(�tm) + ∆Ψ(�tm−1)). (65)
+Thus, (47) with a Crank-Nicolson scheme and with v = �
+θm becomes
+(∂θm, �
+θm) + µ(∇�
+θm, ∇�
+θm) = 1
+µ(∂um
+H − ut(�tm− 1
+2 ), �
+θm) − (wm
+I + wm
+II + µwm
+III, �
+θm),
+= 1
+µ(∂um
+H − ut(�tm− 1
+2 ), �
+θm) − (wm
+T , �
+θm),
+(66)
+where wm
+T = wm
+I + wm
+II + µwm
+III. By definition of ∂ (with the coarse time grid), and since the second term in
+(66) is always positive, we get
+(θm, �
+θm) − (θm−1, �
+θm) ≤ ∆tG
+� 1
+µ
+���∂um
+H − ut(�tm− 1
+2 )
+���
+L2(Ω) +
+��wm
+T
+��
+L2(Ω)
+�����
+θm
+���
+L2(Ω) ,
+17
+
+and by definition of �
+θm (64),
+��θm��2
+L2(Ω) −
+���θm−1���
+2
+L2(Ω) ≤ ∆tG
+� 1
+µ
+���∂um
+h − ut(�tm− 1
+2 )
+���
+L2(Ω) +
+��wm
+T
+��
+L2(Ω)
+����θm + θm−1���
+L2(Ω) ,
+so that, after cancellation of a common factor,
+��θm��
+L2(Ω) −
+���θm−1���
+L2(Ω) ≤ ∆tG
+� 1
+µ
+���∂um
+H − ut(�tm− 1
+2 )
+���
+L2(Ω) +
+��wm
+T
+��
+L2(Ω)
+�
+,
+and by recursive application, it entails
+��θm��
+L2(Ω) ≤ ∆tG
+µ
+m
+∑
+j=1
+���∂uj
+H − ut(�tj− 1
+2 )
+���
+L2(Ω)
+�
+��
+�
+T′′
+2,n
++ ∆tG
+m
+∑
+j=1
+���wj
+T
+���
+L2(Ω)
+�
+��
+�
+T′′
+3,n
+.
+(67)
+– To estimate T′′
+2,n, we use the same tricks as before (51). First, we can decompose T′′
+2,n in 2 contributions,
+such that
+∆tG
+µ
+m
+∑
+j=1
+���∂uj
+H − ut(�tj− 1
+2 )
+���
+L2(Ω) ≤ ∆tG
+µ
+m
+∑
+j=1
+���∂uj
+H − ∂Ph
+1 u(�tj)
+���
+L2(Ω)
+�
+��
+�
+���∂θj
+u
+���
+L2(Ω)
++
+���∂Ph
+1 u(�tj) − ut(�tj− 1
+2 )
+���
+L2(Ω)
+�
+��
+�
+���wj
+I,u+wj
+II,u
+���
+L2(Ω)
+,
+where we denote by wm
+I,u, wm
+II,u the same terms respectively defined by wm
+I , wm
+II (65) but with u instead
+of Ψ
+wm
+I,u := (P1
+h − I)∂u(�tm),
+wm
+II,u := ∂u(�tm) − ut(�tm− 1
+2 ).
+(68)
+We now apply Cauchy-Schwarz inequality
+∆tG
+µ
+m
+∑
+j=1
+���∂uj
+H − ut(�tj− 1
+2 )
+���
+L2(Ω) ≤
+√�tm
+µ
+�� m
+∑
+j=1
+∆tG
+���∂θj���
+2
+L2(Ω)
+�1/2
++
+� m
+∑
+j=1
+∆tG
+���wj
+I,u + wj
+II,u
+���
+2
+L2(Ω)
+�1/2�
+,
+(69)
+≤
+√�tm
+µ
+�� m
+∑
+j=1
+∆tG
+���∂θj���
+2
+L2(Ω)
+�1/2
++
+� m
+∑
+j=1
+∆tG
+���wj
+I,u + wj
+II,u
+���
+2
+L2(Ω) +
+���wj
+III,u
+���
+2
+L2(Ω)
+�1/2�
+* To estimate the first term of (69), we use v = ∂θm
+u , and we now have (from the Crank-Nicolson
+scheme on u (11))
+���∂θm
+u
+���
+2
+L2(Ω) + µ(∇�
+θm
+u , ∇∂θm
+u ) = −(wm
+I,u + wm
+II,u + µwm
+III,u, ∂θm
+u ),
+where
+wm
+III,u := ∆u(�tm− 1
+2 ) − 1
+2(∆u(�tm) + ∆u(�tm−1)),
+(70)
+and θm
+u is the discrete version of (31). By definitions of ∂ and �
+θm
+u , it can be rewritten
+���∂θm
+u
+���
+2
+L2(Ω) +
+µ
+2∆tG
+��∇θm
+u
+��2
+L2(Ω) −
+µ
+2∆tF
+���∇θm−1
+u
+���
+2
+L2(Ω) = −(wm
+I,u + wm
+II,u + µwm
+III,u, ∂θm
+u ),
+and it leads to (using Young’s inequality)
+���∂��m
+u
+���
+2
+L2(Ω) +
+µ
+∆tG
+��∇θm
+u
+��2
+L2(Ω) −
+µ
+∆tG
+���∇θm−1
+u
+���
+2
+L2(Ω) ≤
+���wm
+I,u + wm
+II,u + µwm
+III,u
+���
+2
+L2(Ω) .
+(71)
+Now we find, as in (54) (by summing over all time steps in order to obtain a telescoping sum)
+m
+∑
+j=1
+���∂θj
+u
+���
+2
+L2(Ω) ≤
+µ
+∆tG
+���∇θj
+u
+���
+2
+L2(Ω) +
+m
+∑
+j=1
+���wj
+I,u + wj
+II,u + wj
+III,u
+���
+2
+L2(Ω) ,
+(72)
+18
+
+The term∥∇θm
+u ∥2
+L2(Ω) can easily be bounded by repeated application using (71). We find that (since
+the first term of (71) is positive)
+µ
+∆tG
+��∇θm
+u
+��2
+L2(Ω) ≤
+m
+∑
+j=1
+���wj
+I,u + wj
+II,u + µwj
+III,u
+���
+2
+L2(Ω) ,
+and thus (72) gives
+m
+∑
+j=1
+���∂θj
+u
+���
+2
+L2(Ω) ≤ 2
+m
+∑
+j=1
+���wj
+I,u + wj
+II,u + µwj
+III,u
+���
+2
+L2(Ω) ≤ 4
+m
+∑
+j=1
+���wj
+I,u + wj
+II,u
+���
+2
+L2(Ω) +
+���µwj
+III,u
+���
+2
+L2(Ω) ,
+(73)
+therefore we obtain for (69)
+∆tG
+µ
+m
+∑
+j=1
+���∂uj
+H − ut(�tj− 1
+2 )
+���
+L2(Ω) ≤ 3
+�
+�tm
+��∆tG
+µ2
+m
+∑
+j=1
+���wj
+I,u + wj
+II,u
+���
+2
+L2(Ω) +
+m
+∑
+j=1
+∆tG
+���wj
+III,u
+���
+2
+L2(Ω)
+�1/2�
+,
+which yields
+∆tG
+µ
+m
+∑
+j=1
+���∂uj
+H − ut(�tj− 1
+2 )
+���
+L2(Ω) ≤ 3
+�
+�tm
+�� 2
+µ2
+m
+∑
+j=1
+∆tG
+���wj
+I,u
+���
+2
+L2(Ω) + 2
+µ2
+m
+∑
+j=1
+∆tG
+���wj
+II,u
+���
+2
+L2(Ω)
++
+m
+∑
+j=1
+∆tG
+���wj
+III,u
+���
+2
+L2(Ω)
+�1/2�
+.
+(74)
+* Now, we can estimate the right-hand side terms of (74) as done in [45]. We first remark that
+wj
+I,u = wj
+1,u (and the estimate is given by (57) but with the coarse spatial and time grids), so it
+remains to seek bounds for wj
+II,u and wj
+III,u.
+· For wj
+II,u,
+∆tG
+m
+∑
+j=1
+���wj
+II,u
+���
+2
+L2(Ω) =
+1
+∆tG
+m
+∑
+j=1
+���u(�tj) − u(�tj−1) − ∆tG ut(�tj− 1
+2 )
+���
+2
+L2(Ω) ,
+=
+1
+4∆tG
+m
+∑
+j=1
+������
+� �tj− 1
+2
+�tj−1 (s − �tj−1)2uttt(s) +
+� �tj
+�tj− 1
+2 (s − �tj)2uttt(s) ds
+������
+2
+L2(Ω)
+≤ C∆t3
+G
+m
+∑
+j=1
+�����
+� �tj
+�tj−1 uttt(s) ds
+�����
+2
+L2(Ω)
+,
+≤ C∆t4
+G
+m
+∑
+j=1
+� �tj
+�tj−1∥uttt∥2
+L2(Ω) ds, with Cauchy-Schwarz inequality,
+≤ C∆t4
+G
+� �tm
+�t0 ∥uttt∥2
+L2(Ω) ds.
+(75)
+· For wj
+III,u,
+∆tG
+m
+∑
+j=1
+���wj
+III,u
+���
+2
+L2(Ω) = ∆tG
+m
+∑
+j=1
+����∆u(�tj− 1
+2 ) − 1
+2(∆u(�tj) + ∆u(�tj−1))
+����
+2
+L2(Ω)
+,
+= ∆tG
+4
+m
+∑
+j=1
+������
+� �tj− 1
+2
+�tj−1 (tj−1 − s)∆utt(s) ds +
+� �tj
+�tj− 1
+2 (s − �tj)∆utt(s) ds
+������
+2
+L2(Ω)
+,
+≤ C∆t3
+G
+m
+∑
+j=1
+�����
+� �tj
+�tj−1 ∆utt ds
+�����
+2
+L2(Ω)
+,
+≤ C∆t4
+G
+� �tm
+�t0 ∥∆utt∥2
+L2(Ω) ds, by Cauchy-Schwarz inequality.
+(76)
+19
+
+Altogether,
+T′′
+2,n ≤ C
+� H2
+µ
+� � �tm
+0
+∥ut∥2
+H2(Ω) ds
+�1/2
++ ∆t2
+G
+�� 1
+µ
+� �tm
+0
+∥uttt∥2
+L2(Ω)
+�1/2
++
+� � �tm
+0
+∥∆utt∥2
+L2(Ω)
+�1/2
+ds
+��
+.
+(77)
+– To estimate T′′
+3,n defined in (67), we remark that wj
+I = wj
+1 (but with the coarse spatial and time grids),
+and
+* for wj
+II,
+∆tG
+���wj
+II
+���
+L2(Ω) ≤
+���Ψ(�tj) − Ψ(�tj−1) − ∆tGΨt(�tj− 1
+2 )
+���
+L2(Ω) ,
+= 1
+2
+������
+� �tj− 1
+2
+�tj−1 (s − �tj−1)2Ψttt(s) +
+� �tj
+�tj− 1
+2 (s − �tj)2Ψttt(s) ds
+������
+L2(Ω)
+≤ C∆t2
+G
+� �tj
+�tj−1∥Ψttt∥L2(Ω) ds.
+(78)
+* Finally, for wj
+III,
+∆tG
+���wj
+III
+���
+L2(Ω) = ∆tG
+����Ψ(�tj− 1
+2 ) − 1
+2(Ψ(�tj) + Ψ(�tj−1))
+����
+L2(Ω)
+≤ C∆t2
+G
+� �tj
+�tj−1∥∆Ψtt∥L2(Ω) ds.
+(79)
+Altogether,
+T′′
+3,n ≤ CH2
+� �tm
+0
+∥Ψt∥H2(Ω) ds + C∆t2
+G
+� �tm
+0
+�
+∥Ψttt∥L2(Ω) + µ∥∆Ψtt∥L2(Ω)
+�
+ds,
+(80)
+which concludes the proof (combining (67), (77) and (80)).
+In analogy with the previous work on parabolic equations, we define
+�
+ΨH
+n = I2
+n[Ψm
+H](µ),
+for n = 1, . . . , NT,
+(81)
+with I2
+n defined by (12) as the quadratic interpolation in time of the coarse solution at time tn ∈ Im = [�tm−1,�tm]
+defined on [�tm−2,�tm] from the values Ψm−2
+H
+, Ψm−1
+H
+, and Ψm
+H, for all m = 2, . . . , MT. For tn ∈ I1 = [�t0,�t1], we use
+the same parabola defined by the values Ψ0
+H, Ψ1
+H, ψ2
+H as the one used over [�t1,�t2]. Note that, as before, we could
+have chosen another quadratic interpolation.
+Corollary 2.10 (of Theorem 2.9). Under the assumptions of Theorem 2.9, let um
+H be the fully-discretized solution (11) on
+the coarse mesh TH. Let Ψ and Ψm
+H be the corresponding sensitivities, respectively given by (23) and by (63). Let �
+ΨH
+n be
+the quadratic interpolation of the coarse solution Ψm
+H given by (81). Then,
+∀n = 0, . . . , NT,
+����
+Ψh
+n − Ψ(tn)
+���
+L2(Ω) ≤ CH2����Ψ0���
+H2(Ω) +
+� tn
+0 ∥Ψt∥H2(Ω) ds + C(µ)
+� � tn
+0 ∥ut∥2
+H2(Ω) ds
+�1/2�
++ C∆t2
+G
+� � tn
+0 ∥Ψttt∥L2(Ω) ds +
+� � tn
+0 ∥∆utt∥2
+L2(Ω) ds
+�1/2
++ C(µ)
+�� � tn
+0 ∥uttt∥2
+L2(Ω) ds ]1/2 +
+� tn
+0 ∥∆Ψtt∥L2(Ω) ds
+��
+.
+In the next section, we proceed with the adjoint state formulation.
+2.4
+Sensitivity analysis: The adjoint problem.
+The adjoint method may be seen as an inverse method, where the goal is to retrieve the optimal parameter of
+an objective function F. The objective function will have a different meaning whether the goal is to retrieve the
+parameters from several measures (for parameter identification) or if we want to optimize a function depending
+20
+
+on the variables (PDE-constrained optimization). In the first case, F will have the following form (in its fully-
+discretized form)
+F(µ) = 1
+2
+NT
+∑
+n=1
+��un
+h(µ) − un��2
+L2(Ω)
+�
+��
+�
+∥err(tn; µ)∥
+2
+L2(Ω)
+,
+(82)
+where un refer to the measures, which may be noisy (here for simplicity we consider the case of measures on the
+variables although it may be given by other outputs). In the second setting, it will be written
+F(µ) =
+NT
+∑
+n=1
+gnun
+h(µ),
+(83)
+with gn some suitable weights. Note that by differentiating F with respect to the parameters µp, p = 1, . . . , P, we
+can observe the influence on the objective function of the input parameters through the normalized sensitivity
+coefficients (also called elasticity of P) [4]
+Sk = ∂F
+∂µk
+(µ) ×
+µk
+F(µ), k = 1, . . . , P.
+(84)
+2.4.1
+The continuous setting.
+• Let us for instance consider the first case outlined above, given in the continuous version by
+F(µ) = 1
+2
+� T
+0
+��err(t; µ)
+��2
+L2(Ω) .
+(85)
+• To minimize F under the constraint that u is the solution of our model problem (3), we consider the
+following Lagrangian with (χ, ϕ) the Lagrangian multipliers
+L(u, χ, ϕ; µ) = F(µ) +
+� T
+0 (χ, (∇ · (A(µ)∇u) + f − ut)) ds +
+� T
+0 (ϕ, u)L2(∂Ω) ds,
+(86)
+where
+– χ ∈ V is the multiplier associated to the constraint “u is a solution of (3)”,
+– ϕ ∈ R is the multiplier associated to the constraint of the Dirichlet boundary condition on ∂Ω. Since
+here we consider homogeneous condition, we just impose ϕ = 0.
+• Differentiating L with respect to the parameter µp, for p = 1, . . . , P, we obtain the following adjoint system
+in its variational form (see A for more details)
+�
+�
+�
+�
+�
+Find χ(t) ∈ V for t ∈ [0, T] such that
+(χt(t), v) = −( ∂err
+∂u (t; µ), v) + (A(µ)∇χ(t), ∇v), ∀v ∈ V, t < T,
+χ(·, T) = 0, in Ω.
+(87)
+• After solving (89) with the parameter µ, one can compute dF
+dµp by noticing that
+dF
+dµp
+= dL
+dµp
+=
+� T
+0
+�
+χ, ∇ · ( ∂A
+∂µp
+(µ)∇u)
+�
+ds, from (124).
+(88)
+Remark 2.11. For a stable implementation, one may have to add a regularization term depending of the parameter to
+the cost function F(p).
+21
+
+2.4.2
+Discretized setting.
+In analogy with the direct method, we first discretize the system in space, and then we apply an Euler scheme
+with the fine grids and a Crank-Nicolson scheme with the coarse ones. The semi-discretized version on Th writes
+�
+�
+�
+�
+�
+�
+�
+Find χh(t) ∈ Vh for t ∈ [0, T] such that
+(χh,t(t), vh) − a(χh, vh; µ) = −( ∂errh
+∂uh (t; µ), vh), ∀vh ∈ Vh, t < T,
+χh(·, T) = 0, in Ω.
+(89)
+With the fully-discretized version, on the fine grids, the adjoint system becomes in its variational formulation
+�
+�
+�
+�
+�
+�
+�
+Find χn
+h ∈ Vh for n ∈ {0, . . . , NT} such that
+(∂χn
+h, vh) − a(χh, vh; µ) = −(un
+h − un, vh), ∀n = 0, . . . , NT − 1,
+χNT
+h (·) = 0.
+(90)
+Note that to compute ∂errn
+h
+∂uh , we need the fine solutions un
+h and the measures. As for the state variable (11), we also
+compute the adjoint on the coarse mesh with the Crank-Nicolson scheme,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find χm
+H ∈ VH for m ∈ {0, . . . , MT} such that
+(∂χm
+H, vH) − a( 1
+2(χm
+H + χm−1
+H
+), vH; µ) = − 1
+2
+�
+(um
+H − um, vH) + (um−1
+H
+− um−1, vH)
+�
+), ∀vH ∈ VH, ∀m = 0, . . . , MT − 1, (91)
+χMT
+H (·) = 0.
+Finally, note that the problems (90) and (92) are well-posed, since they are solved backward in time (see [16] for
+precisions in the general setting of time-dependent PDEs).
+The next section adapts the NIRB two-grid algorithm in the context of sensitivity analysis.
+3
+NIRB algorithms applied to sensitivity analysis
+3.1
+On the direct problem.
+3.1.1
+NIRB algorithm.
+Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and Ψp(µ) its sensitivity with respect to
+the parameter µp, p = 1, . . . , P. We consider P parameters of interest. In this context, we use the following
+offline/online decomposition for the NIRB procedure:
+• “Offline part”
+1. For a set of training parameters (�µi)i=1,··· ,Np,train, we define Gp,train =
+∪
+i∈{1,...,Np,train}�µi. Then, through
+a greedy algorithm 1, we adequately choose the parameters of the RB. During this procedure, we
+compute fine fully-discretized solutions {Ψn
+p,h(�µi)}i∈{1,...Nµ,p}, n={0,...,NT} (Nµ,p ≤ Np,train) with the HF
+solver, by solving either (44) or the following problem (where un
+h in (44) has been replaced by its NIRB
+approximation uN,n
+Hh or by its rectified version Rn
+u[uN,n
+Hh ] obtained from the algorithm of section 2.2)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find Ψn
+p,h ∈ Vh for n ∈ {0, . . . , NT} such that
+(∂Ψn
+p,h, vh) + a(Ψn
+p,h, vh; �µ) = −( ∂A
+∂µp (µ)∇uN,n
+Hh (µ), ∇vh) for n = {1, . . . , NT},
+(92)
+Ψ0
+p,h(·) = P1
+hΨ0
+p(·).
+The term −( ∂A
+∂µp (µ)∇uN,n
+Hh (µ), ∇vh) in (92) is replaced by −( ∂A
+∂µp (µ)∇Rn
+u[uN,n
+Hh ](µ), ∇vh) in case of the
+rectification post-treatment. Note that un
+h can directly be used (as in (44)) since this step belongs to the
+offline part of the NIRB algorithm. However, if the number of parameters required for the initial RB is
+22
+
+lower than the number of parameters needed for the sensitivities RB or if one combine the sensitivities
+with an optimization algorithm, it may be convenient to employ (92) instead of (44).
+In analogy to section 2.2, a few time steps may be selected for each parameter of the RB, and thus,
+we obtain Np L2 orthogonal RB (time-independent) functions, denoted (ζh
+p,i)i=1,...,Np, and the reduced
+spaces X
+Np
+p,h := Span{ζh
+p,1, . . . , ζh
+p,Np} for p = 1, . . . , P.
+2. Then, for each p, we solve the eigenvalue problem (14) on X
+Np
+p,h:
+�
+�
+�
+�
+�
+Find ζh ∈ X
+Np
+p,h, and λ ∈ R such that:
+∀v ∈ X
+Np
+p,h,
+�
+Ω ∇ζh · ∇v dx = λ
+�
+Ω ζh · v dx.
+(93)
+For each parameter p ∈ {1, . . . , P}, we get an increasing sequence of eigenvalues λp
+i , and eigenfunc-
+tions (ζh
+p,i)i=1,··· ,Np, orthonormalized in L2(Ω) and orthogonalized in H1(Ω).
+3. As in the offline step 3 from section 2.2, we enhance the NIRB approximation with a rectification post-
+processing. Thus, we introduce the rectification matrices, denoted Rp,n
+Ψ . They are associated to the
+sensitivities problem (44), defined for each p ∈ {1, . . . , P} and each fine time step n ∈ {1, . . . , NT}, and
+constructed from coarse snapshots, generated by solving (63) and whose parameters are the same as
+for the fine snapshots.
+Thus, for all n = 1, . . . , NT and all p = 1, . . . , P, we compute the vectors
+Rp,n
+Ψ,i = ((Ap,n)TAp,n + δpINp)−1(Ap,n)TBp,n
+i
+,
+i = 1, · · · , Np,
+(94)
+where
+∀i = 1, · · · , Np,
+and
+∀�µk ∈ Gp,
+Ap,n
+k,i =
+�
+Ω
+�
+Ψp,H
+n(�µk) · ζh
+p,i dx,
+(95)
+Bp,n
+k,i =
+�
+Ω Ψn
+p,h(�µk) · ζh
+p,i dx,
+(96)
+and where INp refers to the identity matrix and δp is a regularization term (note that we used (81) for
+�
+Ψp,H
+n(�µk)).
+Remark 3.1. In general, Np,train < Np and the parameter δp is required for the inversion of (Ap,n)TAp,n.
+• “Online part”
+The online part of the algorithm is much faster than a double HF evaluation (to seek the sensitivity Ψn
+p,h,
+we also need the solution un
+h with a HF evaluation).
+4. Indeed, we first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time
+step m = 0, . . . , MT using (11).
+5. Then, for each p = 1, . . . , P, we solve the coarse associated sensitivity problems (63) with the same
+parameter µ, at each time step m = 0, . . . , MT.
+6. We quadratically interpolate in time the coarse solution Ψm
+p,H on the fine time grid with (81).
+7. Then, we linearly interpolate �
+Ψp,H
+n(µ) on the fine mesh in order to compute the L2-inner product with
+the basis functions. The approximation used in the two-grid method is
+For n = 0, . . . , NT,
+Ψ
+Np,n
+p,Hh(µ) :=
+Np
+∑
+i=1
+(�
+Ψp,H
+n(µ), ζh
+p,i) ζh
+p,i,
+(97)
+and with the rectification post-treatment step, it becomes
+Rp,n
+Ψ [ΨN
+p,Hh](µ) :=
+Np
+∑
+i,j=1
+Rp,n
+ij
+(�
+Ψp,H
+n(µ), ζh
+p,j) ζh
+p,i,
+(98)
+where Rp,n
+Ψ
+is the rectification matrix at time tn, given by (94).
+23
+
+In the next section, we propose an adaptation of this algorithm with a new post-treatment which reduces the
+online computational time.
+3.1.2
+New NIRB algorithm for the direct problem.
+The main drawback of the algorithm described in the previous section is that it requires 1 + P coarse systems in
+the online part (see the steps 4 and 5 in section 3.1.1). The online portion of the new algorithm described below
+only requires the resolution of two coarse problems, regardless the number of parameters of interest. We refer to
+the following offline/online decomposition:
+• “Offline part”
+1. For a parameter training set Gtrain, we compute the RB functions of the initial problem, denoted
+(Φh
+i )i=1,...,N and generates XN
+h by the steps 1-2 of 2.2 (see algorithm 1).
+2. As before, from the training sets Gp,train, we generate the reduced spaces X
+Np
+p,h, for p = 1, . . . , P using
+steps 1 and 2 of section 3.1.1, and at the end of this part, we obtain Np RB functions (time-independent),
+denoted (ζh
+p,i)i=1,...,Np for each p = 1, . . . , P. We introduce GT defined by
+GT := Gtrain ∩ Gp,train,
+(99)
+and Nµ,T the number of parameters in GT.
+3. We use the fact that the sensitivities are directly derived from the initial solutions, and we consider
+new rectification matrices, denoted �Rp,n and defined for each p ∈ {1, . . . , P} and each fine time step
+n ∈ {1, . . . , NT}. In this new post-treatment, they are constructed from coarse snapshots of the initial
+solution, generated by solving (11) and whose parameters are the same as for the fine sensitivities,
+generated by solving (63).
+Thus, for all n = 1, . . . , NT and all p = 1, . . . , P, we compute the vectors
+�Rp,n
+i
+= ((An)TAn + δIN)−1(An)TBp,n
+i
+,
+i = 1, · · · , Np,
+(100)
+where this time
+∀�µk ∈ GT,
+An
+k,i =
+�
+Ω
+�
+uH
+n(�µk) · Φh
+i dx, ∀i = 1, · · · , N,
+(101)
+Bp,n
+k,i =
+�
+Ω Ψn
+p,h(�µk) · ζh
+p,i dx, ∀i = 1, · · · , Np,
+(102)
+and where IN refers to the identity matrix and δ is a regularization term (required for the inversion of
+(An)TAn). Note that �
+uH
+n(�µk) is the quadratic interpolation given by (12). We highlight the fact that
+this step requires that GT ̸= ∅ (99).
+• “Online step”
+4. We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m =
+0, . . . , MT using (11).
+5. We quadratically interpolate in time the coarse solution um
+H on the fine time grid with (12).
+6. Then, we linearly interpolate �
+uH
+n(µ) on the fine mesh in order to compute the L2-inner product with
+the basis functions. The new NIRB approximation is given by
+�Rp,n[ΨN
+p,Hh](µ) :=
+Np
+∑
+i=1
+N
+∑
+j=1
+�Rp,n
+ij
+( �
+uH
+n(µ), Φh
+j ) ζh
+p,i,
+(103)
+where �Rp,n is the rectification matrix at time tn, given by (100).
+24
+
+3.2
+On the adjoint formulation.
+The adjoint formulation requires some modifications of the NIRB algorithm compared to section 3.1.1. Since in
+(88), for all n ∈ {0, . . . , NT}, the fine solution un
+h(µ) is required to obtain the sensitivities on F, it follows that
+here we have to compute two reductions: one for the initial solution u and one for the adjoint χ. As a matter of
+fact, in (3.1.1), the RB generation for u was optional.
+So let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and χ(µ) its adjoint given by (89). In this
+setting, we use the following offline/online decomposition for the NIRB procedure:
+• “Offline part”
+1. During the offline stage, we first construct the reduced space XN
+h and the RB function (Φh
+1, . . . , Φh
+N)
+with the steps 1-2 of section 2.2.
+2. Then, we use steps 1-2 of section 3.1.1, but instead of solving (44) on the sensitivities, we generate the
+reduced space XN1
+1
+by solving the adjoint problem on the fine mesh (90).
+Thus, for a set of training parameters (�µi)i=1,··· ,N1,train, we define G1,train =
+∪
+i∈{1,...,N1,train}�µi.
+Then,
+through a greedy procedure 1, we adequately choose the parameters of the RB. During this proce-
+dure, we compute fine fully-discretized solutions {χn
+h(�µi)}i∈{1,...Nµ,1}, n={0,...,NT} (Nµ,1 ≤ N1,train) with
+the HF solver, by solving either (90) or the following problem (where un
+h in (90) has been replaced by
+its NIRB approximation uN,n
+Hh or by its rectified version Rn
+u[uN,n
+Hh ] obtained from the algorithm of section
+2.2)
+�
+�
+�
+�
+�
+�
+�
+Find χn
+h ∈ Vh for n ∈ {0, . . . , NT} such that
+(∂χn
+h, vh) − a(χh, vh; µ) = −(uN,n
+Hh − un, vh), ∀n = 0, . . . , NT − 1,
+χNT
+h (·) = 0.
+(104)
+The term −(uN,n
+Hh (µ) − un, vh) in (104) is replaced by −(Rn
+u[uN,n
+Hh ](µ) − un, vh) in case of the rectification
+post-treatment. In practice, since in step 1 a RB for un
+h has already been generated, it is more convenient
+to employ (104) instead of (90).
+In analogy to section 2.2, a few time steps may be selected for each parameter of the RB, and thus,
+we obtain N1 L2 orthogonal RB (time-independent) functions, denoted (ξh
+i )i=1,...,N1, and the reduced
+space XN1
+h
+:= Span{ξh
+1, . . . , ξh
+N1}.
+3. Then, we solve the eigenvalue problem (14) on XN1
+h :
+�
+�
+�
+�
+�
+Find ξh ∈ XN1
+h , and λ ∈ R such that:
+∀v ∈ XN1
+h ,
+�
+Ω ∇ξh · ∇v dx = λ
+�
+Ω ξh · v dx.
+(105)
+We get an increasing sequence of eigenvalues λi, and eigenfunctions (ξh
+i )i=1,··· ,N1, orthonormalized in
+L2(Ω) and orthogonalized in H1(Ω).
+4. As in the offline step 3 from section 3.1.1, we enhance the NIRB approximation with a rectifica-
+tion post-processing. Thus, we introduce a rectification matrix, denoted Rn
+χ for each fine time step
+n ∈ {1, . . . , NT}. It is associated to the adjoint problem (90) and constructed from coarse snapshots,
+generated by solving (92) and whose parameters are the same as for the fine snapshots.
+Thus, for all n = 1, . . . , NT, we compute the vectors
+Rn
+χ,i = ((An)TAn + δIN1)−1(An)TBn
+i ,
+i = 1, · · · , N1,
+(106)
+where
+∀i = 1, · · · , N1,
+and
+∀�µk ∈ Gp,
+An
+k,i =
+�
+Ω
+�
+χH
+n(�µk) · ξh
+i dx,
+(107)
+Bn
+k,i =
+�
+Ω χn
+h(�µk) · ξh
+i dx,
+(108)
+25
+
+and where IN1 refers to the identity matrix and δp is a regularization term required for the inversion
+of (An)TAn (note that we used (81) for �
+χH
+n(�µk)).
+• “Online part”
+4. We first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step
+m = 0, . . . , MT using (11).
+5. Then, we solve the coarse associated adjoint problem (92) with the same parameter µ, at each time
+step m = 0, . . . , MT.
+6. We quadratically interpolate in time the coarse solution χm
+H on the fine time grid with (81).
+7. Then, we linearly interpolate �
+χH
+n(µ) on the fine mesh in order to compute the L2-inner product with
+the basis functions. The approximation used for the adjoint in the two-grid method is
+For n = 0, . . . , NT,
+χN1,n
+Hh (µ) :=
+N1
+∑
+i=1
+( �
+χH
+n(µ), ξh
+i ) ξh
+i ,
+(109)
+and with the rectification post-treatment step, it becomes
+Rn
+χ[χN1
+Hh](µ) :=
+N1
+∑
+i,j=1
+Rn
+χ,ij ( �
+χH
+n(µ), ξh
+j ) ξh
+i ,
+(110)
+where Rn
+χ is the rectification matrix at time tn, given by (106).
+8. Then, we use the steps 5 and 6 of section 2.2 in order to obtain a NIRB approximation for u(µ) from
+the coarse solution um
+H given by step 4 of this online part.
+9. Finally, the sensitivities NIRB approximations of F are given by
+for p = 1, . . . , P, [ ∂F
+∂µp
+]N1
+Hh(µ) :=
+tn
+∑
+j=1
+∆tF
+�
+χN1,j
+Hh , ∇ · ( ∂A
+∂µp
+(µ)∇uN,j
+Hh)
+�
+, from (88),
+(111)
+and with the rectification post-treatment step, it becomes
+for p = 1, . . . , P, Rχ[[ ∂F
+∂µp
+]N1
+Hh](µ) :=
+tn
+∑
+j=1
+∆tF
+�
+Rj
+χ[χN1,j
+Hh ](µ), ∇ · ( ∂A
+∂µp
+(µ)∇Rj
+u[uN,j
+Hh](µ))
+�
+.
+(112)
+The next section gives our main result on the NIRB two-grid method error estimate in the context of sensitivity
+analysis.
+4
+NIRB error estimate on the sensitivities
+Main result
+Our main result is the following theorem.
+Theorem 4.1. (NIRB error estimate for the sensitivities.) Let A(µ) = µ Id, with µ ∈ R+∗ , and let us consider the problem
+3 with its exact solution u(x, t; µ), and the full discretized solution un
+h(x; µ) to the problem 10. Let Ψ(x, t; µ) and Ψn
+h(x; µ)
+respectively by the corresponding sensitivities , given by (23) and (44). Let (ζh
+i )i=1,...,N1 be the L2-orthonormalized and
+H1-orthogonalized RB generated with the greedy algorithm 1 through the NIRB algorithm 3.1.1. Let us consider the NIRB
+approximation,
+For n = 0, . . . , NT,
+ΨN,n
+Hh (µ) :=
+N1
+∑
+i=1
+(�
+ΨH
+n(µ), ζh
+i ) ζh
+i ,
+(113)
+where �
+ΨH
+n(µ) is given by (81). Then, the following estimate holds
+∀n = 0, . . . , NT,
+���Ψ(tn)(µ) − ΨN,n
+Hh (µ)
+���
+H1(Ω) ≤ ε(N) + C1(µ)h + C2(µ, N)H2 + C3(µ)∆tF + C4(µ, N)∆t2
+G,
+(114)
+where C1, C2, C3 and C4 are constants independent of h and H, ∆tF and ∆tG. The term ε depends on the Kolmogorov
+N-width and measures the error given by (21).
+26
+
+If H is such as H2 ∼ h, ∆t2
+G ∼ ∆tF, and ε(N) is small enough, with C2(µ, N) and C4(µ, N) not too large, it
+results in an error estimate in O(h + ∆tF). Theorem 4.1 then states that we recover optimal error estimates in
+L∞(0, T; H1(Ω)). We now go on with the proof of Theorem 4.1.
+Proof. The NIRB approximation at time step n = 0, . . . , NT, for a new parameter µ ∈ G is defined by (97). Thus,
+the triangle inequality gives
+���Ψ(tn)(µ) − ΨN,n
+Hh (µ)
+���
+H1(Ω) ≤
+��Ψ(tn)(µ) − Ψn
+h(µ)
+��
+H1(Ω) +
+���Ψn
+h(µ) − ΨN,n
+hh (µ)
+���
+H1(Ω) +
+���ΨN,n
+hh (µ) − ΨN,n
+Hh (µ)
+���
+H1(Ω)
+=: T1 + T2 + T3,
+(115)
+where ΨN1,n
+hh
+(µ) =
+N1
+∑
+i=1
+(Ψn
+h(µ), ζh
+i ) ζh
+i .
+• The first term T1 may be estimated using the inequality given by Theorem 2.8, such that
+��Ψ(tn)(µ) − Ψn
+h(µ)
+��
+H1(Ω) ≤ C(µ) (h + ∆tF).
+(116)
+• We then denote by S′
+h = {Ψn
+h(µ, t), µ ∈ G, n = 0, . . . NT} the set of all the sensitivities . For our model
+problem, this manifold has a low complexity.
+It means that for an accuracy ε = ε(N) related to the
+Kolmogorov N-width of the manifold S′
+h, for any µ ∈ G, and any n ∈ 0, . . . , NT, T2 is bounded by ε which
+depends on the Kolmogorov N-width.
+T2 =
+������
+Ψn
+h(µ) −
+N1
+∑
+i=1
+(Ψn
+h(µ), ζh
+i ) ζh
+i
+������
+H1(Ω)
+≤ ε(N).
+(117)
+• Since (ζh
+i )i=1,...,N1 is a family of L2 and H1 orthogonalized RB functions (see [19] for only L2 orthonormalized
+RB functions)
+���ΨN,n
+hh − ΨN,n
+Hh
+���
+2
+H1(Ω) =
+N1
+∑
+i=1
+|(Ψn
+h(µ) − �
+ΨH
+n(µ), ζh
+i )|2���ζh
+i
+���
+2
+H1(Ω) ,
+(118)
+where �
+ΨH
+n(µ) is the quadratic interpolation of the coarse snapshots on time tn, ∀n = 0, . . . , NT, defined by
+(81). From the RB orthonormalization in L2, the equation (105) yields
+���ζh
+i
+���
+2
+H1 :=
+���∇ζh
+i
+���
+2
+L2(Ω) = λi
+���ζh
+i
+���
+2
+L2(Ω) = λi ≤ max
+i=1,··· ,Nλi = λN,
+(119)
+such that the equation (118) leads to
+���ΨN,n
+hh − ΨN,n
+Hh
+���
+2
+H1(Ω) ≤ CλN
+���Ψn
+h(µ) − �
+ΨH
+n(µ)
+���
+2
+L2(Ω) .
+(120)
+Now by definition of �
+ΨH
+n(µ) and by corollary 2.10 and Theorem 2.7, for tn ∈ Im,
+���Ψn
+h(µ) − �
+ΨH
+n(µ)
+���
+L2(Ω) ≤ C(µ)(H2 + ∆t2
+G + h2 + ∆tF),
+(121)
+and we end up for equation (120) with
+���ΨN,n
+hh − ΨN,n
+Hh
+���
+H1(Ω) ≤ C(µ)
+�
+λN(H2 + ∆t2
+G + h2 + ∆tF),
+(122)
+where C(µ) does not depend on N. Combining these estimates (116), (117) and (122) concludes the proof
+and yields the estimate (114).
+27
+
+Figure 1: H1
+0 NIRB errors
+5
+Numerical results.
+In this section, we have applied the NIRB algorithms on several numerical tests. We have implemented both
+schemes (Euler and RK2) using FreeFem++ (version 4.9) [26] to compute the fine and coarse snapshots, and the
+solutions have been stored in VTK format.
+Then we have applied the plain NIRB and the NIRB rectified algorithms with python, in order to highlight
+the non-intrusive side of the two-grid method (as in [19]). After saving the NIRB approximations with Paraview
+module on Python, the errors have been computed with FreeFem++.
+5.1
+On the heat equation.
+We have solved (3) and (23) on the parameter set G = [0.5, 9.5], with u0 solution of Poisson’s equation −∆u0 = f
+and Φ0 = 0. We have retrieved several snapshots on t = [0, 2] (note that the coarse time grid must belong to the
+interval of the fine one), and tried our algorithms on several size of meshes, always with ∆tF ≃ h and ∆tG ≃ H
+(both schemes are stables), and such that h = H2.
+• We have taken 18 parameters in G for the RB construction such that µi = 0.5i, i = 1, . . . , 19, i ̸= 2 and a
+reference solution to problem (92), with µ = 1 and its mesh and time step such that hre f ≃ ∆tF,re f = 0.001.
+In figure Figure 1, we present the errors of the FEM solutions and compare them to the one obtained with
+the NIRB algorithm with the rectification to observe the convergence rate.
+A
+Derivation of the adjoint for the heat equation.
+In this appendix, we recall the main steps to derive the adjoint of our model problem, in order to compute
+( ∂F
+∂µk )k=1,...,P. For a more general problem, we refer to [44] in case of FEM.
+28
+
+FEM H relative errors
+NIRB H relative error with H = V h
+0.80-
+0.80-
+h
+h
+FEM coarse error
+NiRB+rectification
+0.50 -
+0.50-
+FEM fine error
+0.20 -
+0.20 
+Error (log
+Error (log 
+0.05-
+0.05
+10-2
+10-1
+10-2
+10-1
+h (size of the fine mesh)
+h (size of the fine mesh)• We consider the Lagrangian formulation (86), denoted by L.
+• Differentiating L with respect to the parameter µp, for p = 1, . . . , P, we obtain
+dL
+dµp
+(u, χ, ζ; µ) =
+� T
+0
+� �
+Ω
+derr
+dµp
+(µ) dx +
+�
+χ, d[∇ · (A(µ)∇u) + f − ut]
+dµp
+��
+ds.
+(123)
+In our setting, the objective does not depend directly on the parameter µp. The time and the parameter
+derivatives can commute ( d
+dt
+�
+du
+dµp
+�
+=
+d
+dµp
+�
+du
+dt
+�
+), and since f is independent of µ, the term linked to f
+vanishes. Therefore, using the chain rule, it may be rewritten
+dL
+dµp
+(u, χ, ζ; µ) =
+� T
+0
+��∂err
+∂u , Ψp
+�
++
+�
+χ, ∇ · ( ∂A
+∂µp
+(µ)∇u)
+�
++
+�
+χ, ∂[∇ · (A(µ)∇u)]
+∂u
+Ψp
+�
+−
+�
+χ, Ψp,t
+�
+�
+��
+�
+TIBP
+�
+ds,
+(124)
+where
+Ψp(t, x; µ) := ∂u
+∂µp
+(t, x; µ).
+As we saw before, a classical forward sensitivity computation would require P + 1 systems of PDEs to
+solve. Here, we want to avoid calculating the sensitivities of the state variables. To do so, the strategy of the
+adjoint method is to factorize all the terms depending on Ψp, and to impose them to vanish by adequately
+choosing χ (which is arbitrary). By IBP on TIBP,
+� T
+0
+�
+Ω χ · Ψp,t dx ds =
+�
+Ω
+�
+χ(T) · Ψp(T) − χ(0) · Ψp(0)
+�
+dx −
+� T
+0
+�
+Ω χt · Ψp dx ds ,
+and choosing χ(T) = 0, and since in our example, u0 does not depend on µ, it yields
+dL
+dµp
+(u, χ, ζ; µ) =
+� T
+0
+��∂err
+∂u , Ψp
+�
++
+�
+χ, ∇ · ( ∂A
+∂µp
+(µ)∇u)
+�
++
+�
+χ, ∂[∇ · (A(µ)∇u)]
+∂u
+Ψp
+�
++
+�
+χt, Ψp
+����
+ds.
+Thus, we want the following term to vanish
+� T
+0
+�
+Ω
+�∂err
+∂u · Ψp + χ∂[∇ · (A(µ)∇u)]
+∂u
+· Ψp
+�
+��
+�
+TGF
++χt · Ψp
+�
+dx ds.
+(125)
+Now, applying Green’s formula twice, we have
+� T
+0
+�
+Ω TGF dx ds =
+� T
+0
+�
+Ω χ∇ · (A(µ)∇Ψp) dxds =
+� T
+0
+�
+−
+�
+Ω A(µ)∇χ · ∇Ψp dx +
+�
+∂Ω A(µ)χ · ∇nΨp dσ
+�
+ds ,
+=
+� T
+0
+� �
+Ω ∇ · (A(µ)∇χ) · Ψp dx −
+�
+∂Ω A(µ)∇nχ · Ψp dσ +
+�
+∂Ω A(µ)χ · ∇nΨp dσ
+�
+ds ,
+with ∇n(·) the normal derivative. Therefore, from the initial boundary conditions, since ∀t ≥ 0, ∀µ ∈ G,
+u(t) = 0 on ∂Ω, we also have Ψp(t) = 0 on ∂Ω and by imposing χ = 0 on ∂Ω, (125) becomes
+� T
+0
+�
+Ω
+��∂err
+∂u + ∇ · (A(µ)∇χ) + χt
+�
+· Ψp
+�
+dxds = 0,
+and this equation leads us to the following adjoint state problem
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find χ ∈ V such that
+χt = − ∂err
+∂u − ∇ · (A(µ)∇χ), in Ω × [0, T[,
+χ(x, T) = 0, in Ω,
+χ(x, t) = 0, on ∂Ω × [0, T[.
+(126)
+29
+
+Acknowledgment
+This work is supported by the SPP2311 program. We would like to give special thanks to Ole Burghardt for his
+precious help on Automatic Differentiation.
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+

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@@ -0,0 +1,835 @@
+Compact and scalable polarimetric self-
+coherent receiver using dielectric metasurface 
+GO SOMA,1,4 YOSHIRO NOMOTO,2 TOSHIMASA UMEZAWA,3 YUKI YOSHIDA,3 
+YOSHIAKI NAKANO,1 AND TAKUO TANEMURA1,5 
+1School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 
+2Central Research Laboratory, Hamamatsu Photonics K.K. 5000 Hirakuchi, Hamakita-ku, Hamamatsu 
+City, Shizuoka, Japan 
+3National Institute of Information and Communications Technologies (NICT), 4-2-1 Nukui-Kitamachi, 
+Koganei, Tokyo, Japan 
+4soma@hotaka.t.u-tokyo.ac.jp, 5tanemura@ee.t.u-tokyo.ac.jp 
+Abstract: The polarimetric self-coherent system using a direct-detection-based Stokes-vector 
+receiver (SVR) is a promising technology to meet both the cost and capacity requirements of 
+the short-reach optical interconnects. However, conventional SVRs require a number of optical 
+components to detect the state of polarization at high speed, resulting in substantially more 
+complicated receiver configurations compared with the current intensity-modulation-direct-
+detection (IMDD) counterparts. Here, we demonstrate a simple and compact polarimetric self-
+coherent receiver based on a thin dielectric metasurface and a photodetector array (PDA). With 
+a single 1.05-m-thick metasurface device fabricated on a compact silicon-on-quartz chip, we 
+implement functionalities of all the necessary passive components: a 1×3 splitter, three 
+polarization beam splitters with different polarization bases, and six focusing lenses. Combined 
+with a high-speed PDA, we demonstrate self-coherent transmission of 20-GBd 16-ary 
+quadrature amplitude modulation (16QAM) and 50-GBd quadrature phase-shift keying 
+(QPSK) signals over a 25-km single-mode fiber. Owing to the surface-normal configuration, it 
+can easily be scaled to receive spatially multiplexed channels from a multicore fiber or a fiber 
+bundle, enabling compact and low-cost receiver modules for the future highly parallelized self-
+coherent systems. 
+ 
+1. Introduction 
+Rapid spread of cloud computing, high-vision video streaming, and 5G mobile services has led 
+to a steady increase in information traffic in the datacenter interconnects and access networks 
+[1]. While intensity-modulation direct-detection (IMDD) formats such as 4-level pulse 
+amplitude modulation (PAM4) are employed in the current short-reach optical links, scaling 
+these IMDD-based transceivers beyond Tb/s is challenging due to the limited spectral 
+efficiency and severe signal distortion caused by the chromatic dispersion of fibers. On the 
+other hand, the digital coherent systems used in metro and long-haul networks can easily 
+expand the capacity by utilizing the full four-dimensional signal space of light and complete 
+compensation of linear impairments through digital signal processing (DSP). However, 
+substantially higher cost, complexity, and power consumption of coherent transceivers have 
+hindered their deployment in short-reach optical interconnects and access networks. 
+To address these issues, the self-coherent transmission scheme has emerged as a promising 
+approach that bridges the gap between the conventional IMDD and coherent systems [2-7]. In 
+this scheme, a continuous-wave (CW) tone is transmitted together with a high-capacity 
+coherent signal, which are mixed at a direct-detection-based receiver to recover the complex 
+optical field of the signal. Unlike the full coherent systems, this scheme eliminates the need for 
+a local oscillator (LO) laser at the receiver side as well as the stringent requirement of using 
+wavelength-tuned narrow-linewidth laser sources, suggesting that substantially low-cost broad-
+
+linewidth uncooled lasers can be used [4]. In addition, since the impacts of laser phase noise 
+and frequency offsets are mitigated, the computational cost of DSP can be reduced significantly 
+[7, 8]. The self-coherent systems thus enable low-cost, low-power-consumption, yet high-
+capacity data transmission, required in the future datacenter interconnects and access networks. 
+Among several variations of implementing self-coherent systems, the polarimetric scheme 
+using a Stokes-vector receiver (SVR) [9-11] has an advantage in terms of simplicity. In this 
+scheme, the coherent signal is transmitted on a single polarization state, together with a CW 
+tone on the orthogonal polarization state. By retrieving the Stokes parameters 𝐒 = [𝑆1, 𝑆2, 𝑆3] 
+at the receiver side, the in-phase-and-quadrature (IQ) signal is demodulated through the DSP 
+after compensating for the effects of polarization rotation, chromatic dispersion, and other 
+signal distortions. To date, a number of high-speed polarimetric self-coherent transmission 
+experiments have been reported, where the SVRs were implemented using off-the-shelf 
+discrete components [3, 10-12]. Toward practical use, integrated waveguide-based SVRs were 
+also realized on Si [13, 14] and InP [15-18]. More recently, surface-normal SVRs were 
+demonstrated using nanophotonic circuits [19, 20] and liquid crystal gratings [21] with external 
+photodetectors (PDs). Compared with the conventional low-cost IMDD receivers, however, 
+these devices still suffer from a large fiber-to-chip coupling loss and/or need for external lenses 
+to focus light to PDs. 
+In this paper, we demonstrate high-speed polarimetric self-coherent signal detection using 
+a compact surface-normal SVR, composed of a metasurface-based polarization-sorting device 
+and a high-speed two-dimensional photodetector array (2D-PDA). A metasurface is a two-
+dimensional array of subwavelength structures that can locally change the intensity, phase, and 
+polarization of input light [22]. Unlike the previous works on metasurface-based polarimeters 
+for imaging and sensing applications [23-27], our device enables efficient coupling of a self-
+coherent optical signal from a single-mode fiber (SMF) and lens-less focusing to six high-speed 
+PDs. More specifically, by superimposing three types of meta-atom arrays, it implements the 
+functionalities of all the necessary passive components, namely a 1×3 splitter, three polarization 
+beam splitters (PBSs) with different polarization bases, and six lenses, inside a single ultrathin 
+device. Combined with an InP/InGaAs-based 2D-PDA chip, we demonstrate penalty-free 
+transmission of polarimetric self-coherent signals over a 25-km SMF in various formats such 
+as 20-GBd 16-ary quadrature amplitude modulation (16QAM) and 50-GBd quadrature phase-
+shift keying (QPSK). Owing to the surface-normal configuration with the embedded focusing 
+functionality, highly efficient lens-free coupling to the 2D-PDA is achieved. The demonstrated 
+SVR, therefore, has a comparable complexity as a conventional low-cost IMDD receiver that 
+fits in a compact receiver optical subassembly (ROSA). Moreover, it can readily be extended 
+to receive spatially multiplexed channels from a multicore fiber (MCF) or a fiber bundle, which 
+are expected in the future >Tb/s highly parallelized optical interconnects [28-31]. 
+ 
+2. Device concept 
+The schematic of the proposed surface-normal SVR is illustrated in Fig. 1(a). The light from 
+an SMF is incident to a thin metasurface-based polarization-sorting device, which is designed 
+to provide the same functionality as a conventional polarimeter shown in the inset. Namely, it 
+splits the light into three paths, resolves each of them to the orthogonal components in three 
+different polarization bases, and focuses them to six PDs integrated on a 2D-PDA chip. Unlike 
+previously demonstrated metasurface-based polarimeters [23-27], our proposed SVR 
+implements the 13 splitter and six metalenses as well to enable direct coupling from an SMF 
+to a high-speed 2D-PDA. As a result, the entire device can fit inside a compact ROSA module, 
+comparable to the current IMDD receivers. Moreover, owing to the surface-normal 
+configuration, this scheme can easily be scaled to receive multiple spatial channels without 
+increasing the number of components by simply replacing the input SMF to a MCF or a fiber 
+bundle and using a larger-scale PDA [32] as shown in Fig. 1(b). 
+
+To enable three operations in parallel using a single metasurface layer, we adopt the spatial 
+multiplexing method [33, 34]; three independently designed meta-atom arrays are 
+superimposed as shown by MA1 (red), MA2 (blue), and MA3 (green) in Fig. 1(c). The phase 
+profile 𝜑(𝑥, 𝑦) of MA1 is designed to focus the x-polarized component of light to PDx and the 
+y-polarized component to PDy at the focal plane as shown in the inset. Similarly, MA2 and 
+MA3 function as PBSs with embedded metalenses for the ±45° polarization basis (a/b) and the 
+right/left-handed circular (RHC/LHC) polarization basis (r/l), respectively, and focus 
+respective components to PDa,b and PDr,l. The Stokes vector 𝐒 ≡ (𝑆1, 𝑆2, 𝑆3)𝑇 can then be 
+derived by taking the difference of the photocurrent signals as 𝑆1  = 𝐼x  − 𝐼y, 𝑆2 = 𝐼a − 𝐼b, and 
+𝑆3 = 𝐼r − 𝐼l, where 𝐼p is the photocurrent at PDp. We should note that this scheme with three 
+balanced PDs without polarizers offers the maximum receiver sensitivity among various SVR 
+configurations [35] and is advantageous compared with the previous demonstrations that 
+employ a non-optimal polarization basis [19-21]. 
+ 
+   
+  
+Fig. 1. Surface-normal SVR based on superimposed meta-atom arrays. (a) Schematic illustration 
+of the receiver module. A single metasurface device implements all the necessary passive optical 
+components of the equivalent circuit as shown in the inset. MS: metasurface. PDA: 
+photodetector array. IC: integrated circuit. HWP: half-wave plate. QWP: quarter-wave plate. 
+PBS: polarization beam splitter. (b) Scalable configuration to receive multiple input channels 
+from a MCF or fiber bundle.  (c) Functionality and configuration of the designed metasurface. 
+The incident light from the SMF is split into three paths and focused to six PDs located at 
+different positions according to the input state of polarization. The superimposed meta-atom 
+arrays (MA1, 2, and 3) operate as PBSs and metalenses for x/y linear, ±45° linear, and RHC/LHC 
+polarization bases, respectively. 
+ 
+ 
+
+ROSA
+MS
+2D-PDA
+Atfocal plane
+2D-PDA
+PDr
+PD,
+S
+PDp
+PDx
+PDa
+PDy
+Si
+K
+K
+Quartz
+MS
+K
+MA1
+PMA2
+MA3
+ROSA
+MS
+2D-PDA
+ICF
+r
+ber
+ndle
+a3. Metasurface design and fabrication 
+As the dielectric metasurface, we employ 1050-nm-high elliptical Si nanoposts on a quartz 
+layer. The phase of the transmitted light and its polarization dependence can be controlled by 
+changing the lengths of two principal axes (𝐷𝑢, 𝐷𝑣) and the in-plane rotation angle θ of each 
+nanopost as defined in Fig. 2(a) [22]. Here, in each meta-atom array, MA1-3, we adopt the 
+triangular lattice with a sub-wavelength lattice constant of Λ = 700√3 nm, so that the non-
+zero-order diffraction is prohibited. Then, three meta-atom arrays are superimposed by shifting 
+their positions by 𝑎 = 700 nm to form the overall metasurface, as shown in Fig. 1(c). 
+First, we set θ to 0 and simulate the transmission characteristics of uniform nanopost array 
+for the x- and y-polarized light at a wavelength of 1550 nm by the rigorous coupled-wave 
+analysis (RCWA) method [36]. From the simulated results, we first derive 𝑡𝑢(𝐷𝑢, 𝐷𝑣) and 
+𝑡𝑣(𝐷𝑢, 𝐷𝑣), which denote the complex transmittance for the x- and y-polarized light as a function 
+of 𝐷𝑢 and 𝐷𝑣. Then, we derive the required (𝐷𝑢, 𝐷𝑣) that provides a phase shift of (𝜑𝑢, 𝜑𝑣) for 
+each polarization component. The results are plotted in Fig. 2(b) (see Section S1 of Supplement 1 
+for details). The amplitude of transmittance for each case is also shown in Fig. 2(c). We can confirm 
+that by setting the dimensions of the ellipse appropriately, arbitrary phase shifts for x- and y-
+polarized components can be achieved with high transmittance.  
+By rotating the elliptical nanoposts by θ as shown in Fig. 2(a), such birefringence can be applied 
+to any linear polarization basis oriented at an arbitrary angle [37]. We should note that the phase 
+shifts and amplitudes of transmission are nearly insensitive to θ [22] and similar results as 
+shown in Fig. 2(b) and 2(c) are obtained for all θ. This is because the light is strongly confined 
+inside each Si nanopost, so that the optical coupling among neighboring meta-atoms has only 
+minor influence on the transmission. 
+We can also provide arbitrary phase shifts to orthogonal circular-polarization states by using the 
+geometric phase shift of meta-atoms [38]. First, we judiciously select 𝐷𝑢 and 𝐷𝑣 to satisfy 𝜑𝑣 =
+𝜑𝑢 + 𝜋, so that each nanopost operates as a half-wave plate. In this case, input RHC and LHC states 
+are converted to LHC and RHC, respectively. In addition, their phases after transmission are written 
+as (𝜑𝑟, 𝜑𝑙) = (𝜑𝑢 + 2𝜃, 𝜑𝑢 − 2𝜃) (see Section S2 of Supplement 1 for the derivation). Therefore, 
+𝐷𝑢 and 𝐷𝑣 of each nanopost are selected to obtain desired 𝜑𝑢 (=(𝜑𝑟 + 𝜑𝑙)/2) while satisfying the 
+condition 𝜑𝑣 = 𝜑𝑢 + 𝜋. The angle 𝜃 is also determined to be (𝜑𝑟 − 𝜑𝑙)/4. 
+To realize the function of a metalens, each meta-atom array needs to impart a spatially 
+dependent phase profile given as [39] 
+ 
+𝜑(𝑥, 𝑦) = −
+2𝜋
+𝜆 (√(𝑥 − 𝑥0)2 + (𝑦 − 𝑦0)2 + 𝑓2 − 𝑓), 
+(1) 
+where (𝑥0, 𝑦0) is the in-plane position of the focal point, 𝑓 is the focal length, and 𝜆 is the 
+operating wavelength. In this work, we set 𝜆 = 1550 nm, 𝑓 = 10 mm, and the diameter of the 
+entire metasurface area to be 2 mm, corresponding to the numerical aperture (NA) of ~0.10. 
+The six focal points are arranged on a regular hexagon with a spacing of 60 µm, which are 
+matched to the positions of the high-speed 2D-PDA used in our self-coherent experiments. 
+Under these conditions, the phase profiles required for MA1, 2, and 3 are determined as shown 
+in Fig. 2(d). Note that a rather large (2 mm) metasurface is used in this work due to the limitation 
+in reducing the focal length 𝑓 in the current optical setup. In a fully packaged module as shown 
+in Fig. 1(a), we can readily shrink the entire area of the metasurface to a few tens of micrometers 
+by reducing 𝑓 and designing the geometrical parameters of each nanopost to satisfy the required 
+phase profiles given by Eq. (1). 
+The designed metasurface was fabricated using a silicon-on-quartz (SOQ) substrate with a 
+1050-nm-thick Si layer. The nanopost patterns were defined by electron-beam lithography with 
+ZEP520A resist. Then, the patterns were transferred to the Si layer by inductively-coupled-
+plasma reactive-ion etching (ICP-RIE) using SF6, C4F8, and O2. An optical microscope image 
+and scanning electron microscopy (SEM) images of the fabricated metasurface are shown in 
+Fig. 2(e)-(g). 
+
+ 
+ 
+Fig. 2. Metasurface design and fabrication. (a) Schematic of a periodic array of Si nano-posts 
+placed at the vertices of a triangular lattice with a lattice constant 𝑎 of 700 nm. The transmission 
+of x- and y-polarized light is simulated for various axes lengths (𝐷𝑢, 𝐷𝑣) of the elliptical posts. 
+(b) Required (𝐷𝑢, 𝐷𝑣) to obtain phase shifts (𝜑𝑢, 𝜑𝑣) for x- and y-polarized light. For ease of 
+fabrication, the ranges of 𝐷𝑢 and 𝐷𝑣 are limited from 100 nm to 650 nm. (c) The amplitude of 
+transmittance for each case in (b) as a function of (𝜑𝑢, 𝜑𝑣). (d) Required phase profiles for MA1, 
+2, and 3. (e) Optical microscope image and (f, g) SEM images of the fabricated device. In (g), 
+the image is false-colored to distinguish MA1, 2, and 3. 
+ 
+4. Static characterization of the fabricated metasurface 
+We first characterized the fabricated metasurface by observing the intensity distribution at the 
+focal plane for various input states of polarization (SOPs). The experimental setup is shown in 
+Fig. 3(a). A CW light with a wavelength of 1550 nm was incident to the metasurface. The SOP 
+was modified by rotating a half-wave plate (HWP) and a quarter-wave plate (QWP). The image 
+at the focal plane was magnified at 50 times by a 4-f lens system and captured by an InGaAs 
+camera. From the detected intensity values at the six focal positions, the Stokes vector was 
+retrieved as described in Section 2. To enable quantitative measurement of the focused power, 
+we inserted a flip mirror and detected the total power by a bucket PD after spatially filtering 
+the focused beam at each target position using an iris. 
+Figure 3(b) shows the observed intensity distributions when the input Stokes vector is set 
+to (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1). We can confirm that the incident light is focused to the 
+six well-defined points by transmitting through the metasurface. Moreover, its intensity 
+distribution changes with the SOP; x/y linear, 45 linear, and RHC/LHC components of light 
+are focused to the designed positions as expected. Figure 3(c) shows the retrieved Stokes 
+         
+        
+ =  0 0   
+ = 700   
+  
+  
+      
+  
+ 
+ 
+ 
+ 
+ 
+      
+ 
+      
+ 
+   
+  
+ 
+       
+  
+ 
+       
+   
+   
+ 
+  
+ 
+       
+  
+ 
+       
+  
+ 
+       
+  
+ 
+       
+  
+ 
+       
+  
+ 
+       
+             
+    
+          
+  
+  
+  2
+  2
+   
+   
+    
+     
+      
+   
+   
+   
+ 
+  
+ 
+  
+ 
+  
+   
+   
+   
+   ,   
+   ,   
+   ,   
+   ,   
+   ,   
+   ,   
+   
+           
+           
+           
+ 
+
+Raith
+Mag-20.00KX
+SE2
+EHT = 10.00KV
+WD-17.$mmRaith
+Mag
+EHT = 10.00 KV
+0=20.2mmvectors on the Poincaré sphere. The average error 〈|Δ𝐒|〉 is as small as 0.028. Figure 3(d) shows 
+the measured focusing efficiencies to the six positions. Subtracting the 4.8-dB intrinsic loss due 
+to the 13 splitter [see Fig. 1(a)], the excess loss is around 6.1 dB, whereas the crosstalk to the 
+orthogonal PD position is suppressed by 13-20 dB. While this excess loss is already comparable 
+to the coupling and propagation losses of the previously reported waveguide-based SVRs [13-
+18], we expect further improvement by applying anti-reflection coating at the silica surface, 
+improving the fabrication processes to minimize the errors, and by adopting advanced 
+algorithms in designing the metasurface that take into account the nonzero interactions between 
+adjacent meta-atoms [40, 41].  
+ 
+ 
+ 
+Fig. 3. Experimental characterization of the fabricated metasurface. (a) Schematic of the optical 
+setup. The flip mirror is used to switch between capturing the intensity distribution at the focal 
+plane and measuring the power of each focused beam. TLS: tunable laser source. PC: 
+polarization controller. VOA: variable optical attenuator. FC: fiber collimator. Pol.: polarizer. 
+HWP: half-wave plate. QWP: quarter-wave plate. MS: metasurface. M: flip mirror. PD: 
+photodetector. (b) Measured intensity distributions at the focal plane for different input SOPs. 
+(c) Retrieved and input Stokes vectors on the Poincaré sphere. (d) Measured focusing efficiency 
+to each PD. The intrinsic loss due to splitting into three paths is shown by a green line. The input 
+polarization is labeled on the top of each bar. 
+ 
+5. Self-coherent signal transmission experiment 
+We then performed the polarimetric self-coherent signal transmission experiment using the 
+fabricated metasurface. The experimental setup is shown in Fig. 4. We employed a 19-pixel 
+2D-PDA with InP/InGaAs-based p-i-n structure [42], from which six PDs were used as shown 
+in Fig. 4(b). Each PD had a diameter of 30 µm, the measured bandwidth above 10 GHz, and 
+the responsivity of 0.3 A/W. The 2D-PDA chip was packaged with the radio-frequency (RF) 
+coaxial connectors connected to each PD. The 2D-PDA was placed at the focal distance of 10 
+mm from the metasurface as shown in Fig. 4(c). This distance was merely limited by the current 
+setup and should be reduced to a sub-millimeter scale in a practical fully packaged module, 
+which can be comparable to current IMDD receiver modules. 
+A CW light at a wavelength of 1550 nm was generated from a tunable laser source (TLS) 
+and split into two ports, which served as the signal and the pilot tone ports. At the signal port, 
+a LiNbO3 IQ modulator was used to generate a high-speed coherent optical signal. The Nyquist 
+filter was applied to the driving electrical signals from an arbitrary waveform generator (AWG). 
+The modulated optical signal was then combined with the pilot tone by a polarization beam 
+combiner (PBC). The optical power of the pilot tone was adjusted by a variable optical 
+  
+  
+    
+  
+   
+     
+         
+     
+    
+      
+      
+   
+     
+ 
+ 
+                
+   
+   
+  
+ 2
+ 3
+     
+         
+   
+  
+ 
+    
+   
+   
+   
+   
+   
+   
+   
+   
+   
+   
+   
+   
+ 
+               
+| 
+| 
+| 
+| 
+| 
+| 
+| 
+| 
+| 
+| 
+| 
+| 
+              
+
+attenuator (VOA), so that their powers were nearly balanced. The self-coherent signal was then 
+transmitted over a 25-km SMF. At the receiver side, the optical signal-to-noise ratio (OSNR) 
+was controlled using another VOA, followed by an erbium-doped fiber amplifier (EDFA) and 
+an optical bandpass filter (OBPF). The electrical signals from the six PDs of the PDA were 
+amplified by differential RF amplifiers and then captured by a real-time oscilloscope (OSC). 
+At a baudrate beyond 20 GBd, we could not use the balanced PD (B-PD) configuration due to 
+the residual skew inside the PDA module. In these cases, we employed four single-ended PDs 
+(S-PDs), where the electrical signals from PDx, PDy, PDa, and PDl were independently captured 
+by a four-channel real-time oscilloscope, so that the skew could be calibrated during DSP. By 
+comparing the results using two configurations, the use of four S-PDs was validated (see 
+Section S3 of Supplement 1 for details). To equalize and reconstruct the original IQ signal, we 
+employed offline DSP with the 2×3 and 2×4 real-valued multi-input-multi-output (MIMO) 
+equalizers [43, 44] for three-B-PD and four-S-PD configurations, respectively. 
+Figures 5(a)-(c) show the BER curves and the constellations for 15-GBd 16QAM signals, 
+measured using the three-B-PD configuration. We can confirm that BERs well below the hard-
+decision forward error correction (HD-FEC) threshold are obtained with a negligible penalty 
+even after 25-km transmission. Figures 5(d)-(f) show the results for 20-GBd 16QAM and 50-
+GBd QPSK signals, measured by the four-S-PD configuration. Once again, BERs below the 
+HD-FEC threshold are obtained. Finally, Fig. 6 shows the measured BER curves and 
+constellation diagrams of 15-GBd 16QAM signal at 1540-nm and 1565-nm wavelengths, 
+demonstrating the wideband operation of our designed metasurface. While the baudrate in this 
+work was limited by the bandwidth of the 2D-PDA, beyond-100-GBd transmission should be 
+possible by using higher-speed surface-normal PDs with bandwidth exceeding 50 GHz [45, 46]. 
+ 
+ 
+ 
+Fig. 4. Self-coherent transmission experiment using the fabricated metasurface and 2D-PDA. (a) 
+Experimental setup. AWG: arbitrary waveform generator. PBC: polarization beam combiner. 
+EDFA: erbium-doped fiber amplifier. OBPF: optical bandpass filter. Osc: oscilloscope. In the 
+insets, three-B-PD and four-S-PD configurations are depicted. (b) Optical microscope image of 
+the fabricated 19-pixel 2D-PDA. The six circled PDs were used in this experiment. (c) 
+Photograph of the receiver. 
+ 
+ 
+MS
+2D-PDA
+(a)
+TLS
+IQ mod.
+PBC
+AWG
+FC
+1:9
+coupler
+EDFA OBPF
+VOA
+VOA
+Real-time Osc
+MS
+2D-PDA
+(c)
+diff. Amp.
+Real-time Osc
+4 S-PDs
+3 B-PDs
+25 km
+PDa
+PDb
+PDr
+PDl
+PDx
+PDy
+100 μ 
+(b)
+
+ 
+Fig. 5. Experimental results of self-coherent signal transmission at a wavelength of 1550 nm. 
+(a)-(c) Measured BER curves and constellation diagrams of 15-GBd 16QAM signals before 
+(b2b) and after 25-km transmission using the three-B-PD configuration. (d)-(f) Measured BER 
+curves and constellation diagrams of 20-GBd 16QAM and 50-GBd QPSK signals after 25-km 
+transmission using the four-S-PD configuration. 
+ 
+ 
+Fig. 6. Experimental results of self-coherent signal transmission at wavelengths of (a) 1540 nm 
+and (b) 1565 nm. (a, b) Measured BER curves of 15-GBd 16QAM signals before (b2b) and after 
+25-km transmission using the three-B-PD configuration. The insets represent the retrieved 
+constellation diagrams. 
+ 
+6. Conclusion 
+We have proposed and demonstrated a surface-normal SVR using a dielectric metasurface and 
+2D-PDA for the high-speed polarimetric self-coherent systems. Three independently designed 
+meta-atom arrays based on Si nanoposts were superimposed onto a single thin metasurface 
+layer to implement both the polarization-sorting and focusing functions simultaneously. Using 
+a compact metasurface chip fabricated on a SOQ substrate, we demonstrated 25-km 
+transmission of 20-GBd 16QAM and 50-GBd QPSK self-coherent signals. The operating 
+baudrate was merely limited by the 2D-PDA, so that higher-capacity transmission should be 
+possible by using a PDA with a broader bandwidth. Owing to the unique surface-normal 
+configuration with the embedded lens array functionality, a compact receiver module with 
+comparable size and complexity as the conventional IMDD receivers can be realized. Moreover, 
+it can easily be extended to receive spatially multiplexed channels by simply replacing the SMF 
+16
+18
+20
+22
+24
+26
+28
+10-5
+10-4
+10-3
+10-2
+10-1
+OSNR (dB)
+BER
+10-5
+10-4
+10-3
+10-2
+10-1
+BER
+16
+18
+20
+22
+24
+26
+28
+OSNR (dB)
+50-GBd QPSK
+20-GBd 16QAM
+15-GBd 16QAM
+(a)
+(d)
+25km
+b2b
+25 km
+25 km
+(b)
+(c)
+(e)
+(f)
+7% HD-FEC
+7% HD-FEC
+25 km
+b2b
+25 km
+b2b
+22
+24
+26
+28
+30
+10-5
+10-4
+10-3
+10-2
+10-1
+OSNR (dB)
+BER
+14
+16
+18
+20
+22
+10-3
+10-2
+10-1
+OSNR (dB)
+BER
+(a)
+(b)
+1540 nm
+1565 nm
+25 km
+b2b
+25 km
+b2b
+
+to a MCF and employing a larger-scale integrated PDA technology [32]. This work would, 
+therefore, pave the way toward realizing cost-effective receivers for the future >Tb/s spatially 
+multiplexed optical interconnects. 
+Funding. National Institute of Information and Communications Technology (NICT). 
+Acknowledgments. This work was obtained from the commissioned research 03601 by National Institute of 
+Information and Communications Technology (NICT), Japan. Portions of this work were presented at the Optical Fiber 
+Communications Conference (OFC) in 2022, M4J.5. A part of the device fabrication was conducted at the cleanroom 
+facilities of d.lab in the University of Tokyo, supported by MEXT Nanotechnology Platform, Japan. The authors also 
+thank all the technical staff at Advanced ICT device laboratory in NICT for supporting the PDA device fabrication. 
+G.S. acknowledges the financial support from Optics and Advanced Laser Science by Innovative Funds for Students 
+(OASIS) and World-leading Innovative Graduate Study Program - Quantum Science and Technology Fellowship 
+Program (WINGS-QSTEP). 
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+"Polarization-insensitive metalenses at visible wavelengths," Nano Lett. 16, 7229–7234 (2016). 
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+in Proc. Eur. Conf. Opt. Commun. (ECOC) 2018, paper Th2.31. 
+43. Y. Mori, C. Zhang, and K. Kikuchi, "Novel configuration of finite-impulse-response filters tolerant to carrier-
+phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation 
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+for optical quadrature amplitude modulation," IEEE Photonics J. 5, 7800110 (2013). 
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+in Proc. OptoElectron. and Commun. Conf. (OECC) and Int. Conf. on Photon. in Switching and Computing 
+(PSC) 2022, paper WD2-2. 
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+band photonic wireless communication,” J. Lightwave Technol. 34, 3138–3147 (2016). 
+
+Compact and scalable polarimetric self-
+coherent receiver using dielectric 
+metasurface: Supplementary Information 
+GO SOMA,1,4 YOSHIRO NOMOTO,2 TOSHIMASA UMEZAWA,3 YUKI YOSHIDA,3 
+YOSHIAKI NAKANO,1 AND TAKUO TANEMURA1,5 
+1School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 
+2Central Research Laboratory, Hamamatsu Photonics K.K. 5000 Hirakuchi, Hamakita-ku, Hamamatsu 
+City, Shizuoka, Japan 
+3National Institute of Information and Communications Technologies (NICT), 4-2-1 Nukui-Kitamachi, 
+Koganei, Tokyo, Japan 
+4soma@hotaka.t.u-tokyo.ac.jp, 5tanemura@ee.t.u-tokyo.ac.jp 
+ 
+S1. Derivation of 𝑫𝒖(𝝋𝒖, 𝝋𝒗) and 𝑫𝒗(𝝋𝒖, 𝝋𝒗) 
+From RCWA simulation, we obtain 𝑡𝑢(𝐷𝑢, 𝐷𝑣) and 𝑡𝑣(𝐷𝑢, 𝐷𝑣), which describe the complex 
+transmittance through an array of elliptical Si nanoposts with the principal axis lengths of 𝐷𝑢 
+and 𝐷𝑣 for the linearly polarized light along 𝑢 and 𝑣 axes, respectively. The simulated intensity 
+|𝑡𝑢|2 and |𝑡𝑣|2 and the phase arg(𝑡𝑢) and arg(𝑡𝑣)  are shown in Fig. S1(a) and (b). From these 
+results, we derive the required 𝐷𝑢(𝜑𝑢, 𝜑𝑣)  and 𝐷𝑣(𝜑𝑢, 𝜑𝑣)  to obtain desired phase shifts 
+(𝜑𝑢, 𝜑𝑣), by using the following the equation [1]: 
+(𝐷𝑢(𝜑𝑢, 𝜑𝑣), 𝐷𝑣(𝜑𝑢, 𝜑𝑣)) = arg min
+(𝐷𝑢, 𝐷𝑣)
+[|𝑡𝑢(𝐷𝑢, 𝐷𝑣) − 𝑒𝑖𝜑𝑢|
+2 + |𝑡𝑣(𝐷𝑢, 𝐷𝑣) − 𝑒𝑖𝜑𝑣|
+2]. 
+ 
+ 
+Fig. S1. Simulated intensity (a) and phase (b) of transmission coefficients as a function of 𝐷𝑢 
+and 𝐷𝑣. 
+0
+0.7
+(μm)
+0
+0.7
+(μm)
+0
+1
+0
+0.7
+0
+0.7
+0
+0.7
+0
+0.7
+0
+0.7
+0
+0.7
+(μm)
+(μm)
+(μm)
+(μm)
+(μm)
+(μm)
+Transmittance
+Phase (rad)
+(a)
+(b)
+
+S2. Derivation of phase shift by a meta-atom array for circularly polarized light 
+The Jones matrix, describing the transmittance through a lossless Si nanopost array can be 
+written as 
+𝐉 = 𝐑(𝜃) (𝑒𝑖𝜑𝑢
+0
+0
+𝑒𝑖𝜑𝑣) 𝐑(−𝜃) = (𝑒𝑖𝜑𝑢 cos2 𝜃 + 𝑒𝑖𝜑𝑣 sin2 𝜃
+(𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣) sin 𝜃 cos 𝜃
+(𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣) sin 𝜃 cos 𝜃
+𝑒𝑖𝜑𝑢 sin2 𝜃 + 𝑒𝑖𝜑𝑣 cos2 𝜃). 
+Here, 𝜑𝑢 and 𝜑𝑣 represent the phase shifts for the polarization components along the principal 
+axes of the elliptical nanoposts and 𝐑(𝜃) is a rotation matrix with a rotation angle of 𝜃. Here 
+we assume that the input lightwave is circularly-polarized and its Jones vector is written as 
+𝑬𝑟,𝑙 = 1/√2(1, ±𝑖)𝑇. Then, the output Jones vector is written as 
+𝐉𝑬𝑟,𝑙 = 𝑒𝑖𝜑𝑢 + 𝑒𝑖𝜑𝑣
+2
+ 𝑬𝑟,𝑙 + 𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣
+2
+ 𝑒±𝑖2𝜃𝑬𝑙,𝑟. 
+Therefore, when the meta-atom functions as a half-wave plate, i.e., 𝜑𝑣 = 𝜑𝑢 + 𝜋, the output 
+Jones vector becomes 𝑒𝑖(𝜑𝑢±2𝜃)𝑬𝑙,𝑟. In other words, the phase shifts given to the right-handed 
+and left-handed circularly-polarized waves are (𝜑𝑟, 𝜑𝑙) = (𝜑𝑢 + 2𝜃, 𝜑𝑢 − 2𝜃) , while the 
+output polarization handedness is reversed. 
+ 
+S3. Comparison between three-B-PD and four-S-PD configurations 
+To compare the results obtained by three-B-PD and four-S-PD configurations, we performed 
+the self-coherent transmission experiment with 15-GBd 16QAM signals using the two 
+experimental setups shown in Fig. 4(a). The measured BER curves are shown in Fig. S2. Since 
+the BER was limited by the optical signal-to-noise ratio (OSNR), identical results were 
+obtained in two cases. 
+ 
+ 
+Fig. S2. Measured BER curves of 15-GBd 16QAM signals using the setup with three-B-PD and 
+four-S-PD configurations. 
+ 
+References 
+1. 
+A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and 
+polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10, 937–943 
+(2015). 
+    
+  
+  
+  
+  
+   
+        
+         
+        
+         
+        
+    
+    
+    
+    
+

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CtE2T4oBgHgl3EQf9Ant/content/tmp_files/2301.04225v1.pdf.txt

@@ -0,0 +1,2539 @@
+1
+Inferring Gene Regulatory Neural Networks for
+Bacterial Decision Making in Biofilms
+Samitha Somathilaka, Student, IEEE, Daniel P. Martins, Member, IEEE, Xu Li, Yusong Li,
+Sasitharan Balasubramaniam, Senior Member, IEEE,
+Abstract—Bacterial cells are sensitive to a range of external sig-
+nals used to learn the environment. These incoming external sig-
+nals are then processed using a Gene Regulatory Network (GRN),
+exhibiting similarities to modern computing algorithms. An in-
+depth analysis of gene expression dynamics suggests an inherited
+Gene Regulatory Neural Network (GRNN) behavior within the
+GRN that enables the cellular decision-making based on received
+signals from the environment and neighbor cells. In this study,
+we extract a sub-network of Pseudomonas aeruginosa GRN that
+is associated with one virulence factor: pyocyanin production as a
+use case to investigate the GRNN behaviors. Further, using Graph
+Neural Network (GNN) architecture, we model a single species
+biofilm to reveal the role of GRNN dynamics on ecosystem-wide
+decision-making. Varying environmental conditions, we prove
+that the extracted GRNN computes input signals similar to
+natural decision-making process of the cell. Identifying of neural
+network behaviors in GRNs may lead to more accurate bacterial
+cell activity predictive models for many applications, including
+human health-related problems and agricultural applications.
+Further, this model can produce data on causal relationships
+throughout the network, enabling the possibility of designing
+tailor-made infection-controlling mechanisms. More interestingly,
+these GRNNs can perform computational tasks for bio-hybrid
+computing systems.
+Index Terms—Gene Regulatory Networks, Graph Neural Net-
+work, Biofilm, Neural Network.
+I. INTRODUCTION
+B
+ACTERIA are well-known for their capability to sense
+external stimuli, for complex information computations
+and for a wide range of responses [1]. The microbes can sense
+numerous external signals, including a plethora of molecules,
+temperatures, pH levels, and the presence of other microorgan-
+isms [2]. The sensed signals then go through the Gene Regu-
+latory Network (GRN), where a large number of parallel and
+sequential molecular signals are collectively processed. The
+GRN is identified as the main computational component of the
+cell [3], which contains about 100 to more than 11000 genes
+Samitha Somathilaka is with VistaMilk Research Centre, Walton Institute
+for Information and Communication Systems Science, Waterford Institute
+of Technology, Waterford, X91 P20H, Ireland and School of Computing,
+University of Nebraska-Lincoln, 104 Schorr Center, 1100 T Street, Lincoln,
+NE, 68588-0150, USA. E-mail: samitha.somathilaka@waltoninstitute.ie.
+Daniel P. Martins are with VistaMilk Research Centre and the Wal-
+ton Institute for Information and Communication Systems Science, Wa-
+terford Institute of Technology, Waterford, X91 P20H, Ireland. E-mail:
+daniel.martins@waltoninstitute.ie.
+Xu Li and Yusong Li are with Department of Civil and Environmental
+Engineering University of Nebraska-Lincoln 900 N. 16th Street Nebraska Hall
+W181, Lincoln, NE 68588-0531 E-mail:xuli,yli7@unl.edu.
+S. Balasubramaniam is with School of Computing, University of Nebraska-
+Lincoln, 104 Schorr Center, 1100 T Street, Lincoln, NE, 68588-0150, USA.
+E-mail:sasi@unl.edu
+Full GNN
+...
+...
+...
+Extracted 
+NN
+GRN
+Biofilm
+...
+Fig. 1: Illustration of the Gene Regulatory Neural Networks
+(GRNN) extraction and the implementation of the GNN to
+model the biofilm. The diffusion of molecules from one cell
+to another is modeled as a vector, where mq represents the
+concentration of the qth molecular signal.
+(the largest genome identified so far belongs to Sorangium
+cellulosum strain So0157-2) [4]. Despite the absence of neural
+components, the computational process through GRN allows
+the bacteria to actuate through various mechanisms, such as
+molecular production, motility, physiological state changes and
+even sophisticated social behaviors. Understanding the natural
+computing mechanism of cells can lead to progression of
+key areas of machine learning in bioinformatics, including
+prediction of biological processes, prevention of diseases and
+personalized treatment [5].
+Bacterial cells are equipped with various regulatory sys-
+tems, such as single/two/multi-component systems including
+Quorum sensing (QS), to respond to environmental stimuli.
+The receptors and transporters on cell membranes can react
+and transport extracellular molecules, which subsequently in-
+teract with respective genes. In turn, the GRN is triggered
+to go through a complex non-linear computational process in
+response to the input signals. In the literature, it has been
+suggested that the computational process through the GRN of
+a bacterial cell comprises a hidden neural network (NN)-like
+architecture [6], [7]. This indicates that, even though bacterial
+cells can be categorized as non-neural organisms, they perform
+neural decision-making processes through the GRN. This re-
+sults in recent attention towards Molecular Machine Learning
+systems, where AI and ML are developed using molecular
+systems [8]. In these systems, several neural components can
+be identified in GRNs, in which genes may be regarded as
+arXiv:2301.04225v1  [q-bio.MN]  10 Jan 2023
+
+2
+C4- 
+RhlR
+rhlI
+3OC-
+LasR
+Phosphate
+Fe(II)
+BqsR
+pqsABCDE
+phz2
+phz1
+PQS-
+PQSR
+PhoB
+pqsR
+rhlR
+lasR
+LasI
+pqsH
+HHQ-
+PQSR
+?
+3OC
+C4
+PqsH
+HHQ
+PqsR
+LasR
+RhlR
+PQS
+AlgR
+czcR
+PhZ2 PhZ1
+3OC
+C4
+PqsH
+HHQ
+PqsR LasR
+RhlR
+(a)
+hn11
+hn12
+PhoB
+hn13
+BqSR
+hn14
+AlgR
+hn21
+hn22
+hn23
+pqsABCDE
+czcR
+rhlI
+pqsR lasR
+lasI
+rhlR
+hn31
+phz2
+phz1
+hn24
+pqsH
+(b)
+GNN model
+Inter-cellular diffusion
+GRN
+NN
+...
+...
+...
+...
+(c)
+Fig. 2: Extraction of a GRNN considering a specific sub-network of the GRN where a) is the two-component systems (TCSs)
+and QS network that is associated with the pyocyanin production, b) is the derived GRNN that is equipped with hypothetical
+nodes (hns) without affecting its computation process to form a symmetric network structure and c) is the conversion of real
+biofilm to the suggested in-silico model.
+computational units or neurons, transcription regulatory factors
+as weights/biases and proteins/second messenger Molecular
+Communications (MC) as neuron-to-neuron interactions. Ow-
+ing to a large number of genes and the interactions in a GRN,
+it is possible to infer sub-networks with NN behaviors that
+we term Gene Regulatory Neural Networks (GRNN). The
+non-linear computing of genes results from various factors
+that expand through multi-omics layers, including proteomic,
+transcriptomic and metabolomic data (further explained in
+Section II-A). In contrast, the GRNN is a pure NN of genes
+with summarized non-linearity stemmed from multi-omics
+layers with weights/biases.
+Identification of GRNNs can be used to model the decision-
+making process of the cell precisely, especially considering
+simultaneous multiple MC inputs or outputs. However, due
+to the limited understanding and data availability, it is still
+impossible to model the complete GRN with its NN-like
+behaviors. Therefore, this study uses a GRNN of Pseudomonas
+aeruginosa that is associated with PhoR-PhoB and BqsS-
+BqsR two-component systems (TCSs) and three QS systems
+related to pyocyanin production as a use case to explore the
+NN-like behaviors. Although a single bacterium can do a
+massive amount of computing, they prefer living in biofilms.
+Hence, in order to understand the biofilm decision-making
+mechanism, we extend this single-cell computational model
+to an ecosystem level by designing an in-silico single species
+biofilm with inter-cellular MC signaling as shown in Fig. 1.
+The contributions of this study are as follows:
+• Extracting a GRNN: Due to the complexity and insuf-
+ficient understanding of the gene expression dynamics of
+the full GRN, we only focus on a sub-network associated
+with pyocyanin production (shown in Fig. 2a) to inves-
+tigate the NN-like computational behavior of the GRN.
+Further, the genes of extracted sub-network are arranged
+following a NN structure that comprises input, hidden
+and output layers, as shown in Fig. 2b.
+• Modeling a biofilm as a GNN: The GRNN only repre-
+sents the single-cell activities. To model the biofilm-wide
+decision-making process, we use a Graph Neural Network
+(GNN). First, we create a graph network of the bacterial
+cell and convert it to a GNN by embedding each node
+with the extracted GRNN as the update function. Second,
+the diffusion-based MCs between bacterial cells in the
+biofilm are encoded as the message-passing protocol of
+the GNN, as shown in Fig. 2c.
+• Exploring the behaviors of the GRNN and intra-
+cellular MC dynamics to predict cell decisions: The
+output of the GRNN is evaluated by comparing it with
+the transcriptomic and pyocyanin production data from
+the literature. Finally, an edge-level analysis of the GRNN
+is conducted to explore the causal relationships between
+gene expression and pyocyanin production.
+This paper is organized as follows: Section II explains
+the background of bacterial decision-making in two levels:
+cellular-level in Section II-A and population-level in Section
+II-B, while the background on the P. aeruginosa is introduced
+in Section II-C. Section III is dedicated to explaining the model
+design of cellular and population levels. The results related
+to model validation and the intergenic intra-cellular signaling
+pattern analysis are presented in Section IV and the study is
+concluded in Section V.
+II. BACKGROUND
+As the model expands through single cellular and biofilm-
+wide decision-making layers, this section provides the back-
+ground of how a bacterium uses the GRN to make decisions
+and how bacterial cells make decisions in biofilms. Moreover,
+we briefly discuss the cellular activities of the Pseudomonas
+aeruginosa as it is the use case species of this study.
+A. Decision-Making Process of an Individual Cell
+Prokaryotic cells are capable of sensing the environment
+through multiple mechanisms, including TCSs that have been
+widely studied and it is one of the focal points of this
+study. The concentrations of molecular-input signals from
+the extracellular environment influence the bacterial activities
+at the cellular and ecosystem levels [9]. Apart from the
+extracellular signals of nutrients, it is evident that the QS
+input signals have a diverse set of regulative mechanisms in
+biofilm-wide characteristics, including size and shape [10].
+These input signals undergo a computational process through
+the GRN, exhibiting a complex decision-making mechanism.
+Past studies have explored and suggested this underpinning
+computational mechanism in a plethora of directions, such
+
+3
+Prom
+Op
+Enh
+GeneA
+...
+Sil
+GeneB
+...
+Fig. 3: Illustration of gene expression regulators that are
+considered the weight influencers of the edges of GRNN.
+Here, the α(σ), α(∼σ), α(T F ), α(Rep), α(eT F ) and α(sT F )
+are relative concentrations of sigma factors, anti-sigma factors,
+transcription factors (TFs), repressors, enhancer-binding TFs
+and silencer-binding TFs respectively. Moreover, β(P rom),
+β(Op), β(Enh), and β(Sil) are the binding affinities of the pro-
+moter, operator, enhancer and silencers regions respectively.
+as using differential equations [11] and probabilistic Boolean
+networks [12] and logic circuit [13]. All of these models
+mainly infer that the bacterial cells can make decisions not
+just based on the single input-output combinations, but they
+can integrate several incoming signals non-linearly to produce
+outputs.
+The studies that focus on differences in gene expression
+levels suggest that a hidden weight behavior controls the
+impact of one gene on another [6]. This weight behavior
+emerges through several elements, such as the number of
+transcription factors that induce the expression, the affinity
+of the transcription factor binding site, and machinery such
+as thermoregulators and enhancers/silencers [14], [15]. Fig.
+3 depicts a set of factors influencing the weight between
+genes. The weight of an edge between two genes has a
+higher dynamicity as it is combinedly determined by several
+of these factors. Based on environmental conditions, the GRN
+of the bacterial cell adapts various weights to increase the
+survivability and repress unnecessary cellular functions to
+preserve energy. An example of such regulation is shown in
+Fig. 4 where a P. aeruginosa cell uses a thermoregulator to
+regulate the QS behaviors. Fig. 4a has a set of relative weights
+based on cellar activities in an environment at 37 ◦C, while
+Fig. 4b represents weights at 30 ◦C. The weights between the
+hn21 and rhlR are different in two conditions, and these cellar
+activities are further explained in [14].
+B. Biofilm Decision-Making
+Even though an individual cell is capable of sensing,
+computing, and actuating, the majority of bacterial cells live
+in biofilms, where the survivability is significantly increased
+compared to their planktonic state. Biofilm formation can
+cause biofouling and corrosion in water supply and industrial
+systems [16]. However, biofilms formation can be desirable
+in many situations, for example, bioreactors in wastewater
+treatment [17], bioremediation of contaminated groundwater
+[18], [19], where biofilms serve as platforms for biogeochem-
+ical reactions. A massive number of factors can influence
+biofilm formation, including substratum surface geometrical
+characteristics, diversity of species constituting the biofilm,
+hydrodynamic conditions, nutrient availability, and especially
+communication patterns [20] where the TCS and QS play
+rhlI
+BqsR
+pqsABCDE
+phz2
+phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+LasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+hn22
+AlgR
+czcR
+1
+-1
+0
+(a)
+rhlI
+BqsR
+pqsABCDE
+phz2
+phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+LasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+hn22
+AlgR
+czcR
+1
+-1
+0
+(b)
+Fig. 4: Two GRNN setups with different weights associated
+with two environmental conditions. a) is the relative weight
+setup of P. aeruginosa cell in 37 ◦C and b) is in 30 ◦C.
+significant roles. A TCS comprises a histidine kinase that
+is the sensor for specific stimulus and a cognate response
+regulator that initiates expressions of a set of genes [21].
+Hence, in each stage, essential functions can be traced back
+to their gene expression upon a response to the input signals
+detected by bacterial cells. For instance, in the first stage
+of biofilm formation, the attachment of bacteria to a surface
+is associated with sensing a suitable surface and altering
+the activities of the flagella. In the next stage, rhamnolipids
+production is associated with ferric iron Fe3+ availability in
+the environment, mostly sensed through BqsS-BqsR TCSs.
+Further, Fe3+ was identified as a regulator of pqsA, pqsR, and
+pqsE gene expressions that are associated with the production
+of two critical components for the formation of microcolonies:
+eDNA and EPS [22]. Similarly, in the final stage, the dis-
+persion process can also be traced back to a specific set
+of gene regulations, including bdlA an rbdA [23], [24]. An
+understanding of the underlying decision-making process of
+bacteria may enable us to control their cellular activities.
+C. Pseudomonas Aeruginosa
+The main reason for selecting P. aeruginosa in this work
+lies in its alarming role in human health. For example, this
+species is the main cause of death in cystic fibrosis patients
+[25]. P. aeruginosa is a gram-negative opportunistic pathogen
+with a range of virulence factors, including pyocyanin and
+cytotoxin secretion [26]. These secreted molecules can lead to
+complications such as respiratory tract ciliary dysfunction and
+induce proinflammatory and oxidative effects damaging the
+host cells [27]. The biofilms are being formed on more than
+90% endotracheal tubes implanted in patients who are getting
+assisted ventilation, causing upper respiratory tract infections
+[28]. In addition, another important reason for targeting P.
+aeruginosa is the data availability for the GRN structure [29],
+pathways [30], genome [31], transcriptome [32] and data from
+mutagenesis studies [33], [34]. Compared to the complexity of
+the GRN, the amount of data and information available on the
+
+4
+C4- 
+RhlR
+3OC-
+LasR
+PQS-
+PQSR
+HHQ-
+PQSR
+3OC
+C4
+PqsH
+HHQ
+PqsR
+LasR
+RhlR
+PQS
+Fig. 5: Illustrations of intra-cellular metabolite interaction.
+gene-to-gene interactions and expression patterns is insuffi-
+cient to develop an accurate full in-silico model. Therefore,
+we chose a set of specific genes that are associated with QS,
+TCS, and pyocyanin production.
+III. SYSTEM DESIGN
+This section explains the system design in two main phases,
+extracting a NN-like architecture from the GRN targeting the
+set of genes and creating a model of the biofilm ecosystem.
+A. Extracting Natural Neural Network from GRN
+We first fetch the structure of the GRN graph from Reg-
+ulomePA [29] database that contains only the existence of
+interactions and their types (positive or negative). As the next
+step, using information from the past studies [35]–[38], we
+identified the genes involved in the Las, Rhl and PQS QS
+systems, PhoR-PhoB and BqsS-BqsR TCSs, and pyocyanin
+production to derive the sub-network of GRN as shown in
+Fig 2a. We further explored the expression dynamics using
+transcriptomic data [39], [40] where we observed the non-
+linearity in computations that are difficult to capture with
+existing approaches such as logic circuits, etc. [6], making
+the NN approach more suitable. However, a NN model with a
+black box that is trained on a large amount of transcriptomic
+data records to do computations similar to the GRN has a
+number of limitations, especially in understanding the core
+of the computational process [41]. Our model does not use a
+conventional NN model; instead, we extract a NN from the in-
+teraction patterns of the GRN, which we consider a pre-trained
+GRNN. In this sub-network, we observed that the lengths of
+expression pathways are not equal. For example, the path from
+PhoR-PhoB to the phz2 gene has two hops, but the path from
+the BqsS-BqsR system to the rhlR gene only has one hop. The
+extracted network has the structure of a random NN. Hence,
+we transform this GRNN to Gene Regulatory Feedforward
+Neural Network by introducing hypothetical nodes (hns) that
+do not affect the behaviors of the GRNN as shown in Fig 2b.
+In this transformation, we decide the number of hidden layers
+based on the maximum number of hops in gene expression
+pathways. In our network, the maximum number of hops is
+two, which determines the number of hidden layers as one,
+and then the number of hops of all the pathways is leveled
+by introducing hns. If a hn is introduced between a source
+and target genes, the edge weights from the source node to
+the hn and from hn to the target node are made “1” so that
+the hn does not have an influence on the regulation of genes.
+Moreover, if a gene does not induce another in the network,
+the weight of the edge between that pair is made “0”.
+Here, we summarize multiple factors of interaction into
+a weight that determines the transcriptional regulation of
+a particular gene. This regulation process occurs when the
+gene products get bound to the promoter region of another,
+influencing the transcriptional machinery. Hence, we observe
+this regulation process of a target gene as a multi-layered
+model that relies on the products of a set of source genes, the
+interaction between gene products, and the diffusion dynamics
+within the cell. Creating a framework to infer an absolute
+weight value using all the above factors is a highly complex
+task. In order to infer weight, one method is to train a NN
+model with the same structure as the GRN using a series of
+transcriptomic data. However, this approach also has numerous
+challenges, such as the lack of a sufficient amount of data in
+similar environments.
+Therefore, we estimate a set of relative weights based on
+genomic, transcriptomic, and proteomic explanations of each
+interaction from the literature. The weights were further fine-
+tuned using the transcriptomics data. A relative weight value
+of an edge can be considered a summarizing of multi-layer
+transcriptional-translation to represent the impact of the source
+gene on a target gene.
+In this computational process, we identify another layer of
+interactions that occur within the cell. The produced molecules
+by the considered TCs network go through a set of metabolic
+interactions that are crucial for the functionality of the cell.
+Since our primary goal is to explore the NN behaviors of
+GRN, we model these inter-cellular chemical reactions as a
+separate process, keeping the gene-to-gene interactions and
+metabolic interactions in two different layers. To model the
+complete pyocyanin production functionality of the cell, we
+use the inter-cellular molecular interactions shown in Fig 5.
+Here, RhlR is a transcriptional regulator of P. aeruginosa that
+forms a complex by getting attached to its cognate inducer
+C4-HSL and then binds to the promoter regions of relevant
+genes [42]. Similarly, LasR transcriptional regulator protein
+and 3-oxo-C12-HSL (3OC), and PqsR with PQS and HHQ
+form complexes and get involved in the regulation of a range
+of genes [43], [44]. Further, C10H10O6 in the environment are
+converted by the P. aeruginosa cells in multiple steps using
+the products of the GRNN we consider. First, C10H10O6
+is converted into phenazine-1-carboxylic using the enzymes
+of pqsABCDEFG genes. Later, phenazine-1-carboxylic was
+converted into 5-Methylphenazine-1-carboxylate, and finally,
+5-Methylphenazine-1-carboxylate into Pyocyanin by PhzM
+and PhzS, respectively [45].
+Molecular accumulation within a bacterial cell can be
+considered its memory module where certain intra-cellular
+interactions occurs. Therefore, we define an internal memory
+matrix IM as,
+IM(t) =
+im1
+im2
+...
+imJ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+B1
+C(t)
+(1,im1)
+C(t)
+(1,im2)
+...
+C(t)
+(1,imJ)
+B2
+C(t)
+(2,im1)
+C(t)
+(2,im2)
+...
+C(t)
+(2,imJ)
+...
+...
+...
+...
+...
+BP
+C(t)
+(P,im1)
+C(t)
+(P,im2)
+...
+C(t)
+(P,imJ)
+,
+(1)
+where the concentration of the internal molecule imj is
+
+5
+C(t)
+(i,imj).
+GRNN process molecular signals from the environment and
+other cells. Hence, we used the approach of GNN as a scalable
+mechanism to model the MCs and biofilm wide decision-
+making process. The extreme computational power demand of
+modeling the diffusion-based MCs of each cell is also avoided
+by using this approach.
+B. Graph Neural Network Modeling of Biofilm
+First, the biofilm is created as a graph network of bacterial
+cells where each node is a representation of a cell, and an edge
+between two nodes is a MC channel. We convert the graph
+network into a Graph Neural Network (GNN) in three steps: 1)
+embedding the extracted GRNN of pyocyanin production into
+each node as the update function, 2) encoding the diffusion-
+based cell-to-cell MC channels as the message passing scheme,
+and 3) creating an aggregation function at the reception of
+molecular messages by a node as shown in Fig. 6. Next, we
+define feature vectors of each node of the GNN to represent
+the gene expression profile of the individual cell at a given
+time. Subsequently, considering L is the number of genes in
+the GRNN, P is the number of bacterial cells in the biofilm
+and b(t)
+(i,gl) is the expression of gene gl by the bacteria Bi,
+we derive the following matrix FV(t) that represents all the
+feature vectors of the GNN at time t.
+FV(t) =
+g1
+g2
+...
+gL
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+B1
+b(t)
+(1,g1)
+b(t)
+(1,g2)
+...
+b(t)
+(1,gL)
+B2
+b(t)
+(2,g1)
+b(t)
+(2,g2)
+...
+b(t)
+(2,gL)
+...
+...
+...
+...
+...
+BP
+b(t)
+(P,g1)
+b(t)
+(P,g2)
+...
+b(t)
+(P,gL)
+(2)
+The computational output of the GRNN of each node results
+in the secretion of a set of molecules that are considered
+messages in our GNN model as illustrated in the Fig. 7.
+When the number of molecular species considered in the
+network is Q and output mq molecular message from bacterial
+cell Bi at TS t is msg(t)
+(i,mq), we derive the matrix
+MSG(t) =
+m1
+m2
+...
+mQ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+B1
+msg(t)
+(1,m1)
+msg(t)
+(1,m2)
+...
+msg(t)
+(1,mQ)
+B2
+msg(t)
+(2,m1)
+msg(t)
+(2,m2)
+...
+msg(t)
+(2,mQ)
+...
+...
+...
+...
+...
+BP
+msg(t)
+(P,m1)
+msg(t)
+(P,m2)
+...
+msg(t)
+(P,mQ)
+.
+(3)
+Further, we use a static diffusion coefficients vector
+D = {Dm1, Dm2, ..., DmQ},
+(4)
+where Dmq is diffusion coefficient of molecular species mq.
+Gene expression profile of bacterial cell b1 at time step t
+(a)
+B2
+B5
+B6
+B1
+B3
+B7
+B2
+B5
+B6
+B1
+B3
+B7
+B2
+B5
+B6
+B1
+B3
+B7
+B2
+B5
+B6
+B1
+B3
+B7
+t=0
+t=1
+t=2
+t=T
+...
+(b)
+Fig. 6: Illustration of the GNN components where a) is a
+snapshot of the bacterial network that has the gene expression
+profile as the feature vector. Further, this gene expression
+pattern of a cell is encoded to a message of secreted molecules
+where MC plays a crucial role. Moreover, b) shows the
+temporal behavior of the GNN, that the output of one graph
+snapshot influences the next.
+...
+...
+Fig. 7: The process of one GRNN outputs reaching another
+GRNN as molecular messages.
+We define another matrix ED that contains the euclidean
+distances between bacterial cells in the biofilm as follows
+ED =
+B1
+B2
+...
+BP
+�
+�
+�
+�
+�
+�
+�
+�
+B1
+d(1,1)
+d(1,2)
+...
+d(1,P )
+B2
+d(2,1)
+d(2,2)
+...
+d(2,P )
+...
+...
+...
+...
+...
+BP
+d(P,1)
+d(P,2)
+...
+d(P,P )
+(5)
+where di,j is the euclidean distance between the ith and jth
+cells.
+
+6
+TABLE I: Parameters utilised in the system development
+Parameter
+Value
+Description
+No. of cells
+2000
+The number of cells is limited due to the memory availability of the server.
+No. of genes
+13
+The network only consists of the gene that are directly associated with QS, PhoR-PhoB and BqsS-BqsR
+TCSs, and pyocyanin production.
+No. internal memory molecules
+16
+The set of molecules that involved in QS, PhoR-PhoB and BqsS-BqsR TCSs,and pyocyanin production.
+No. messenger molecules
+4
+The number of molecules that were exchanged between cells in the sub network.
+Dimensions of the environment
+20x20x20µm
+The dimensions were fixed considering the average sizes of P. aeruginosa biofilms and computational
+demand of the model.
+Duration
+150 TSs
+The number of TSs can be modified to explore the cellular and ecosystem level activities. For this
+experiment we fixed a TS to represent 30mins.
+No. iterations per setup
+10
+Considering the stochasticity ranging from the gene expression to ecosystem-wide communications, the
+experiments were iterated 10 times.
+The feature vector of ith bacterial cell at the TS t + 1 is
+then modeled as,
+FV(t+1)
+i
+= GRNNi(MSG(t)
+i
++ S(t)
+i )
+(6)
+where MSG(t)
+i
+is the message generated by the same cell
+in the previous TS. The GRNNi is the extracted GRNN
+that is the update function in the GNN learning process
+and S(t)
+i
+= R(t)
+i
++ K(t)
+i , is the aggregate function. In the
+aggregation component, the R(t)
+i
+is the incoming signals from
+peer bacterial cells and K(t+1)
+(i:mq) is the external molecule input
+vector at the location of Bi and the TS t that is expressed as
+K(t+1)
+i
+=
+�
+K(t+1)
+i:m1 , K(t+1)
+i:m2 , ..., K(t+1)
+i:mQ
+�
+.
+(7)
+In order to compute R(t+1)
+i
+, we use a matrix Yi;
+Yi =
+↔
+1 [Q×1] ×EDi, where
+↔
+1 [Q×1] is an all-ones matrix of
+dimension Q × 1. The ˆg matrix is then defined as follows,
+ˆg(D⊺, Y, t) =
+�
+����
+g(Dm1, d(i,1), t)
+g(Dm1, d(i,2), t)
+...
+g(Dm1, d(i,P ), t)
+g(Dm2, d(i,1), t)
+g(Dm2, d(i,2), t)
+...
+g(Dm2, d(i,P ), t)
+...
+...
+...
+...
+g(DmQ, d(i,1), t)
+g(DmQ, d(i,2), t)
+...
+g(DmQ, d(i,P ), t)
+�
+���� .
+(8)
+In the above matrix, g(Dml, d(i,j), t) is the Green’s function
+of the diffusion equation as shown below,
+ˆG(Dml, d(i,j), t) =
+1
+(4πDmlt)
+3
+2 exp
+�
+−
+d2
+(i,j)
+4Dmlt
+�
+.
+(9)
+Further, the incoming signal vector R(t+1)
+i
+is denoted as
+below,
+R(t+1)
+i
+= diag
+�
+ˆg(D⊺, Y, t) × MSG(t)�
+.
+(10)
+Further, we equip our model with a 3-D environment
+to compensate for the noise element and external molecule
+inputs. Environment-layer is designed as a 3-D grid of voxels
+that can store precise information on external nutrients (simi-
+larly to our previous model in [46]). The diffusion of nutrient
+molecules through the medium is modeled as a random-walk
+process. This layer allows us to enrich the model with the
+dynamics of nutrient accessibility of bacterial cells due to
+diffusion variations between the medium and the Extracellular
+Polymeric Substance (EPS).
+The bacterial cells in the ecosystem also perform their
+own computing tasks individually, resulting in a massively
+parallel processing framework. Hence, we use the python-cuda
+platform to make our model closer to the parallel processing
+architecture of the biofilm, where we dedicate a GPU block for
+each bacterial cell and the threads of each block for the matrix
+multiplication of the GRNN computation associated with the
+particular cell. Additionally, due to the massive number of
+iterative components in the model, the computational power
+demand faces significant challenges with serial programming
+making parallelization the best match for the model.
+IV. SIMULATIONS
+In this section, we first explain the simulation setup and
+then discuss the results of gene expression and molecular
+production dynamics to prove the accuracy of the extracted
+GRNN, emphasizing that it works similarly to the real GRN.
+Later, we use computing through the GRNN to explain certain
+activities of the biofilm.
+A. Simulation Setup
+As our interest is to investigate the NN-like computational
+process, we do not model the formation process of the biofilm,
+but we only remodel a completely formed biofilm and disre-
+garding the maturation and dispersion stages. In this model, we
+consider the biofilm as a static 3-D structure of bacterial cells.
+Hence, we first place bacterial cells randomly in the model in a
+paraboloid shape using the equation, z < x2
+5 + y2
+5 +20 where x,
+y and z are the components of 3-D Cartesian coordinates. This
+paraboloid shape is chosen to make the spacial arrangement of
+the cells close to real biofilm while keeping the cell placement
+process mathematically simple. Within this 3-D biofilm region,
+we model the diffusivity according to DB/Daq = 0.4, which
+is the mean relative diffusion [47] where DB and Daq are
+the average molecular diffusion coefficients of the biofilm
+and pure water, respectively. Further, to start the simulation
+at a stage where the biofilm is fully formed and the MC is
+already taking place, we filled the internal memory vector
+of each cell with the average molecular level at the initial
+TS. Each bacterial cell will use the initial signals from the
+internal memory and use its GRNN to process and update the
+feature vector for the next TS. Table I presents the parameter
+descriptions and values used for the simulation. As shown in
+Table I, the model runs for 150 TSs, generating data on a
+range of functions for the system. For instance, this model can
+produce data on feature vector of each cell, MC between cells,
+
+7
+Cell count %
+0
+20
+Phos. Concentration %
+40
+60
+0
+20
+40
+60
+80
+100
+80
+60
+40
+20
+0
+Time steps
+(a)
+Cell count %
+0
+20
+Phos. Concentration %
+40
+60
+0
+20
+40
+60
+80
+100
+80
+60
+40
+20
+0
+Time steps
+(b)
+Fig. 8: The nutrient accessibility variations of cells is ex-
+pressed in two different environment conditions: a) low phos-
+phate and b) high phosphate concentrations.
+LP_WD
+HP_WD
+0
+25 50 75 100 125 150
+100
+80
+60
+40
+20
+0
+TS
+Rel. pyocyanin acc.
+(a)
+0
+25 50 75 100125150
+100
+80
+60
+40
+20
+0
+TS
+Rel. pyocyanin acc.
+LP_lasR
+HP_lasR
+(b)
+LP_phoB
+HP_phoB
+0
+25 50 75 100 125 150
+100
+80
+60
+40
+20
+0
+TS
+Rel. pyocyanin acc.
+(c)
+LP_lasRphoB
+HP_lasRphoB
+0
+25 50 75 100 125 150
+100
+80
+60
+40
+20
+0
+TS
+Rel. pyocyanin acc.
+(d)
+Fig. 9: Relative Pyocyanin accumulation of four different
+biofilms of a) WD, b) lasR∆, c) phob∆ and d) lasR∆phob∆
+in both low and high phosphate levels.
+molecular consumption by cells, secretion to the environment,
+and nutrient accessibility of cells for each TS.
+In order to prove that our GRNN computes similarly to
+the natural bacterial cell and collective behaviors of the cells
+are the same as the natural biofilm, we conduct a series of
+experiments. We explore the GRNN computation and biofilm
+activities under High Phosphate (HP) and Low Phosphate (LP)
+levels using eight experimental setups as follows, 1) wild-
+type bacteria (WD) in LP, 2) lasR mutant (lasR∆) in LP, 3)
+phoB mutant (phoB∆) in LP, 4) lasR & PhoB double mutant
+(LasR∆PhoB∆) in LP, 5) WD in HP, 6) lasR∆ in HP, 7)
+PhoB∆ in HP and LasR∆PhoB∆ in HP. While the WD uses
+the full GRNN, lasR∆ is created by making the weight of
+the link between hn22 and lasR as “0”. Further, the GRNN of
+phoB∆ is created by making the weights of links from PhoB
+to hn23 and PhoB to pqsABCDE also “0”.
+B. Model Validation
+First, we show the nutrient accessibility variation in the
+biofilm through Fig 8. The cells in the biofilm core have
+less accessibility while the cells closer to the periphery have
+more access to nutrients due to variations in diffusion between
+100
+80
+60
+40
+20
+0
+WD
+lasR
+phoB
+lasRphoB
+Model 
+Wet-lab
+HP to LP Pyocyanin  
+production ratio (%)
+Fig. 10: Evaluation of the model accuracy by comparing HP
+to LP pyocyanin production ratio with wet-lab data from [48].
+the environment and the EPS. Fig 8a shows that when a low
+phosphate concentration is introduced to the environment, the
+direct access to the nutrient by the cells is limited. After the
+TS = 10, around 60% of cells have access to 20% of the
+nutrient concentration. Further, Fig 8b shows that the increased
+nutrient introduction to the environment positively reflects on
+the accessibility. This accessibility plays a role mainly in the
+deviation of gene expression patterns resulting in phenotypic
+differentiations that is further analyzed in Section IV-C.
+Comparing the predictions of molecular production through
+GRNN computing with the wet-lab experimental data from
+the literature, we are able to prove that the components of
+the GRN work similarly to a NN. Fig. 9 shows the pyocyanin
+accumulation variations of the environment in the eight setups
+mentioned earlier as results of decision-making of the GRNN.
+Production of pyocyanin of the WD P. aeruginosa biofilms
+is high in LP, compared to the HP environments as shown
+in Fig: 9a. Further, the same pattern can be observed in the
+lasR∆ biofilms, but with a significantly increased pyocyanin
+production in LP as shown in Fig. 9b. The phob∆ and
+LasR∆phob∆ biofilms produce a reduced level of pyocyanin
+compared to WD and LasR∆ that are shown in Fig. 9c and
+Fig. 9d respectively. We then present a comparison between
+GRNN prediction and wet-lab experimental data [48] as ratios
+of HP to LP in Fig. 10. The differences between pyocyanin
+production through GRNN in HP and LP condition for all the
+four setups in Fig. 10 are fairly close to the wet-lab data. In
+the WD setup, the difference between the GRNN model and
+wet-lab data only has around 5% difference, while deviations
+around 10% can be observed in lasR∆ and phoB∆. The most
+significant deviation around 20% of pyocyanin production
+difference is visible in lasR∆phoB∆ that is caused by the lack
+of interaction from other gene expression pathways, as we only
+extracted a sub-network portion of the GRN. Therefore, these
+results prove that the extracted GRNN behaves similarly to
+the GRN dynamics.
+We further prove that the GRNN computing process per-
+forms similarly to the GRN by comparing the gene expression
+behaviors of the model with the wet-lab data [48] as shown in
+Fig. 11. First, we show the expression dynamics of genes lasI,
+pqsA and rhlR of WD in LP in Fig. 11a, Fig. 11b and Fig. 11c
+respectively. All the figures depict that gene expression levels
+are higher in LP compared to HP until around TS = 100.
+Beyond that point, relative gene expression levels are close to
+zero as the the environment run out of nutrients. Moreover, the
+differences in gene expression levels predicted by the GRNN
+
+12
+10
+8
+6
+4
+2
+0
+120
+100
+80
+0
+10
+60
+20
+30
+40
+40
+50
+20
+60
+70
+0100
+Relative Pyocyanin Accumulation
+LP WD
+HP WD
+80
+60
+40
+20
+0
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)100
+Relative Pyocyanin Accumulation
+80
+60
+40
+20
+LP lasR△
+HP lasR△
+0
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)100
+LP PhoB△
+HP PhoB△
+80
+60
+40
+20
+0
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)100
+LP LASR△ PHOB△
+HP LASRA PHOBA
+80
+60
+40
+20
+0
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)100
+Model
+80
+Real
+60
+40
+20
+0
+WD
+lasRA
+PHOBAlasRA PHOBA8
+7
+6
+5
+4
+3
+2
+1
+0
+120
+100
+80
+0
+10
+60
+20
+30
+40
+40
+50
+20
+60
+70
+08
+0
+25
+50
+75
+100
+125
+150
+8
+7
+6
+5
+4
+3
+2
+Relative lasI expression
+POA1_LP
+POA1_HP
+TS
+(a)
+0
+25
+50
+75
+100
+125
+150
+POA1_LP
+POA1_HP
+TS
+12
+10
+8
+6
+4
+2
+0
+Relative pqsA expression
+(b)
+0
+25
+50
+75
+100
+125
+150
+6
+5
+4
+3
+2
+1
+0
+Relative rhlR expression
+POA1_LP
+POA1_HP
+TS
+(c)
+100
+80
+60
+40
+20
+0
+Model 
+Wet-lab
+lasI
+pqsA
+rhlR
+HP to LP gene  
+expression ratio (%)
+(d)
+Fig. 11: Expression levels of three different genes to that were used to prove the accuracy of the GRNN: a) lasI, b) pqsA, c)
+rhlR expression levels in LP and HP and d) comparison between GRNN computing results and wet-lab data.
+rhlI
+BqsR
+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+lasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+hn22
+AlgR
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+TS
+Genes
+pqsABCDE
+czcR
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+pqsR
+lasR
+lasI
+lasR
+pqsH
+phz2
+phz1
+0 3 6 9 12151821242730333639424548
+(a)
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+phz2 phz1
+PhoB
+hn11
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+lasR
+lasI
+pqsH
+hn31
+hn12
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+hn14
+hn23
+hn24
+hn21
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+AlgR
+czcR
+TS
+0 3 6 9 12151821242730333639424548
+(b)
+rhlI
+BqsR
+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+lasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+hn22
+AlgR
+czcR
+TS
+0 3 6 9 12151821242730333639424548
+(c)
+Fig. 12: Gene expression and associated information flow variations in GRNNs of a) WD b) lasR∆ and c) phoB∆ in LP.
+rhlI
+BqsR
+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+lasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+hn22
+AlgR
+czcR
+TS
+Genes
+pqsABCDE
+czcR
+rhlI
+pqsR
+lasR
+lasI
+lasR
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+phz2
+phz1
+0 3 6 9 12151821242730333639424548
+(a)
+TS
+rhlI
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+pqsABCDE
+phz2 phz1
+PhoB
+hn11
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+rhlR
+lasR
+lasI
+pqsH
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+hn21
+hn22
+AlgR
+czcR
+0 3 6 9 12151821242730333639424548
+(b)
+rhlI
+BqsR
+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+lasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+AlgR
+czcR
+hn22
+TS
+0 3 6 9 12151821242730333639424548
+(c)
+Fig. 13: Gene expression and associated information flow variations in GRNNs of a) WD b) lasR∆ and c) phoB∆ in HP.
+computing for LP and HP are also compared with the wet-
+lab data in Fig. 11d. In this comparison, it is evident that
+the predicted gene expression differences of all three genes
+are close to the wet-lab data with only around 10% variation.
+The performance similarities between the GRNN and real cell
+activities once again prove that the GRN has underpinning
+NN-like behaviors.
+C. Analysis of GRNN Computing
+Fig. 12 and Fig. 13 are used to show the diverse information
+flow of the GRNN that cause the variations in pyocyanin
+
+Lasl
+8
+POA1 LP
+Relative Gene expression
+¥7
+POA1 HP
+2
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)PqsA
+12
+POA1 LP
+Relative Gene expression
+POA1 HP
+10
+8
+6
+4
+2
+0
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)RhiR
+6
+POA1 LP
+Relative Gene expression
+¥5
+POA1 HP
+¥32
+1
+0
+0
+25
+50
+75
+100
+125
+150
+Timesteps(Hrs)100
+Model
+80
+Wet-lab
+60
+40
+20
+0
+WD
+lasRA
+PHOBApqSABCDE
+10
+CZCR
+rhll 
+8
+pqsR
+lasR-
+6
+Lasl
+rhIR -
+4
+pqsH
+phz2
+2
+phz1
+0
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+TSPqSABCDE
+10
+CZCR
+hll -
+8
+pqsR -
+lasR
+6
+Lasl -
+rhIR -
+4
+pqsH
+phz2 -
+2
+phz1
+0
+J
+9 12151821242730333639424548
+TSpqSABCDE
+5
+CZCR
+rhll :
+pqsR 
+lasR -
+3
+Lasl -
+mIR -
+2
+pqsH -
+phz2
+phzl
+0
+9 12151821242730333639424548
+TS8
+PqSABCDE
+CZCR
+rhli
+6
+pqsR -
+lasR-
+Lasl -
+4
+rhIR -
+pqsH -
+2
+phz2
+phz1
+0
+9 12151821242730333639424548
+TSPqSABCDE
+CZCR
+6
+rhll -
+5
+pqsR -
+lasR-
+4
+Lasl-
+3
+mIR -
+pqsH -
+2
+phz2-
+1
+phz1
+0
+9 12151821242730333639424548
+TSPqSABCDE
+CZCR
+6
+rhll -
+5
+pqsR -
+lasR-
+4
+Lasl-
+3
+mIR -
+pqsH -
+2
+phz2-
+1
+phz1
+0
+9 12151821242730333639424548
+TS9
+5
+4
+3
+2
+1
+0
+5
+4
+3
+2
+1
+0
+5
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+1
+0
+X
+rhlI
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+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+lasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+hn22
+AlgR
+czcR
+rhlI
+BqsR
+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
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+hn31
+hn12
+hn13
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+hn23
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+AlgR
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+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
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+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+AlgR
+czcR
+hn22
+rhlI
+BqsR
+pqsABCDE
+phz2 phz1
+PhoB
+hn11
+pqsR
+rhlR
+lasR
+lasI
+pqsH
+hn31
+hn12
+hn13
+hn14
+hn23
+hn24
+hn21
+AlgR
+czcR
+hn22
+20.0
+20.0
+15.0
+10.0
+5.0
+0.0
+15.0
+10.0
+5.0
+0.0
+20.0
+15.0
+10.0
+5.0
+0.0
+Y
+Z
+TS
+pqsABCDE
+czcR
+rhlI
+pqsR
+lasR
+lasI
+lasR
+phz2
+phz1
+0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48
+pqsH
+TS
+pqsABCDE
+czcR
+rhlI
+pqsR
+lasR
+lasI
+lasR
+phz2
+phz1
+0 3 6 9 12151821 24 273033 3639 42 4548
+pqsH
+TS
+pqsABCDE
+czcR
+rhlI
+pqsR
+lasR
+lasI
+lasR
+phz2
+phz1
+0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48
+pqsH
+TS
+pqsABCDE
+czcR
+rhlI
+pqsR
+lasR
+lasI
+lasR
+phz2
+phz1
+0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48
+pqsH
+5
+4
+3
+2
+1
+0
+(a)
+(b)
+(c)
+(d)
+Fig. 14: Illustration of GRNN information flow variations concerning the particular positions of cells within the biofilm. We
+selected four cells at a) [10, 10, 0] – close to the attached surface, b) [10, 10, 5]- close to the periphery, c) [7, 10, 13] – at
+the center and d) [3, 15, 0] – close to the attached surface and the periphery of the biofilm.
+production in LP and HP conditions, respectively. Here, we
+use gene expression profiles extracted from one bacterial cell
+located at (7, 9, 2)µm in the Cartesian coordinates that is in
+the middle region of the biofilm with limited access to the
+nutrients. First, the gene expression variations of WD, lasR∆,
+and phob∆ bacterial cells in LP (Fig. 12) and HP (Fig. 13)
+are shown for TS < 50. Next, the information flow through
+the GRNN is illustrated above each expression profile at time
+TS = 20, where the variations will be discussed. In Fig. 12a,
+impact of the inputs 3OC-LasR and phosphate cause higher
+expression levels of the nodes hn12 and phoB in the input
+layer that cascade the nodes phZ1, phZ2, pqsR, lasR, 3OC,
+rhlR and PqsH in the output layer at TS = 20. Fig. 12b has
+significantly higher pqsA operon expression levels compared
+to HP conditions (Fig. 13b), reflecting higher pyocyanin pro-
+duction that can be seen in Fig. 9b. Nevertheless, the reduced
+gene expression levels, except pqsA operon, of lasR∆ biofilm
+in both LP (Fig. 12b) and HP (Fig. 13b) conditions compared
+to the other setups emphasize that the inputs via inter-cellular
+MC significantly alter GRNN computing outputs. In contrast,
+only a smaller gene expression difference can be observed
+between the two setups of phob∆ in LP (Fig. 12c and phob∆
+in HP (Fig. 13c) resulting in minimized pyocyanin production
+differences as shown earlier in Fig. 9c.
+The GRNN model supports the understanding of the gene
+expression variations due to factors such as nutrient acces-
+sibility, where in our case is a single species biofilm. Fig.
+14 depicts the variability in the gene expression levels for
+four different locations of the biofilm at TS = 3. Fig. 14a
+and Fig. 14b are the gene expression profiles and the signal
+flow through GRNN pairs of two cells located close to the
+attached surface and the center of the biofilm. The phosphate
+accessibility for these two locations is limited. Hence, edges
+from phob have a higher information flow compared to the
+other two cells near the periphery of the biofilm, which can be
+observed in Fig. 14c and Fig. 14d. The microbes in the center
+(Fig. 14a) and the bottom (Fig. 14b) mainly have access to the
+inter-cellular MCs, while the other two bacteria have direct
+access to the extracellular phosphate.
+This GRNN produced data can further be used to understand
+the spatial and temporal dynamics of phenotypic clustering
+of gene expressions which is important in predicting and
+diagnosis of diseases [49]. Fig. 15 shows the phenotypic
+variation of WD biofilm in LP. Fig. 15a shows the number of
+cluster variations over the first 30 TSs when the significant
+phenotypic changes of the biofilm is evident. At around
+TS = 9 and TS = 10, the bacterial cells have the most diverse
+expression patterns due to the highest extracellular nutrient
+penetration (can be seen in Fig. 8a) to the biofilm and inter-
+cellular communications. Here we use four TSs (TS = 5 - Fig.
+15b, TS = 15 - Fig. 15c, TS = 23 - Fig. 15d and TS = 30 -
+Fig. 15e) to analyze this phenotypic differentiation. Each pair
+of Uniform Manifold Approximation and Projection (UMAP)
+plot and diagram of cell locations of each cluster explain how
+nutrient accessibility contribute to the phenotypic clustering.
+Although at TS = 5 (Fig. 15) the average number of clusters
+is over four, there are only two major clusters that can be
+observed with higher proportions, as shown in the pie chart.
+Among the two major clusters (blue and green) of Fig. 15b,
+the bacteria in the blue cluster can mostly be found in the
+
+8
+PqSABCDE
+CZCR
+rhli
+6
+pqsR -
+lasR-
+Lasl -
+4
+rhIR -
+pqsH -
+2
+phz2
+phz1
+0
+912151821242730333639424548
+TS8
+pqsABCDE
+CZCR
+rhll -
+6
+pqsR -
+lasR -
+Lasl -
+4
+rhIR -
+pqsH -
+2
+phz2-
+phzl
+-0
+0
+6
+912151821242730333639424548
+TSpqsABCDE
+5
+CZCR
+rhll -
+4
+pqsR 
+lasR -
+3
+Lasl -
+mIR -
+2
+pqsH -
+phz2
+phzl
+0
+9 12151821242730333639424548
+TS17.5
+15.0
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+20.0 17.5 15.0 12.5 10.0
+7.5
+5.0
+2.5
+X Label
+YLabel
+0.0PqSABCDE
+5
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+rhll :
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+lasR -
+3
+Lasl -
+mIR -
+2
+pqsH -
+phz2
+phzl
+0
+9 12151821242730333639424548
+TS10
+5
+15
+TS = 5
+TS = 15
+TS = 25
+TS = 30
+30
+20
+10
+0
+-10
+-20
+-30
+-20
+-10
+0
+10
+20
+30
+30
+20
+10
+0
+-10
+-20
+-30
+-20
+-10
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+10
+20
+30
+30
+20
+10
+0
+-10
+-20
+-30
+-20
+-10
+0
+10
+20
+30
+30
+20
+10
+0
+-10
+-20
+-30
+-20
+-10
+0
+10
+20
+30
+20
+15
+10
+5
+0
+20
+15
+10
+5
+0
+0
+5
+10
+15
+20
+X
+Y
+Z
+20
+15
+10
+5
+0
+20
+15
+10
+5
+0
+0
+5
+10
+15
+20
+X
+Y
+Z
+20
+15
+10
+5
+0
+20
+15
+10
+5
+0
+0
+5
+10
+15
+20
+X
+Y
+Z
+20
+15
+10
+5
+0
+20
+15
+10
+5
+0
+0
+5
+10
+15
+20
+X
+Y
+Z
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+20
+16
+12
+8
+4
+0
+No. of cluster
+(b)
+(c)
+(d)
+(e)
+(a)
+UMAP2
+UMAP2
+UMAP2
+UMAP2
+UMAP1
+UMAP1
+UMAP1
+UMAP 1
+TS
+Fig. 15: Illustration of GRNN-driven phenotypic cluster formation behaviors. a) shows the number of clusters (with their
+proportions via pie charts) for TS < 30, b), c), d) and e) are pairs of the UMAP clustering based on gene expressions of cells
+and their locations in the biofilm at TS = 5, TS = 15, TS = 23 and TS = 30, respectively.
+center of the biofilm, while the green cluster cells are close
+to the periphery. Fig. 15c and Fig. 15d have more clusters
+as the nutrient accessibility among cells is high. In contrast,
+due to the lack of nutrients in the biofilm, a limited number
+of clusters can be seen in the biofilm after around TS = 30,
+which can be observed Fig. 15e.
+V. CONCLUSION
+The past literature has captured the non-linear signal com-
+puting mechanisms of Bacterial GRNs, suggesting under-
+pinning NN behaviors. This study extracts a GRNN with
+summarized multi-omics gene expression regulation mecha-
+nisms as weights that can further analyze gene expression
+dynamics, design predictive models, or even conduct in-vivo
+computational tasks. We used P. aeruginosa single species
+biofilm as a use case and extracted relevant gene expression
+data from databases such as RegulomePA and transcriptomic
+data from databases including GEO. Due to the complexity
+of the GRN and expression dynamics, we only considered a
+smaller sub-network of the GRN as a GRNN that is associated
+with QS, iron and phosphate inputs, and pyocyanin produc-
+tion. Considering this GRNN, we modeled the computation
+process that drives cellular decision-making mechanism. As
+bacteria live in ecosystems in general where intra-cellular
+communication play a significant role in cellular activities,
+an in-silico biofilm is modeled using GNN to further analyze
+the biofilm-wide decision-making. A comparison between
+the GRNN generated data and the transcriptomic data from
+the literature exhibits that the GRN behaves similarly to a
+NN. Hence, this model can explore the causal relationships
+between gene regulation and cellular activities, predict the
+future behaviors of the biofilm as well as conduct bio-hybrid
+computing tasks. Further, in the GRNN extraction phase, we
+were able to identify the possibility of modeling more network
+structures with various number of input nodes, hidden layers,
+and output nodes. In addition, GRN components including
+auto regulated genes and bidirectional intergenic interactions
+hints the possibility of extracting more sophisticated types of
+GRNNs such as Recurrent NN and Residual NN in the future.
+The idea of extracting sub-networks as NNs can lead to more
+intriguing intra-cellular distributed computing. Further, this
+model can be extended to multi-species ecosystems for more
+advanced predictive models as well as distributed computing
+architectures combining various NNs.
+REFERENCES
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+17.5
+15.0
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+17.5 15.0 12.5 10.0
+7.5
+5.0
+2.5
+X Label
+YLabelUMAP/DBSCAN
+30
+20
+10
+0
+-10
+-20
+-30
+-20
+10
+0
+10
+20
+30UMAP/DBSCAN
+30
+20
+10
+0
+-10 
+-20
+-30
+-20
+10
+0
+10
+20
+3012
+13
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+2
+27
+29
+Timesteps(Hrs)Timestep: 5
+0
+431
+5
+2Timestep: 15
+5
+3
+2
+0Timestep: 23
+2
+10
+3Timestep: 30
+1
+0
+217.5
+15.0
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+17.5 15.0 12.5 10.0
+7.5
+5.0
+2.5
+X Label
+YLabelUMAP/DBSCAN
+30
+20
+10 
+0
+10
+-20
+-30
+-20
+10
+0
+10
+20 
+3017.5
+15.0
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+17.515.0
+012.510.0
+7.5
+5.0
+2.5
+X Label
+Y Label17.5
+15.0
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+17.5 15.0 12.5 10.0
+7.5
+5.0
+2.5
+X Label
+Y LabelUMAP/DBSCAN
+30
+20
+10 
+.
+0
+-10
+-20
+-30
+-20
+10
+0
+10
+20 
+3011
+[4] M. Land, L. Hauser, S.-R. Jun, I. Nookaew, M. R. Leuze, T.-H.
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+THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+1
+Conformal Loss-Controlling Prediction
+Di Wang, Ping Wang, Zhong Ji, Xiaojun Yang, Hongyue Li
+Abstract—Conformal prediction is a learning framework con-
+trolling prediction coverage of prediction sets, which can be
+built on any learning algorithm for point prediction. This work
+proposes a learning framework named conformal loss-controlling
+prediction, which extends conformal prediction to the situation
+where the value of a loss function needs to be controlled.
+Different from existing works about risk-controlling prediction
+sets and conformal risk control with the purpose of controlling
+the expected values of loss functions, the proposed approach
+in this paper focuses on the loss for any test object, which is
+an extension of conformal prediction from miscoverage loss to
+some general loss. The controlling guarantee is proved under the
+assumption of exchangeability of data in finite-sample cases and
+the framework is tested empirically for classification with a class-
+varying loss and statistical postprocessing of numerical weather
+forecasting applications, which are introduced as point-wise
+classification and point-wise regression problems. All theoretical
+analysis and experimental results confirm the effectiveness of our
+loss-controlling approach.
+Index Terms—Conformal prediction, Loss-controlling predic-
+tion, Finite-sample guarantee, Weather forecasting.
+I. INTRODUCTION
+Prediction sets convey uncertainty or confidence information
+for users, which is more preferred than prediction points,
+especially for sensitive applications such as medicine, finance
+and weather forecasting [1] [2] [3]. Conformal prediction
+(CP) is a learning framework providing prediction sets for
+test labels, which guarantees the finite-sample coverage at a
+user-preset level under the assumption of exchangeability of
+data samples [4]. This property of validity has been proved
+both theoretically and empirically in many works and applied
+to many areas [5] [6]. Besides, many researches extend CP
+to more general cases, such as conformal prediction for
+multi-label learning [7] [8], functional data [9] [10], few-shot
+learning [11] and distribution shift [12] [13].
+However, existing CP methods mainly make promise about
+the coverage of prediction sets, which limits its use to other
+broad applications concerning controlling general losses. To
+tackle this issue, two works have been proposed recently.
+[14] employs DeepSets to estimate the expected value or
+This work has been submitted to the IEEE for possible publication.
+Copyright may be transferred without notice, after which this version may
+no longer be accessible.
+This work was supported by the National Natural Science Foundation of
+China under Grant 62106169. (Corresponding author: Hongyue Li)
+Di Wang and Zhong Ji are with School of Electrical and Information Engi-
+neering, Tianjin University, Tianjin 300072, China, and also with Tianjin Key
+Laboratory of Brain-inspired Intelligence Technology, School of Electrical and
+Information Engineering, Tianjin University, Tianjin 300072, China. (email:
+wangdi2015@tju.edu.cn; jizhong@tju.edu.cn;).
+Ping Wang and Hongyue Li are with School of Electrical and In-
+formation Engineering, Tianjin University, Tianjin 300072, China. (email:
+wangps@tju.edu.cn; lihongyue@tju.edu.cn).
+Xiaojun Yang is with Tianjin Meteorological Observatory, Tianjin 300074,
+China. (email: boluo0127@yeah.net)
+the cumulative distribution function of the number of false
+positives, and then use calibration data to control the number
+of false positives of prediction sets. Conformal risk control
+(CRC) [15] extends CP to prediction tasks of controlling
+the expected value of a general loss based on finding the
+optimal parameter for nested prediction sets. The spirit is to
+employ calibration data to obtain the information of the upper
+bound of the expected value of the loss function at hand and
+control the expected value for the test object, whose main idea
+was originally proposed from their pioneer work named risk-
+controlling prediction sets [16].
+This paper extends CP to the situation where the value of a
+general loss function needs to be controlled, which has been
+not considered in the literature to our best knowledge. This
+situation is reasonable and practical since one may only care
+about the loss value for a specific test object, just like the
+coverage guarantee made by CP. Our approach is similar to
+CRC with the main difference being that we focus on finding
+the optimal parameter for nested prediction sets to control the
+loss. Therefore, we also concentrate on inductive conformal
+prediction [17] or split conformal prediction [18] process like
+CRC.
+Recall that inductive conformal prediction makes the cov-
+erage guarantee as follows,
+P
+�
+Yn+1 ∈ C(n)
+1−δ(Xn+1)
+�
+≥ 1 − δ,
+where δ is the significance level preset by users, C(n)
+1−δ
+is the set predictor made by CP based on n calibration
+data {(Xi, Yi)}n
+i=1, (Xn+1, Yn+1) is the test feature-response
+pair, and the randomness is from both {(Xi, Yi)}n
+i=1 and
+(Xn+1, Yn+1). By comparison, conformal loss-controlling
+prediction (CLCP), the learning framework proposed in this
+paper, provides the controlling guarantee as follows,
+P
+�
+L
+�
+Yn+1, Cλ∗(Xn+1)
+�
+≤ α
+�
+≥ 1 − δ,
+where L is a loss function satisfying some monotonic con-
+ditions as in [15], α is the preset level of loss, Cλ(·) is the
+prediction set usually constructed by an underlying algorithm
+and a parameter λ. The optimal λ∗ is obtained based on α,
+δ and calibration data. The controlling guarantee needs two
+levels α and δ to be chosen by users, which is similar with
+that in [16], i.e., CLCP guarantees that the prediction loss
+is not greater than α with high probability 1 − δ when δ is
+small such as 0.1. If L is replaced by false positive for multi-
+label classification, the controlling guarantee above is also the
+(α, δ)-FP validity defined in Definition 4.2 in [14].
+We prove the controlling guarantee for distribution-free and
+finite-sample settings with the assumption of exchangeability
+arXiv:2301.02424v1  [cs.LG]  6 Jan 2023
+
+THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+2
+of data samples. The main idea is that we find the λ∗ to make
+the 1 − δ quantile of the loss values on calibration data not
+greater than α, which is inspired by CRC focusing on making
+the mean of the loss values not greater than α. Furthermore,
+we test our approach in classification with a class-varying loss
+introduced in [16], and postprocessing of numerical weather
+forecasts, which we consider as point-wise classification and
+point-wise regression problems. The experimental results em-
+pirically confirm the theoretical guarantee we prove in this
+paper.
+In summary, the main contributions of this paper are:
+• A learning framework named conformal loss-controlling
+prediction (CLCP) is proposed for controlling the pre-
+diction loss for the test object. The approach is simple
+to implement and can be built on any machine learning
+algorithm for point prediction.
+• The controlling guarantee is proved mathematically for
+finite-sample cases with the exchangeability assumption,
+without any further assumption for data distribution.
+• The controlling guarantee is empirically verified by clas-
+sification with a class-varying loss and weather forecast-
+ing problems, which confirms the effectiveness of CLCP.
+The rest of this paper is organized as follows. Section II
+reviews inductive conformal prediction and conformal risk
+control. Section III introduces conformal loss-controlling pre-
+diction and its theoretical guarantee. Section IV conducts
+experiments to test the proposed method and the conclusions
+are drawn in Section V.
+II. INDUCTIVE CONFORMAL PREDICTION AND
+CONFORMAL RISK CONTROL
+This section reviews inductive conformal prediction and
+conformal risk control. Throughout this paper, {(Xi, Yi)}n+1
+i=1
+denotes n + 1 data drawn exchangeably from PXY
+on
+X × Y, where {(Xi, Yi)}n
+i=1 is the calibration dataset and
+(Xn+1, Yn+1) is the test object-response pair. We use lower-
+case letter (xi, yi) to represent the realization of (Xi, Yi).
+The set-valued function and loss function considered in
+this paper are the same as those in [15] and [16], which we
+formally introduce as follows. Let Cλ : X → Y′ be a set-
+valued function with a parameter λ ∈ R, where Y′ represents
+some space of sets and R is the set of real numbers. Taking
+single-label classification for example, Y′ can be the power
+set of Y. For binary image segmentation, Y′ can be equal to
+Y as the space of all possible results of image segmentation,
+where the sets here stand for all of the pixels of positive class
+for the image.
+We also introduce the nesting property for prediction sets
+and losses as in [16] as follows. For each realization of input
+object x, we assume that Cλ(x) satisfies the following nesting
+property:
+λ1 < λ2 =⇒ Cλ1(x) ⊆ Cλ2(x).
+(1)
+Let L : Y × Y′ → R be a loss function respecting the nesting
+property for each realization of response y:
+C1 ⊆ C2 ⊆ Y′ =⇒ L(y, C2) ≤ L(y, C1) ≤ B,
+(2)
+where B is the upper bound of the loss function.
+A. Inductive Conformal Prediction
+Inductive conformal prediction (ICP) is a computationally
+efficient version of the original conformal prediction approach.
+It starts with any measurable function named nonconformity
+measure A : X × Y → R and obtain n nonconformity scores
+as
+Ai = A(Xi, Yi),
+for i = 1, · · · , n. Then, with the exchangeable assumption and
+a preset δ ∈ (0, 1), one can conclude that
+P
+�
+A(Xn+1, Yn+1) ≤ Q(n)
+1−δ
+�
+≥ 1 − δ,
+where Q(n)
+1−δ is the 1 − δ quantile of {Ai}n
+i=1 ∪ {∞} [12].
+Therefore, the prediction set made by ICP is
+C(n)
+1−δ(Xn+1) = {y : A(Xn+1, y) ≤ Q(n)
+1−δ},
+which satisfies
+P
+�
+Yn+1 ∈ C(n)
+1−δ(Xn+1)
+�
+≥ 1 − δ.
+The nonconformity measure A is often defined based on
+a point prediction model ˆf learned from some other training
+samples, each of which is also drawn from PXY .
+Here is an example of constructing prediction sets with CP.
+For a classification problem with K classes, one can first train
+a classifier ˆf : X → [0, 1]K with the ith output being the
+estimation of the probability of the ith class, and calculate the
+nonconformity scores as
+A(x, y) = 1 − ˆfk(x),
+where ˆfk is the kth output of ˆf(x), if y stands for the kth
+class. Therefore, the corresponding prediction set for an input
+object x is
+C(n)
+1−δ(x) = {k : ˆfk(x) ≥ 1 − Q(n)
+1−δ},
+which indicates that k ∈ C(n)
+1−δ(x) if the estimated probability
+of kth class is not less than 1 − Q(n)
+1−δ.
+B. Conformal Risk Control
+Different from conformal prediction, CRC starts with a set-
+valued function with the nesting property, whose approach is
+inspired by nested conformal prediction [19] and was first
+proposed in the researches about risk-controlling prediction
+sets.
+Assume one has a way of constructing a set-valued function
+Cλ with the nesting property of formula (1). Given a loss
+function L with the nesting property of formula (2), the
+purpose of CRC is to find λ∗ such that
+E
+�
+L(Yn+1, Cλ∗(Xn+1))
+�
+≤ α,
+(3)
+i.e., the expected loss or the risk is not greater than α.
+To do so, CRC first calculates Li as
+Li(λ) = L(Yi, Cλ(Xi)),
+(4)
+
+THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+3
+with the fact that Li(λ) is a monotone decreasing function of
+λ based on the nesting properties. Then, CRC searches for λ∗
+using the following equation,
+λ∗ = inf
+�
+λ :
+n
+n + 1
+ˆRn(λ) +
+B
+n + 1 ≤ α
+�
+,
+where ˆRn(λ) = (L1(λ) + · · · + Ln(λ))/n is an estimation of
+the risk on calibration data and B is introduced to make the
+estimation not overconfident. Also, to make λ∗ well defined,
+one should assume that ˆRn(λ) is right continuous.
+These two steps of CRC are too simple that one may
+surprise about its theoretical conclusion that with the assump-
+tion of exchangeability of data samples, the prediction set
+Cλ∗(Xn+1) obtained by CRC satisfies formula (3), which has
+been also proved empirically in [15]. CRC extends CP from
+controlling the expected value of miscoverage loss to some
+general loss, which can be applied to the cases where Y is
+beyond real numbers or vectors, such as images, fields and
+even graphs.
+After tackling the theoretical issue, the problem for CRC
+is how to construct Cλ. Here, we also give an example of a
+classification problem with K classes. In fact, with the same
+notations of the example in Section II-A, CRC can construct
+the prediction set as
+Cλ(x) = {k : ˆfk(x) ≥ 1 − λ}.
+Therefore, as long as L satisfies formula (2), such as L is the
+indicator of miscoverage, CRC guarantees to control the risk
+as formula (3).
+III. CONFORMAL LOSS-CONTROLLING PREDICTION AND
+ITS THEORETICAL ANALYSIS
+This section introduces the approach of CLCP and its
+theoretical analysis. CLCP also has two steps like CRC, and
+the main difference between them is that CLCP focuses on
+whether the estimation of the 1 − δ quantile of the losses is
+not greater than α while CRC concentrates on whether the
+mean of the losses not greater than α. The controlling of the
+1−δ quantile of the losses makes CLCP be able to control the
+value of a general loss by employing the probability inequation
+derived from the exchangeability assumption, which is also
+employed by ICP if the loss is seen as the nonconformity
+score.
+Suppose one has a way of constructing a set-valued function
+Cλ with the nesting property of formula (1), which can be the
+same as that used in CRC. Here, we assume that the parameter
+λ is selected from a discrete set Λ, such as from 0 to 1 with a
+step size 0.01, which avoids us from the assumption of right
+continuous for the loss function in theoretical analysis, and
+is also reasonable since we actually search for λ∗ with some
+step size in practice [15] [16]. Besides, the latest paper about
+risk-controlling prediction also makes this discrete assumption
+for general cases [20]. After determining Cλ(·) and Λ, CLCP
+first calculates Li(λ) on calibration data as formula (4). Then,
+for any preset α ∈ R and δ ∈ (0, 1), CLCP searches for λ∗
+such that
+λ∗ = min
+�
+λ ∈ Λ : Q(n)
+1−δ(λ) ≤ α
+�
+,
+(5)
+with Q(n)
+1−δ(λ) being the 1 − δ quantile of {Li(λ)}n
+i=1 ∪ {B}.
+The approach of CLCP is summarised in Algorithm 1, which
+is easy to implement.
+Algorithm 1 Conformal Loss-Controlling Prediction
+Input:
+Calibration dataset {(xi, yi)}n
+i=1, test input object xn+1,
+the set predictor Cλ satisfies formula (1), the loss function
+L satisfies formula (2), preset α ∈ R and δ ∈ (0, 1).
+Output:
+Predictive set for yn+1.
+1: Based on formula (4), calculate {Li(λ)}n
+i=1.
+2: Search for λ∗ satisfying formula (5).
+3: return Cλ∗(xn+1)
+Next, we introduce the definition of (α, δ)-loss-controlling
+set predictors and then prove our theoretical conclusion about
+CLCP.
+Definition 1. Given a loss function L : Y × Y′ → R and a
+random sample (X, Y ) ∈ X ×Y, a random set-valued function
+C whose realization is in the space of functions X → Y′ is a
+(α, δ)-loss-controlling set predictor if it satisfies that
+P
+�
+L
+�
+Y, C(X)
+�
+≤ α
+�
+≥ 1 − δ,
+where the randomness is both from C and (X, Y ).
+After all these preparations, we can prove in Theorem 1
+that Cλ∗ constructed by CLCP is a (α, δ)-loss-controlling set
+predictor.
+Theorem 1. Suppose {(Xi, Yi)}n+1
+i=1 are n + 1 data drawn
+exchangeably from PXY on X × Y, Cλ : X → Y′ is a set-
+valued function satisfying formula (1) with the parameter λ
+taking values from a discrete set Λ ⊂ R , L : Y × Y′ → R
+is a loss function satisfying formula (2) and Li(λ) is defined
+as formula (4). For any preset α ∈ R, if L also satisfies the
+following conditions,
+min
+λ max
+i
+Li(λ) < α,
+max
+λ
+min
+i
+Li(λ) > α,
+(6)
+then for any δ ∈ (0, 1), we have
+P
+�
+L
+�
+Yn+1, Cλ∗(Xn+1)
+�
+≤ α
+�
+≥ 1 − δ,
+(7)
+where λ∗ is defined as formula (5).
+Proof. Let Q(n+1)
+1−δ (λ) be the 1 − δ quantile of {Li(λ)}n+1
+i=1 ,
+and define ˜λ as
+˜λ = min
+�
+λ ∈ Λ : Q(n+1)
+1−δ (λ) ≤ α
+�
+.
+
+THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+4
+Similarly, let Q(n)
+1−δ(λ) be the 1 − δ quantile of {Li(λ)}n
+i=1 ∪
+{B}, and we have
+λ∗ = min
+�
+λ ∈ Λ : Q(n)
+1−δ(λ) ≤ α
+�
+.
+Since B is the upper bound of Ln+1(λ), by definition, we
+have
+˜λ ≤ λ∗,
+and the conditions introduced in formula (6) is to make ˜λ and
+ˆλ well defined. This leads to
+Ln+1(λ∗) ≤ Ln+1(˜λ),
+(8)
+as Cλ and L satisfy the nesting properties of formula (1) and
+(2).
+Since ˜λ is dependent on the whole dataset {(Xi, Yi)}n+1
+i=1 ,
+{Li(˜λ)}n+1
+i=1 are exchangeable variables, which leads to
+P
+�
+Ln+1(˜λ) ≤ Q(n+1)
+1−δ (˜λ)
+�
+≥ 1 − δ,
+(9)
+as Q(n+1)
+1−δ (˜λ) is just the corresponding 1−δ quantile (See the
+proof of Lemma 1 in [12]).
+Combining the definition of ˜λ, formula (8) and (9), we have
+P
+�
+Ln+1(λ∗) ≤ α
+�
+≥ 1 − δ,
+which completes the proof.
+At the end of this section, we show that CP can be seen
+as a special case of CLCP from the following viewpoint.
+Suppose Cλ is constructed by a nonconformity score A, which
+is defined as
+Cλ(x) = {y : A(x, y) ≤ λ},
+and L is the miscoverage loss such that
+Li(λ) = L(yi, Cλ(xi)) = I{yi /∈ Cλ(xi)},
+where I is the indicator function. In this case, Q(n)
+1−δ(λ) can
+only be 0 or 1 as the loss can only be these two numbers.
+Besides, only α ∈ [0, 1) is meaningful, which means that
+one wants to control the miscoverage. For CLCP, let Λ be
+an arithmetic sequence whose common difference, minimum
+and maximum are ∆, λmin and λmax respectively and set
+α = 0. By definition, λ∗ can be written as
+λ∗ = min
+�
+λ ∈ Λ :
+1
+n + 1
+n
+�
+i=1
+I{ai ≤ λ} ≥ 1 − δ
+�
+= min
+�
+λ ∈ Λ : 1
+n
+n
+�
+i=1
+I{ai ≤ λ} ≥ ⌈(1 − δ)(n + 1)⌉
+n
+�
+,
+where ai = A(xi, yi) is the nonconformity score of the ith
+calibration data for CP. In comparison, referring to [15], the
+optimal ˆλ for CP is
+ˆλ = inf
+�
+λ ∈ R : 1
+n
+n
+�
+i=1
+I{ai ≤ λ} ≥ ⌈(1 − δ)(n + 1)⌉
+n
+�
+.
+Therefore, if λmin < ai < λmax for each i, we have
+|λ∗ − ˆλ| ≤ ∆,
+which implies that the prediction sets of CP and CLCP are
+nearly the same if ∆ is small enough. In summary, if Cλ
+and L have special forms and Λ includes the upper and lower
+bounds of nonconformity scores with ∆ being small enough
+to be ignored, CP can be seen as a special case of CLCP.
+IV. EXPERIMENTS
+This section conducts the experiments to empirically test the
+approach of CLCP. First, we built CLCP for the classification
+problem with a class-varying loss introduced in [16]. Then,
+we focus on two types of weather forecasting applications,
+which can be seen as point-wise classification and point-
+wise regression problems respectively. All experiments were
+coded in Python [21]. The statistical learning methods used in
+Section IV-A were implemented using Scikit-learn [22] and
+the deep learning methods used in Section IV-B and Section
+IV-C were implemented with Pytorch [23].
+A. CLCP for classification with a class-varying loss
+We collected 20 binary or multiclass classification datasets
+from UCI repositories [24] whose information is summarized
+in Table I. The problem is to make the prediction sets of labels
+controlling the following loss
+L(y, C) = LyI{y /∈ C},
+where Ly is the loss for y being not in the prediction set.
+The loss for each label is generated uniformly on (0, 1) like
+[16]. Support vector machine (SVM) [25], neural network
+(NN) [26] and random forests (RF) [27] were employed as
+the underlying algorithms separately to construct prediction
+sets based on CLCP. The prediction set Cλ is constructed as
+Cλ(x) = {k : ˆfk(x) ≥ 1 − λ},
+where ˆfk is the estimated probability of the observation being
+kth class by the corresponding underlying algorithm. For each
+dataset, we used 20% of the data for testing and 80% and 20%
+of the remaining data for training and calibration respectively.
+Based on the training data, we selected the meta-parameters
+with three-fold cross-validation and used the optimal meta-
+parameters to train the classifiers. The regularization parameter
+of SVM was selected from {0.001, 0.01, 0.1, 1, 10, 100}, and
+the learning rate and the epochs of NN were selected from
+{0.001, 0.0001} and {200, 500, 1000}. The number of trees
+of RF were selected from {100, 300, 500} and the partition
+criterion was either gini or entropy. After training, we used the
+trained classifiers and the calibration data to search for λ∗ with
+Algorithm 1 and construct the final set predictors. All of the
+features were normalized to [0, 1] by min–max normalization
+and for each dataset, the experiments were conducted 10 times
+and the average results were recorded.
+The bar plots in Fig. 1 and Fig. 2 show the experimental
+results for 20 public datasets with δ ∈ {0.05, 0.1, 0.15, 0.2}
+and α ∈ {0.1, 0.2}. The results in Fig. 1 concern about the
+
+THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+5
+TABLE I
+DATASETS FROM UCI REPOSITORIES
+Dataset
+Examples
+Dimensionality
+Classes
+bc-wisc-diag
+569
+30
+2
+car
+1728
+6
+4
+chess-kr-kp
+3196
+36
+2
+contrac
+1473
+9
+3
+credit-a
+690
+15
+2
+credit-g
+1000
+20
+2
+ctg-10classes
+2126
+21
+10
+ctg-3classes
+2126
+21
+3
+haberman
+306
+3
+2
+optical
+5620
+62
+10
+phishing-web
+11055
+30
+2
+st-image
+2310
+18
+7
+st-landsat
+6435
+36
+6
+tic-tac-toe
+958
+9
+2
+wall-following
+5456
+24
+4
+waveform
+5000
+21
+3
+waveform-noise
+5000
+40
+3
+wilt
+4839
+5
+2
+wine-quality-red
+1599
+11
+6
+wine-quality-white
+4898
+11
+7
+frequency of the prediction losses being more than α on test
+set, which is the estimated probability of
+P
+�
+L
+�
+Yn+1, Cλ∗(Xn+1)
+�
+> α
+�
+,
+which should be near or lower than δ empirically due to
+formula (7). The bar plots of Fig. 1 demonstrate that the
+frequency of the prediction losses is near or below δ, which
+verifies the conclusion of Theorem 1.
+The bar plots of Fig. 2 show the average sizes of prediction
+sets for different δ, describing the informational efficiency of
+the prediction sets. Changing δ can effectively change the
+average size of prediction sets and changing α may slightly
+change average size (such as the results for wine-quality-red).
+Although many prediction sets are meaningful with average
+sizes being near 1, the prediction sets for the dataset contrac
+may be not usefull, since no matter how to change δ and α,
+the average sizes of the prediction sets are all near or above
+2, whereas the number of classes of contrac is 3. Thus, how
+to construct efficient prediction sets in the learning framework
+of CLCP is worth exploring for further researches.
+Combining Fig. 1 and Fig. 2, we observe that different
+classifiers can perform differently for different datasets, which
+indicates that the underlying algorithm affects the performance
+and the model selection approach is necessary for CLCP.
+B. CLCP for high-impact weather forecasting
+The remaining experiments apply CLCP to weather fore-
+casting problems. Here we concentrate on postprocessing of
+the forecasts made by numerical weather prediction (NWP)
+models [28] [29]. NWP models use equations of atmospheric
+dynamics and estimations of current weather conditions to do
+weather forecasting, which is the mainstream weather fore-
+casting technique nowadays especially for forecasting beyond
+12 hours. Many errors affect the performance of NWP models,
+such as the estimation errors of initial conditions and the
+approximation errors of NWP models, leading to the research
+topic about postprocessing the forecasts of NWP models. Most
+postprocessing methods are built on some learning process,
+which takes the forecasts of NWP models as inputs and the
+observations of weather elements or events as outputs.
+In this paper, we use CLCP to postprocess the ensemble
+forecasts with the control forecast and 50 perturbed forecasts
+issued by the NWP model from European Centre for Medium-
+Range Weather Forecasts (ECMWF) [30], which are obtained
+from the THORPEX Interactive Grand Global Ensemble
+(TIGGE) dataset [31]. We focus on 2-m maximum temperature
+and minimum temperature between the forecast lead times
+of 12nd hour and 36th hour with the forecasts initialized at
+0000 UTC. The forecast fields are grided with the resolution
+of 0.5◦ × 0.5◦ and the corresponding label fields with the
+same resolution are extracted from the ERA5 reanalysis data
+[32]. The area ranges from 109◦E to 122◦E in longitude and
+from 29◦N to 42◦N in latitude, covering the main parts of
+North China, East China and Central China, whose grid size
+is 27 × 27. The ECMWF forecast data and ERA5 reanalysis
+data are collected from 2007 to 2020 (14 years).
+We first consider high-impact weather forecasting, which
+is to forecast whether a high-impact weather exists for each
+grid and can be seen as a point-wise classification problem or
+image segmentation problem for computer vision. The high-
+impact weather we consider is whether the 2-m maximum
+temperature is above 35 °C or the 2-m minimum temperature
+is below −15 °C for each grid. These two cases are treated
+as high temperature weather or low temperature weather in
+China, which make meteorological observatories issue high
+temperature warning or low temperature warning respectively.
+The prediction sets and the loss function used for high-
+impact weather forecasting are the same as those for image
+segmentation in [15]. Taking the ensemble forecast fields of
+the NWP model as input x, the corresponding label y is a set
+of grids having high-impact weather, which can be seen as
+a segmentation problem for high-impact weather. Therefore,
+we first train a segmentation neural network f(x), where
+f(p,q)(x) is the estimated probability of the grid (p, q) having
+high-impact weather. Then the set-valued function Cλ can be
+constructed as
+Cλ(x) = {(p, q) : f(p,q)(x) ≥ 1 − λ},
+(10)
+and the loss function is
+L(y, C) = 1 − |y ∩ C|
+|y|
+,
+(11)
+which measures the ratio of the prediction sets failing to do the
+warning. We use CLCP with the prediction set and the loss
+function above to do high temperature and low temperature
+forecasting respectively.
+1) Dataset for high temperature forecasting: The reanalysis
+fields of 2-m maximum temperature were collected from
+ERA5 and the label fields were calculated based on weather
+the 2-m maximum temperature is above 35 °C. To make the
+loss function take finite values, we only collected the data
+whose label fields have at least one high temperature grid
+to do this empirical study, which resulted in 1200 samples
+in total, i.e., 1200 ensemble forecasts from the NWP model
+of ECMWF and corresponding label fields calculated from
+
+THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+6
+Fig. 1. Bar plots of the frequencies of the prediction losses being more than α vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for classification with a class-varying
+loss. The first row corresponds to α = 0.1 and the second row corresponds to α = 0.2. Different columns represent different classifiers. All bars are near or
+below the preset δ, which confirms the controlling guarantee of CLCP empirically.
+Fig. 2. Bar plots of the average sizes of prediction sets vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for classification with a class-varying loss. The first row
+corresponds to α = 0.1 and the second row corresponds to α = 0.2. Different columns represent different classifiers. The plots demonstrate the information
+in prediction sets. In general, large δ leads to small average size and different classifiers have different informational efficiency.
+ERA5. We name this dataset as HighTemp.
+2) Dataset for low temperature forecasting: The dataset
+for testing CLCP for low temperature weather forecasting
+was constructed in a similar way. The reanalysis fields of
+2-m minimum temperature were collected from ERA5 and
+the label fields were calculated based on weather the 2-m
+minimum temperature is below −15 °C. We only collected
+the data whose label fields have at least one low temperature
+grid to do this empirical study, which resulted in 1233 samples
+in total. We name this dataset as LowTemp.
+For each dataset, the same process was used to conduct the
+experiment as Section IV-A , i.e., all forecasts from the NWP
+model were normalized to [0, 1] by min–max normalization,
+and we used 20% of the data for testing and 80% and 20%
+of the remaining data for training and calibration respectively.
+We employed two fully convolutional neural networks [33]
+for binary image segmentation as our underlying algorithms.
+One was U-Net [34] with the same structure as that in [35],
+whose numbers of hidden feature maps were all set to 32. The
+other was the naive deep neural network (nDNN) with the
+same encoder-decoder structure as the U-Net without skip-
+connections, i.e., the U-Net removing skip-connections. We
+
+α = 0.1 I Classifier = SVM
+α = 0.1 I Classifier = NN
+α = 0.1  Classifier = RF
+0.30
+0.25
+0.20
+Dataset
+ bc-wisc-diag
+car
+chess-kr-kp
+contrac
+credit-a
+0.05
+credit-g
+ctg-10classes
+ctg-3classes
+0.00
+haberman
+α = 0.2 I Classifier = SVM
+α = 0.2 I Classifier = NN
+α = 0.2 I Classifier = RF
+optical
+0.30
+phishing-web
+ st-image
+st-landsat
+0.25
+tic-tac-toe
+α
+wall-following
+waveform
+waveform-noise
+wilt
+wine-quality-red
+wine-quality-white
+0.05
+0.00
+0.15
+0.15
+0.2
+0.1
+0.15
+0.05
+0.1
+0.05
+0.2
+0.05
+0.1
+0.2
+6
+6
+6α = 0.1 I Classifier = SVM
+α = 0.1 I Classifier = NN
+α = 0.1 I Classifier = RF
+3.0
+2.5
+Dataset
+1.5
+bc-wisc-diag
+car
+chess-kr-kp
+1.0
+contrac
+credit-a
+0.5
+credit-g
+ctg-10classes
+ctg-3classes
+0.0
+haberman
+α = 0.2 I Classifier = SVM
+α = 0.2 I Classifier = NN
+α = 0.2 I Classifier = RF
+optical
+phishing-web
+3.0
+st-image
+st-landsat
+tic-tac-toe
+2.5
+ wall-following
+waveform
+2.0
+waveform-noise
+Average s
+wilt
+wine-quality-red
+1.5
+wine-quality-white
+1.0
+0.5
+0.0
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+6
+6
+6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+7
+Fig. 3. Bar plots of the frequencies of the prediction losses being more than α vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for high-impact weather forecasting.
+The first row corresponds to HighTemp and the second row corresponds to LowTemp. Different columns represent different α. All bars are near or below
+the preset δ, which confirms the controlling guarantee of CLCP empirically.
+Fig. 4.
+Boxen plots of the prediction losses vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for high-impact weather forecasting. The first row corresponds to
+HighTemp and the second row corresponds to LowTemp. Different columns represent different α. The loss distributions are controlled by α and δ properly
+to obtain the empirical validity in Fig. 3.
+use these two neural networks to show that the designation of
+the underlying algorithm is necessary for better performance,
+as U-Net fuses multi-scale features and nDNN does not. The
+data for training U-Net and DNN were further partitioned
+to the validation part (10%) for model selection and proper
+training part (90%) for updating the parameters. Adam op-
+timization [36] was used for training. The learning rate was
+set to 0.0001 and the number of epochs was set to 50. After
+training 50 epochs, the model with lowest binary cross entropy
+on validation data was used for formula (10) to construct
+prediction sets, where λ is search from 1 to 0 with step size
+0.01. The experiments of using CLCP for the loss function
+as formula (11) were conducted 10 times and the prediction
+results on test set are shown in Fig. 3, Fig. 4 and Fig. 5.
+Fig. 3 also shows the bar plots of the frequencies of the
+prediction losses being more than α for δ = 0.05, 0.1, 0.15
+and 0.2. Four column stands for the cases where α
+=
+0.05, 0.1, 0.15 and 0.2 respectively. It can be seen that for
+the two datasets HighTemp and LowTemp, all bars are near
+or below the preset δ, which verifies formula (7) empirically.
+Fig. 4 further shows the distributions of the losses for different
+δ and different α using boxen plots, which contain more
+information than box plots by drawing narrow boxes for tails.
+It can be seen that larger α and δ lead to larger losses, which
+is reasonable since large α and δ relax the constraint on
+prediction losses. We measure the informational efficiency of
+the prediction set Cλ∗(x) using its normalized size defined
+as |Cλ∗(x)|/PQ, where P and Q are the numbers of the
+vertical and the horizontal grids respectively. The distributions
+of normalized sizes in Fig. 5 show that U-Net is more
+informationally efficient than nDNN, which indicates that
+designation of the underlying algorithm is important for CLCP.
+Different α and δ lead to different normalized sizes, implying
+the trade-off among the preset loss bound α, confidence level
+1 − δ and informational efficiency of the prediction sets.
+By choosing α and δ properly, the prediction sets of CLCP
+
+Dataset = HighTemp | α = 0.05
+Dataset = HighTemp | α = 0.15
+Dataset = HighTemp I α = 0.2
+Dataset = HighTemp I α = 0.1
+0.30
+0.25
+0.05
+0.00
+Model
+Dataset = LowTemp I α = 0.05
+Dataset = LowTemp I α = 0.1
+Dataset = LowTemp I α = 0.15
+Dataset = LowTemp I α = 0.2
+nDNN
+0.30
+U-Net
+0.25
+0.05
+0.00
+0.05
+0.1
+0.15
+0.2 
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.05
+0.1
+0.15
+0.2 
+0.2
+6
+6
+6Dataset = HighTemp I α = 0.05
+Dataset = HighTemp I α = 0.1
+Dataset = HighTemp I α = 0.15
+Dataset = HighTemp I α = 0.2
+1.0
+0.8
+0.6
+Loss
+0.4
+0.2 
+0.0
+Model
+Dataset = LowTemp I α = 0.05
+Dataset = LowTemp I α = 0.1
+Dataset = LowTemp I α = 0.15
+Dataset = LowTemp I α = 0.2
+nDNN
+1.0
+U-Net
+0.8
+0.6
+Loss
+0.4
+0.2
+0.0
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+6
+6
+6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+8
+Fig. 5. Boxen plots for the distributions of normalized sizes of prediction sets vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for high-impact weather forecasting..
+The first row corresponds to HighTemp and the second row corresponds to LowTemp. Different columns represent different α. U-Net performs better than
+nDNN, which indicates the importance of careful designation of the underlying algorithm.
+can have reasonable sizes. Also, we can see that forecasting
+low temperature is somehow easier than high temperature
+with the fact that for the same α and δ, the normalized
+sizes of forecasting low temperature are distributed lower
+than the ones of forecasting high temperature, indicating the
+need of designation of the underlying algorithms to improve
+performance for forecasting high temperature.
+C. CLCP for maximum temperature and minimum tempera-
+ture forecasting
+This section focuses on using CLCP to forecast the 2-m
+maximum temperature or minimum temperature value for each
+grid, which is a point-wise regression problem or image-to-
+image regression problem. To construct the prediction sets,
+we follow the procedure proposed in [37] and train the neural
+network with 3 output channels jointly predicting the point-
+wise 0.05, 0.5 and 0.95 quantiles of the fields using quantile
+regression [38] [37], which are denoted by f 0.05(x), f 0.5(x)
+and f 0.95(x). Then the prediction set Cλ(x) is equal to
+�
+y : y(p,q) ∈ [f 0.5
+(p,q)(x)−λ∆−
+(p,q)(x), f 0.5
+(p,q)(x)+λ∆+
+(p,q)(x)]
+�
+,
+where
+∆−(x) = max{f 0.5(x) − f 0.05(x), 10−6},
+∆+(x) = max{f 0.95(x) − f 0.5(x), 10−6},
+and max is a point-wise operator making ∆− and ∆+ at least
+10−6. This prediction set is a prediction band for the output
+field, whose prediction interval at grid (p, q) is
+[f 0.5
+(p,q)(x) − λ∆−
+(p,q)(x), f 0.5
+(p,q)(x) + λ∆+
+(p,q)(x)]
+(12)
+with the point-wise width being an increasing function of λ.
+This construction was proposed in [37] for image-to-image
+regression and we use the same loss function in [37] measuring
+miscoverage rate of a prediction band C for a field y, which
+can be formalized as
+L(y, C) =
+1
+PQ
+���
+�
+(p, q) : y(p,q) /∈ C(p,q)
+����,
+(13)
+where C(p,q) is the prediction interval at grid (p, q) for
+prediction band C.
+All of the data collected from 2007 to 2020 were used, lead-
+ing to 4945 samples for each forecasting application and the
+datasets are named as MaxTemp and MinTemp respectively.
+The experimental designation is the same as that in Section
+IV-B, except that we also normalized the label for each grid to
+[0, 1] by min–max normalization, used quantile loss for model
+selection and we searched for λ∗ with two steps. First we
+found two values λ1 and λ2 from {100, 10, 1, 0.1, 0.01, ...}
+such that Q(n)
+1−δ(λ1) ≤ α and Q(n)
+1−δ(λ2) > α. Then we
+searched for λ∗ from 100 values starting with λ1 and ending
+with λ2 using a common step size. The experimental results
+are recorded in Fig. 6, Fig 7 and Fig 8.
+Although the set predictors and the loss function used in
+this section are different from those in Section IV-B, the
+experimental results and conclusions are similar. From Fig. 6,
+we can see that the frequencies of the prediction losses being
+more than α are controlled by δ, which also verifies formula
+(7) empirically. Larger α and δ lead to larger losses, which is
+shown in Fig. 7. Here we use the following average interval
+length
+1
+PQ
+P
+�
+p=1
+Q
+�
+q=1
+λ∗(∆+
+(p,q)(x) − ∆−
+(p,q)(x))
+(14)
+to measure the informational efficiency of the prediction set
+Cλ∗(x) and Fig. 8 also depicts the trade-off among the
+preset loss bound α, confidence level 1 − δ and informational
+efficiency of the prediction sets and indicates that better des-
+ignation of underlying algorithms lead to better performance.
+V. CONCLUSION
+This paper extends conformal prediction to the situation
+where the value of a loss function needs to be controlled,
+
+Dataset = HighTemp I α = 0.05
+Dataset = HighTemp I α = 0.1
+Dataset = HighTemp I α = 0.15
+Dataset = HighTemp I α = 0.2
+1.0
+0.8
+I Size
+Normalized 
+0.6
+0.4
+0.2
+0.0
+Model
+Dataset = LowTemp I α = 0.05
+Dataset = LowTemp I α = 0.1
+Dataset = LowTemp I α = 0.15
+Dataset = LowTemp I α = 0.2
+nDNN
+1.0
+U-Net
+0.8
+0.6
+0.4
+0.2
+0.0
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+0.1
+0.15
+0.2
+0.05
+6
+6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.
+9
+Fig. 6.
+Bar plots of the frequencies of the prediction losses being more than α vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for maximum temperature and
+minimum temperature forecasting. The first row corresponds to MaxTemp and the second row corresponds to MinTemp. Different columns represent different
+α. All bars are near or below the preset δ, which confirms the controlling guarantee of CLCP empirically.
+Fig. 7. Boxen plots of the prediction losses vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for maximum temperature and minimum temperature forecasting. The
+first row corresponds to MaxTemp and the second row corresponds to MinTemp. Different columns represent different α. The loss distributions are controlled
+by α and δ properly to obtain the empirical validity in Fig. 6.
+which is inspired by risk-controlling prediction sets and con-
+formal risk control approaches. The loss-controlling guarantee
+is proved in theory with the assumption of exchangeability
+and is empirically verified for different kinds of applications
+including classification with a class-varying loss and weather
+forecasting. Different from conformal prediction, conformal
+loss-controlling prediction approach proposed in this paper
+has two preset parameters α and δ, which guarantees that the
+prediction loss is not greater than α with confidence 1−δ. Both
+parameters impose restrictions on prediction sets and should
+be set based on specific applications. Despite loss-controlling
+guarantee, informational efficiency of the prediction sets built
+by conformal loss-controlling prediction is highly related to
+underlying algorithms, which has been shown in empirical
+studies. Since this is a rather new topic, the underlying
+algorithms and the way of constructing set predictors are
+inherited from conformal risk control. This leaves the im-
+portant question on how to build informationally efficient set
+predictors in an optimal way, which is one of our further
+researches in the future.
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+Dataset = MaxTemp I α = 0.2
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+Dataset = MinTemp I α = 0.1
+Dataset = MinTemp I α = 0.15
+Dataset = MinTemp I α = 0.2
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+U-Net
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+Dataset = MaxTemp I α = 0.15
+Dataset = MaxTemp I α = 0.2
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+Model
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+Dataset = MinTemp I α = 0.15
+Dataset = MinTemp I α = 0.2
+nDNN
+0.40
+U-Net
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+0.05
+0.1
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+0.2
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+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
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+6
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+arXiv:2301.03369v1  [physics.data-an]  6 Jan 2023
+NOLTA, IEICE
+Paper
+Time series analysis using persistent
+homology of distance matrix
+Takashi Ichinomiya 1,2a)
+1 Gifu University School of Medicine,
+Yanagido 1-1, Gifu, Gifu 501-1194, Japan
+2 The United Graduate School of Drug Discovery and Medical Informatic
+Science, Gifu University,
+Yanagido 1-1, Gifu 501-1194, Japan
+a) tk1miya@gifu-u.ac.jp
+Received December 06, 2022
+Abstract: The analysis of nonlinear dynamics is an important issue in numerous fields of
+science. In this study, we propose a new method to analyze the time series data using persistent
+homology (PH). The key idea is the application of PH to the distance matrix. Using this
+method, we can obtain the topological features embedded in the trajectories. We apply this
+method to the logistic map, R¨ossler system, and electrocardiogram data. The results reveal
+that our method can effectively identify nonlocal characteristics of the attractor and can classify
+data based on the amount of noise.
+Key Words: time-series analysis, persistent homology, dynamical system
+1. Introduction
+The analysis of the nonlinear dynamics is a challenging problem in physics, engineering, and data
+science. Numerous methods for time series analysis, such as Fourier transformation or Kalman fil-
+tering, are based on the theory of linear dynamics, and the performance of these methods is limited.
+To overcome their limitations, several methods, such as Koopman’s mode decomposition[1], phase
+reduction[2], deep learning[3], and reservoir computing[4, 5], have been proposed. However, we often
+meet the situations where these methods are not available. In Koopman’s mode decomposition we
+map the finite-dimensional nonlinear dynamical system into an infinite-dimensional linear dynamical
+system, and investigate the eigenvalues and eigenfunctions in this space. However, the determination
+of these eigenfunctions is often difficult. Phase reduction is a powerful method to investigate the
+oscillatory dynamics, but the application to non-oscillatory systems is limited. Deep learning and
+reservoir computing help us to predict the state in the future, but they do not provide a rationale for
+their prediction.
+In this study, we propose a new method to analyze the dynamics using distance matrix.
+The
+distance matrix D(t, s), defined as the distance between states at time t and s, reveals considerable
+information regarding the dynamics of the system. For example, if D(t, t+T) = 0 for all t, the system
+exhibits a periodic motion with period T. Recurrence plot (RP) is the most useful method for the
+1
+Nonlinear Theory and Its Applications, IEICE, vol. X, no. 0, pp. 1–14
+©IEICE 2023
+DOI: 10.1587/nolta.X.1
+
+time-series analysis using a distance matrix [6]. In this method, the distance matrix is visualized using
+R(t, s) = Θ(ǫ − D(t, s)), where Θ(x) is the Heaviside step function and ǫ is a parameter. Using this
+method, periodic motion can be distinguished from chaotic trajectories. Based on an RP, recurrence
+quantification analysis (RQA), wherein the dynamics are characterized by several quantities, such as
+recurrence rate and determinism, was proposed.
+However, an RP uses only the limited information embedded in the distance matrix. First, an RP
+does not provide information on the nonlocal properties of the trajectory. An RP lists the points that
+are close to each other in phase space and is useful to investigate local properties such as Lyapunov
+exponents. In contrast, it is unsuitable for investigating the global structure of the trajectory. For
+example, determining whether the attractor has a double scroll structure like the Lorenz system or
+an oscillation-like structure similar to the R¨ossler system by RP is difficult. Another problem with
+the use of an RP is that there is no clear rule to select the value of ǫ, and studies have reported that
+the result of the RP is often sensitive to this choice [7].
+In this study, we propose the application of persistent homology (PH)[8], an emerging technique of
+data analysis, to the analysis of the distance matrix. PH is the one of the most popular techniques
+in topological data analysis (TDA). In TDA, we investigate the topological characteristics such as
+number of connected components or holes embedded in the dataset. The theory of PH is still being
+developed and has been successfully applied in various fields, including biophysics[9–11], material
+science[12–14], and image processing[15, 16].
+There have been several proposals to apply PH to time series datasets. The favored approach for
+the time-series analysis using PH is based on delay embedding [17, 18]. In this approach, we first
+create the point clouds in n-dimensional space using Takens’ delay embedding [19] and subsequently
+characterize the state using PH. However, this approach has several difficulties.
+First, we must
+determine how to embed the dataset. There is no general rule to determine the way of embedding,
+though several ideas have been proposed [20–22]. Second, the computational cost increases rapidly
+as the embedding dimension and the size of point cloud increases. It is known that the computation
+time for PH is O(N⌈D/2⌉), where N and D represent the size of the point cloud and dimension of
+the space, respectively [23]. Therefore, the computational cost increases exponentially as D increases.
+In our approach, we can avoid the latter difficulty. Because the distance matrix is represented in
+two-dimensional space, the computation cost of PH is considerably reduced.
+The results of this
+study reveal that using PH, the essential information of a dataset, such as the non-local structure of
+attractors and the amount of noise, can be easily determined.
+The remainder of this study is structured as follows. In Section
+2, we explain our method to
+investigate the distance matrix. In Section 3, we present the results of application of our method to
+three different datasets. First, we present the results of the analysis of the logistic map as a typical
+example of discrete time dynamics. Second, we present the results on the R¨ossler system as an example
+of continuous time dynamics. Third, we discuss the results on the analysis of electrocardiogram (ECG)
+data, as an example of real-world time series data. Finally, we discuss on the possible improvements
+to our method in future and conclude this study.
+2. Method
+The general definition of PH requires further background knowledge of algebraic topology. In this
+section, we explain the PH of filtered cubical complexes of degree 0, which is used in the latter part
+of this study. The readers who want to know the general definition of PH can consult textbooks on
+PH and TDA[24, 25].
+We consider a real-valued filtration function f : Z2 → R. The sublevel set M(θ) is defined by
+M(θ) = {(x, y) ∈ Z2|f(x, y) ≤ θ}.
+(1)
+For example, assume that f(x, y) is given by the table shown in Fig. 1(a). In this case, M(0) is given
+by the gray blocks. When we increase θ, M(θ) also grows, as shown in Fig. 1(b) and (c).
+PH with degree 0 using a sublevel set of f represents the change in connected components in M(θ)
+when θ is varied from −∞ to ∞. Here we say two blocks are “connected” if they share an edge. For
+2
+
+!
+!
+!
+!
+"
+"
+"
+#
+#
+"
+"
+#
+$
+"
+#
+"
+"
+"
+#
+#
+"
+#
+!
+"
+$
+!
+!
+!
+!
+"
+"
+"
+#
+#
+"
+"
+#
+$
+"
+#
+"
+"
+"
+#
+#
+"
+#
+!
+"
+$
+!
+!
+!
+!
+"
+"
+"
+#
+#
+"
+"
+#
+$
+"
+#
+"
+"
+"
+#
+"
+"
+#
+!
+"
+$
+%&'()!"#$
+%*'()!"%$
+%+'()!"&$
+'
+#
+%
+&
+(
+)
+#
+%
+&
+(
+)
+*
+!
+"
+#
+$
++
+,
+!
+"
+#
+$
+!
+"
+#
+$
+%,'
+%-'
+#
+%
+&
+(
+)
+#
+%
+&
+(
+)
+Fig. 1.
+Example of persistent homology using sublevel sets, (a)–(c) represent
+the M(θ) for θ = 0, 1, and 2, respectively.
+The corresponding persistence
+barcode and persistence diagram are shown in (d) and (e).
+example, we have two components in Fig. 1(a): there is one isolated component at (x, y) = (2, 0)
+and one connected component at y = 4, 0 ≤ x ≤ 3. By increasing θ to 1, these two components are
+merged into one large component, and another component appears at (x, y) = (3, 2), as shown in
+Fig. 1(b). Here, we do not say these two components are connected, because although they share two
+corners, they do not share an edge. In this case, we say these two components are “disconnected.”
+When we increase θ to 2, all three components are merged into one large component, as shown
+in Fig. 1(c).
+The theory of PH guarantees that we can define the “birth” and “death” of each
+component. In this example, at θ = 0, two disconnected components are “born.” At θ = 1, these
+components merged into one large component. Hence, we say that one of the components “dies” and
+the other disconnected component at (x, y) = (3, 2) is born. Finally, at θ = 2, these components are
+connected. Further increase in θ does not change the number of disconnected components. Therefore,
+we have three “connected components,” often called “generators,” whose births b and deaths d are
+(b, d) = (0, ∞), (0, 1), (1, 2), respectively.
+There are two major visualization techniques to represent the distribution of generators. One is
+“persistence barcode,” wherein we represent each generator as a “bar” from birth to death, as shown
+in Fig. 1(d). This representation is intuitive when the number of generators is small. For example,
+when the dataset has a periodic structure, we will obtain numerous generators that have the same
+birth and death, and we can easily identify them by persistence barcode. However, when we have
+more than a hundred of generators, the number of bars becomes too large, and gaining any insight
+from the barcode becomes difficult. In this case, a persistence diagram (PD) is better visualization
+method; herein, we make a scatter plot of birth and death, as shown in Fig. 1(e). In a PD, generators
+with infinite death are generally omitted. When the number of generators becomes large, we also use
+a density heatmap of generators, which is also called a PD. In the rest of this study, we use PDs to
+represent the results of our PH analysis.
+The PH allows us to study the local minimum and saddle points of f. For example, we consider
+the case of Fig. 2. In the case of Fig. 2(a), f is a smooth function with two minima. In this case, we
+can define the sublevel set M(θ) as M(θ) = {x ∈ R|f(x) ≤ θ}. If θ is smaller than the saddle value of
+3
+
+!"#$%
+&'($%
+)(*
+)!*
+)+*
+!
+!
+!
+Fig. 2.
+Relation between the form of the filtering function f (upper) and
+corresponding persistence diagram(lower). (a) When f has two local minima,
+we obtain two generators, and only one has finite death. (b) When f is more
+complex and has more local minima, the number of generators increases. (c)
+If the “saddles” between local minima are low, the lifetime of generators de-
+creases. The dashed line in the persistence diagrams indicates the line birth =
+death.
+f, M(θ) has two connected components, and for larger θ, M(θ) has only one connected component.
+Therefore, in this case, there are two generators, and one of these generators has finite death. In
+contrast, if D has numerous local minima as described in Fig. 2(b), we obtain numerous generators
+with finite deaths. Therefore, the number of generators indicates the number of local minima of f.
+Moreover, PH provides information about the height of the saddles. For example, we consider the
+case in Fig. 2(c). Here, f has numerous local minima, but the height of the saddle is lower than in
+Fig. 2(b). In this case, the connected components of M(θ) merge after only a slight increase of θ.
+Therefore, the lifetime, defined as the difference between death and birth, decreases as the height of
+the corresponding saddle decreases.
+In this study, we used a distance matrix D for filtration. We suppose that we have time series data
+xi, i = 1, 2, . . . , K, where i represents the discretized time. From this dataset, we define the distance
+matrix D(i, j) as
+D(i, j) = ||xi − xj||,
+(2)
+where ||· · · || represents L2-norm. D(i, j) is a real-valued function from (i, j) ∈ Z2, and we can apply
+PH using the distance matrix.
+There are several software for PH analysis, which include Gudhi [26], Phat [27], and Javaplex [28].
+In this study, we used Homcloud developed by Obayashi et al [29]. One of the advantages of Homcloud
+compared with other software is that it can provide the “birth position” and “death position” of each
+generator. Using this function, we can obtain the positions of the local minima and the saddles of
+D(i, j). These values are useful for interpreting the result obtained by PH.
+3. Results
+We applied our method to three different datasets. The first example is time-series data obtained
+from the logistic map, and the second one is that obtained from the R¨ossler equations. Finally, we
+analyzed the dataset ECG200, as an example of a real-world dataset.
+3.1 Analysis of the logistic map
+In this subsection, we investigated the distance matrix of the logistic map defined by
+xi+1 = axi(1 − xi),
+(3)
+where 0 < a < 4 is a parameter.
+We calculated xi for i = 1, 2, . . . , 1000 with initial condition
+x0 = 0.832 and performed a PH analysis using data at i = 801, 802, . . . , 1000.
+First, we investigated the case a = 3.4, wherein xi is periodic, shown in Fig. 3(a). In this case, all
+generators had birth 0 and death 0.3901. This result indicated that xi takes only two values, and
+4
+
+!"#
+!$#
+!%#
+!&#
+Fig. 3.
+Persistence diagram for logistic maps: (a) a = 3.4, (b) a = 3.5, (c)
+a = 3.6, and (d) a = 3.7.
+the difference between these two is 0.3901. This is consistent with the fact that the attractor of this
+system is a cycle with period 2: x2i+1 = 0.4520 and x2i = 0.8421. The death time is given by the
+difference between these two values. As the period increases, the number of values that generators
+can take also increases. For example, at a = 3.5, the distribution had 6 peaks, corresponding to the
+fact that the period of the logistic map was 4. We show M(θ) for several θ in Fig.4 in the case of
+a = 3.5. The number of peaks represents the topological information of the attractors.
+Furthermore, the PD in a chaotic region provides topological information of the attractor. In the
+case of a = 3.6, where the logistic map becomes chaotic, the births and deaths of generators are widely
+distributed, as shown in Fig. 3(c). Herein, we found that the distribution had several characteristic
+properties. First, all deaths were larger than 0.188. This property could be explained by the existence
+of a gap in the attractor. At a = 3.6, x takes values from 0.3 to 0.6 and from 0.788 to 0.900, but
+does not take values between 0.601 to 0.787. This property generates a the gap in the distribution of
+deaths. To confirm this suggestion, we show the “death point” of generators whose death is smaller
+than 0.19 as a red line in Fig. 5. In this figure, the points connected by lines give the saddles of
+D(i, j). Evidently, the red lines connect the top point of the lower band and the bottom points of the
+upper band, which supports our claim.
+Additionally, Fig. 3(c) reveals that birth + death < 0.4695. Unfortunately, we have no theory to
+explain this property. Notably, we have an upper limit on birth + death when xmin ≤ xi ≤ xmax
+for all i, because birth ≥ 0 and death ≤ xmax − xmin. However, the fact that the death corresponds
+to the saddle makes the problem difficult. For example, the green line in Fig. 3(e) shows the death
+point pairs with the largest birth + death, 0.4694. This figure shows that one terminal of this line
+is at the upper limit of the upper band, whereas the other terminal stays in the middle of the lower
+band. We have no theoretical explanation why there is no saddle pair of states that provides larger
+birth + death. This is a problem that should be solved in the future.
+In the case of a = 3.7 shown in Fig. 3(d), the upper limit of death+birth appears to remain with
+several exceptional generators, whereas the lower limit disappears.
+The absence of a lower limit
+indicates the elimination of the gap in x found in the case a = 3.6. Summarizing these results, the
+PD gives the topological structure of the attractor, in the cases of both the periodic trajectory and
+chaotic attractor.
+5
+
+Logistic, a=3.5
+0.8
+0.7
+0.6
+103
+0.5
+Death
+0.4
+Value
+102
+0.3
+0.2
+0.1
+101
+0.0
+-0.1
+0.0
+0.2
+0.4
+0.6
+0.8
+BirthLogistic, a=3.4
+0.8
+105
+0.7
+0.6
+0.5
+104
+Death
+0.4
+Value
+0.3
+0.2
+103
+0.1
+0.0
+-0.1
+102
+0.0
+0.2
+0.4
+0.6
+0.8
+BirthLogistic, a=3.6
+0.8
+0.7
+0.6
+0.5
+102
+Death
+0.4
+Value
+0.3
+0.2
+101
+0.1
+0.0
+-0.1
+0.0
+0.2
+0.4
+0.6
+0.8
+BirthLogistic, a=3.7
+0.8
+0.7
+102
+0.6
+0.5
+Death
+0.4
+0.3
+0.2
+0.1
+0.0
+-0.1
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+Birth(a)
+(b)
+(c)
+(d)
+(e)
+Fig. 4.
+Sublevel set M(θ) represented as black areas in the case of a = 3.5.
+(a) θ = 0.01, (b) θ = 0.05, (c) θ = 0.20, (d) θ = 0.35, and (e) θ = 0.40,
+respectively.
+Fig. 5.
+Time series xi for the case of a = 3.6. Red lines indicate the death
+position of generators whose lifetime is smaller than 0.19.
+The green line
+indicates the death position of the generators with the largest birth+death,
+birth = 0.0104 and death = 0.4590.
+6
+
+0.9
+0.8
+0.7
+× 0.6
+0.5
+0.4
+0.3
+800
+825
+850
+875
+900
+925
+950
+975
+1000
+timeM(0.40)
+980
+985
+990
+995
+980
+985
+990
+995M(0.35)
+980
+985
+990
+995
+980
+985
+990
+995M(0.20)
+980
+985
+990
+995
+980
+985
+990
+995
+:M(0.05)
+980
+985
+990
+995
+980
+985
+990
+995M(0.01)
+980
+985
+990
+995
+980
+985
+990
+995!"#
+!$#
+Fig. 6.
+(a) Persistence diagram, and (b) trajectory of R¨ossler system, with
+a = 0.2, b = 1.6, c = 5.7. Red lines in (b) represent the death positions of
+generators whose deaths are larger than 10.0.
+3.2 Analysis of the R¨ossler system
+Next, we investigated the dynamics of the R¨ossler system described by the equations
+dx
+dt = −y − z
+(4)
+dy
+dt = x + ay
+(5)
+dz
+dt = b + xz − cz.
+(6)
+In this study, we set a = 0.2, c = 5.7, and investigated the change in the PD by varying b between
+0.1 and 1.7.
+We calculated (x(t), y(t), z(t)) for 0 ≤ t ≤ 500, and used (x(t), y(t), z(t)) for t =
+400, 400.1, . . . , 499.9 for the calculation of the distance matrix.
+First, we began from the case with b = 1.6. In this case, the PD shown in Fig. 6(a) shows that
+there are two classes of generators: generators in the first group have deaths smaller than 1.0 and
+those in the second group have deaths larger than 10.0. To study the origin of the generators with
+large death, we investigated the “death position” of these generators, which are indicated by the red
+lines in Fig. 6(b). At this parameter, the attractor was the orbit with period 1, and the large death
+value indicated the “diameter” of this orbit.
+Next, we demonstrated the PD for b = 1.2 in Fig. 7(a). The PD was similar to the case of b = 1.6,
+but new generators whose deaths are approximately 4.0 appeared. These generators represent the
+period doubling of the attractor. To reveal the relation between period doubling and the generators,
+we show the trajectories of the system with the birth and death positions of these generators in Fig. 7.
+Evidently, the birth positions of these generators, which are represented by green lines, connect the
+parallel part of the trajectory. These generators dies at the red lines, where these two lines diverge
+along the z axis. This example suggests that the generators with large births and deaths imply the
+existence of a parallel separated trajectory in the attractor. This claim holds true in the chaotic region.
+For example, we show the PD at b = 0.7 in Fig. 8(a); herein, the system had a chaotic attractor.
+The PD shows several clusters of generators with large lifetimes, and these generators represent the
+parallel trajectories. For example, the birth of the generators surrounded by black and green ellipses
+in Fig. 8(a) correspond with the black and green lines in Fig. 8(b), respectively. These birth points
+are the pairs between parallel trajectories in the phase space. These examples suggest that we can
+identify the nonlocal structure of the attractors using PH.
+In the analysis above, we analyzed the dynamics using the snapshots of the system with an interval
+∆t = 0.1. It is natural to ask whether our results are robust against the change of the interval.
+We calculated the PDs for several ∆t, using the trajectory with b = 0.7, 400 ≤ t ≤ 500. Here, we
+notice that the number of snapshots decreases as ∆t increases. The result is shown in Fig. 9. When
+7
+
+Rossler,b=1.60
+12
+103
+10
+8
+Death
+6
+102
+4
+2
+0
+101
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRossler, b=1.60
+3
+2
+Z
+1
+0
+5.0
+2.5
+-5.02.50.0
+0.0
+-2.5y
+2.5
+5.0
+X
+5.0!"#
+!$#
+Fig. 7.
+(a):
+Persistence diagram, and (b) trajectory of R¨ossler system,
+a = 0.2, b = 1.2, c = 5.7.
+Green and red lines in Fig.
+(b) represent the
+birth and death positions of generators whose deaths are between 3.0 and 4.0,
+respectively.
+!"#
+!$#
+Fig. 8.
+(a) Persistence diagram, and (b) trajectory of R¨ossler system, a =
+0.2, b = 0.7, c = 5.7. The birth positions of generators surrounded by black and
+green ellipses in (a) correspond to the black and green lines in (b), respectively.
+8
+
+Rossler,b=0.70
+12
+10
+102
+8
+Death
+6
+4
+101
+2
+0
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRosslerb=0.70
+10
+8
+6
+Z
+4
+2
+0
+5
+5
+0 x
+0
+-5
+y
+-5Rossler, b=1.20
+12
+103
+10
+8
+102
+Death
+6
+4
+101
+2
+0
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRossler, b=1.20
+6
+4
+Z
+2
+5
+0
+0 x
+5.0
+2.5
+0.0
+5.0
+-7.5!"#
+!$#
+!%#
+Fig. 9.
+Persistence diagrams for several interval of snapshots ∆t. (a) ∆t =
+0.2, (b) ∆t = 0.5, and (c)∆t = 1.0, respectively.
+The pararmeters of the
+R¨ossler system are a = 0.2, b = 0.7, c = 5.7
+!"#
+!$#
+Fig. 10.
+Persistence diagrams obtained using the trajectory (a) 300 ≤ t ≤
+500, and (b) 100 ≤ t ≤ 500. The parameters of the R¨ossler system are a =
+0.2, = 0.7, c = 5.7, and the interval of snapshots ∆t = 0.5.
+∆t = 0.2, we obtained the PD shown in Fig. 9(a), which is qualitatively consistent with the case of
+∆t = 0.1 in Fig. 8(a). Further increase of ∆t produced the many “noisy” generators, which made
+the structure of generators unclear. Fig. 9 (b) and (c) represent the PDs where ∆t = 0.5 and 1.0,
+respectively. In the case of ∆t = 0.5, the PD has many noisy generators, but it seems similar to the
+PD with ∆t = 0.1 and 0.2. In the case of ∆t = 1.0, we find no clear cluster of generators. These
+“noisy” generators cannot be removed even if we use a longer time series. Fig. 10 shows the PDs
+when we take longer sequences for ∆t = 0.5. Fig. 10(a) and (b) use two times and four times longer
+sequences than the case of Fig. 9(b), but the “noisy” generators remain. These results suggests that
+a small ∆t is required to apply our method.
+3.3 Analysis of the ECG200 dataset
+In this subsection, we present the analysis of the real time-series dataset ECG200, provided by Ol-
+szewski et al[30]. This dataset includes the 200 ECGs of a heartbeat, which are classified into two
+classes: healthy patients and patients with a myocardial infarction (MI). The dataset is divided into
+100 training samples and 100 test samples. In this study, we only used the dataset for training. The
+dataset was downloaded from the Time Series Classification Repository[31].
+First, we present the examples of ECG data of a healthy patient and a patient with an MI in
+Figs. 11 (a) and (d). In this figure, the ECG signal of an MI patient appears flat, whereas that of
+the healthy patient contains considerable noise. The corresponding PDs are shown in Figs. 11(b)
+and (e).
+In the case of an MI patient shown in Fig. 11 (e), the lifetime of most generators was
+smaller than 0.5. In contrast, the lifetimes in Fig. 11(e) had a large variation, which suggested the
+existence of noise in the data from a healthy patient. In this case, the PH gives more intuition to
+us than the standard technique such as Fourier transformation. For example, we present the results
+of Fourier transformation cn = �
+m x(m) exp
+� 2πimn
+N
+�
+in Fig. 11(c) and (f), where N represents the
+9
+
+Rossler,b=0.70, 100 ≤ t ≤500
+12
+102
+10
+8
+Death
+6
+101
+4
+2
+0
+100
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRossler,b=0.70,300 ≤ t ≤500
+12
+102
+10
+8
+Death
+6
+101
+4
+2
+0
+100
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRossler,b=0.70,△t=1.0
+12
+102
+10
+8
+Death
+6
+101
+4
+2
+0
+100
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRossler,b=0.70.△t=0.5
+12
+10
+101
+8
+Death
+6
+2
+0
+100
+0.0
+2.5
+5.0
+7.5
+10.0
+BirthRossler,b=0.70,△t=0.2
+12
+10
+8
+Death
+6
+101
+4
+2
+0
+100
+0.0
+2.5
+5.0
+7.5
+10.0
+Birth!"#
+!$#
+!%#
+!&#
+!'#
+!(#
+Fig. 11.
+Examples of electrocardiogram data and corresponding persistence
+diagrams and Fourier components.
+Upper: (a) electrocardiogram data, (b)
+persistence diagram, and (c) Fourier components obtained from a healthy pa-
+tients. Lower: (d) electrocardiogram data, (e) persisntence diagram, and (f)
+Fourier components obtained from a patient with myocardial infarction.
+length of sequence. The difference of the coefficients between the healthy patient and the patient with
+MI seems clear, but it is difficult to define a single variable that classify these two classes. Fourier
+transformation is a powerful tool when the time series has some characteristic frequencies, but in this
+case, there is no typical frequency that distinguishes patients with MI from healthy patients. To apply
+Fourier transformation in this problem, we require more complicated methods such as the analysis
+using Bag-of-SFA-Symbols[32].
+Based on this observation, we investigated whether we can use the variance in the lifetimes of the
+generators as an indicator for an MI. First, we studied the distribution of the variance of lifetimes
+for patients with MI and healthy patients. The result is shown in Fig. 12(a). In the case of an MI,
+the distribution peaked around 0.01, whereas in the case of the healthy patients, it peaked around
+0.04 for normal persons. This figure suggests that low variance in lifetimes is the signal of an MI. To
+estimate the performance of the variance of lifetimes as the marker of an MI, we calculated the receiver
+operating characteristic (ROC) curve and the area under the curve (AUC). The ROC curve represents
+the relation between false positive rate (FPR) and true positive rate (TPR). Suppose that we judge
+the ECG whose variance of lifetimes is smaller than p is positive. Then, number of true positive
+(TP), false negative (FN), true negative (TN), and false positive (FP) are defined as the number of
+correctly identified MI patients, misidentified MI patients, correctly identified healthy patients, and
+misidentified healthy patients, respectively. TPR and FPR are defined by
+TPR =
+TP
+TP + FN ,
+(7)
+and
+FPR =
+FP
+FP + TN .
+(8)
+The ROC curve is the plot of (FPR, TPR). AUC, defined as the area under the ROC curve, is
+the standard characteristic of the performance of a quantitative diagnostic test. If AUC = 1.0, we
+have no incorrect identification, whereas if AUC=0.5, the identification is equivalent to a random
+identification.
+In our case, the AUC was 0.811, which implies that the variance has a moderate
+accuracy as an indicator of MI [33]. Compared with this result, Kirchenko et al. classified the same
+dataset using the deep learning of RQA and the RP image, with AUC=0.76 and 0.92, respectively
+[34]. Therefore, our method is better than analysis using RQA, but it does not improve upon the
+analysis of the RP using deep learning. However, we note that the interpretation of our result is
+10
+
+#2: Healthy
+30
+Re(Cn)
+Im(Cn)
+20
+10
+G
+0
+-10
+-20
+-30
+0
+10
+20
+30
+40
+50
+n#1: MI
+Re(Cn)
+40
+Im(Cn)
+20
+0
+-20
+0
+10
+20
+30
+40
+50
+n#1: MI
+2.5
+2.0
+1.5
+101
+Death
+Value
+1.0
+0.5
+0.0
+100
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Birth#2:healthy
+1
+-2
+0
+25
+50
+75#1: MI
+2
+1
+0
+-1
+-2
+0
+25
+50
+75#2: healthy
+2.5
+101
+2.0
+1.5
+Death
+Value
+1.0
+0.5
+0.0
+100
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Birth!"#
+!$#
+Fig. 12.
+(a) Distribution of variance of lifetimes for MI patients and healthy
+patients. (b) Receiver operating curve when we use variance of lifetime as the
+indicator of MI. Area under curve = 0.811.
+much easier than that obtained by deep learning. In summary, our example shows that the variance
+of lifetimes is a promising signal to diagnose MI.
+4. Discussion and Conclusion
+In this study, we proposed a new method for time-series analysis, using PH analysis of the distance
+matrix. We demonstrated the efficacy of our method to understand the structure of attractors in the
+logistic map and the R¨ossler systems; additionally, we demonstrated that this method is applicable
+to real-world datasets such as ECG200. Our method uses the distance matrix, which is represented
+in two-dimensional space, thereby saving computational cost compared with other methods. Thus,
+our method is useful for analyzing dynamics in high-dimensional systems.
+However, PH is not a developed method for data analysis, and there remains room for improvement
+in our method. To conclude this study, we discuss several directions for future studies.
+First, combining this technique with machine learning is a promising approach. To analyze a large
+real-world dataset, machine learning techniques, such as deep learning, must be used. In machine
+learning, vectorizing the characteristic features is essential. However, the number of generators pro-
+duced by PH is not constant, thereby rendering difficulty in the application of standard machine
+learning techniques such as principal component analysis. Moreover, the generators with small life-
+time are often disregarded as meaningless because they are produced by small noise, and the “sig-
+nificance” of each generator must be estimated. To overcome these difficulties, numerous researchers
+have proposed a variety of techniques such as persistence landscape [35], persistence images [36], and
+persistence weighted Gaussian kernel [37]. These techniques combined with our proposed method will
+enable the application of our method to numerous problems.
+Second, investigating the information embedded in different PDs is interesting. In this study, we
+did not use the PDs of degree 1, which provide characteristics of “holes” surrounded by M(θ). The
+generators with degree 1 provide insights on the peaks of the distance matrix, which may give us
+essential insights on the dynamics. Additionally, we can also calculate PDs using dilation-erosion
+filtration [38]. In this approach, first, we make a recurrence plot, and subsequently calculate the
+distance to the “boundary,” the places where black and white cells contact, for each cell. PH analysis
+using this “distance to the boundary” as a filtration function provides a new quantification of the
+recurrence plot. However, to generate a PD with dilation-erosion, we must determine the threshold
+for the recurrence plot. The multi-parameter PH, which is currently studied intensively [39, 40], is
+the PH method with several filtration functions. The application of this method will provide a way
+to combine our method and dilation-erosion based PH analysis.
+Finally, another future task is to mathematically investigate the relation between PD and dynamics.
+In the case of RQA, it is known that the determinism has relation to the positive Lyapunov exponents.
+It would be an interesting to seek the mathematical relation between the PDs and the characteristics
+of the dynamical systems.
+11
+
+1.0
+0.8
+Rate
+Positive
+0.6
+0.4
+True
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FalsePositiveRate0.06
+lifetime
+0.05
+0.04
+of
+variance
+0.03
+0.02
+0.01
+0.00
+MI
+healthy
+target5. Acknowledgemens
+This work is financially supported by JSPS KAKENHI Grant Number JP22K19816.
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+

MtE2T4oBgHgl3EQfBQa7/content/tmp_files/2301.03601v1.pdf.txt

@@ -0,0 +1,2972 @@
+Quark-lepton Yukawa ratios and nucleon decay in
+SU(5) GUTs with type-III seesaw
+Stefan Antusch, Kevin Hinze, and Shaikh Saad
+Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
+E-mail: stefan.antusch@unibas.ch, kevin.hinze@unibas.ch,
+shaikh.saad@unibas.ch
+Abstract:
+We consider an extension of the Georgi-Glashow SU(5) GUT model by a
+45-dimensional scalar and a 24-dimensional fermionic representation, where the latter leads
+to the generation of the observed light neutrino masses via a combination of a type I and a
+type III seesaw mechanism. Within this scenario, we investigate the viability of predictions
+for the ratios between the charged lepton and down-type quark Yukawa couplings, focusing
+on the second and third family. Such predictions can emerge when the relevant entries of the
+Yukawa matrices are generated from single joint GUT operators (i.e. under the condition of
+single operator dominance). We show that three combinations are viable, (i) yτ/yb = 3/2,
+yµ/ys = 9/2, (ii) yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6. We extend these
+possibilities to three toy models, accounting also for the first family masses, and calculate
+their predictions for various nucleon decay rates. We also analyse how the requirement of
+gauge coupling unification constrains the masses of potentially light relic states testable at
+colliders.
+arXiv:2301.03601v1  [hep-ph]  9 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+GUT scenario
+3
+2.1
+Particle content
+3
+2.2
+Neutrino masses
+4
+2.3
+Quark-lepton Yukawa ratios
+5
+2.4
+Toy models
+5
+3
+Numerical procedure
+6
+3.1
+Implementation of the charged fermion Yukawa sector
+6
+3.2
+Implementation of the neutrino sector
+6
+3.3
+GUT scale parameters and low energy observables
+6
+3.4
+Fitting procedure
+7
+4
+Results
+7
+4.1
+Benchmark points
+8
+4.2
+Highest posterior densities
+9
+4.2.1
+Quark-lepton mass ratios
+10
+4.2.2
+Intermediate-scale particle masses
+10
+4.2.3
+Nucleon decay width and GUT scale
+12
+5
+Conclusion
+12
+Appendices
+12
+A Definition of new Yukawa couplings
+12
+B Renormalization group equations
+14
+B.1
+Gauge couplings
+14
+B.2
+Yukawa matrices
+16
+B.3
+Effective neutrino mass operator
+20
+1
+Introduction
+Grand Unified Theories (GUTs) [1–6] are arguably one of the most appealing extensions of
+the Standard Model (SM) of particle physics. In 1974, a simple and elegant GUT based on
+the unifying gauge group SU(5) was proposed by H. Georgi and S. Glashow (GG model)
+[3].
+However, this model is incompatible with the current experimental data for three
+main reasons. Firstly, the GG model does not allow for gauge coupling unification, which
+– 1 –
+
+is a necessary condition for a GUT. Secondly, it predicts massless neutrinos, which is in
+conflict with neutrino oscillation experiments requiring that at least two neutrino should
+be massive [7].
+Thirdly, since the SM Higgs doublet is embedded into a 5-dimensional
+Higgs representation of SU(5), the GG model predicts the GUT scale relation between the
+charged lepton and down-type quark Yukawa matrices
+Ye = Y T
+d .
+(1.1)
+This relation in particular implies a GUT scale unification of the tau and bottom Yukawa
+couplings yτ = yb, as well as a unification of the muon and strange Yukawa couplings
+yµ = ys, which disagrees with the low energy data.
+The first shortcoming requires extending the particle content of the minimal model by
+additional GUT representations and suitably splitting the masses of their component fields
+such that the running gauge couplings meet. The second shortcoming can be addressed
+by introducing SU(5) representations that allow neutrino mass generation at the tree level
+[8–14] or at the loop level [15–20].
+Finally, the third shortcoming can for instance be
+resolved by generating the Yukawa couplings from linear combinations of the renormalisable
+and higher dimensional non-renormalisable operators [21], or at the renormalizable level
+by either introducing a 45-dimensional Higgs field and considering linear combinations of
+couplings between the SM fermions and both the 5- as well as the 45-dimensional Higgs
+field [22], or by introducing vector-like fermions which mix with the SM fermions [23–25].
+However, historically a first and very aesthetic solution for the third problem was
+proposed by H. Georgi and C. Jarlskog (GJ model) in 1979 [26].
+In their model, the
+particle content of the GG model is extended by a 45-dimensional Higgs field (as well as
+by two 5-dimensional Higgs fields). If the 45-dimensional Higgs field couples to the SM
+fermions this gives rise to the GUT scale relation
+Ye = −3Y T
+d .
+(1.2)
+Considering a linear combination of the operators giving the relations (1.1) and (1.2) would,
+on the one hand, solve the shortcoming (as already mentioned above), but, on the other
+hand, predictivity in the Yukawa sector would be lost. Predictivity is however maintained
+if it is ensured that different generations of charged leptons and down-type quarks couple
+to different Higgs fields (which can, for example, be achieved when a family symmetry is
+introduced on top of the gauge symmetry). To achieve predictivity, without referring to
+any particular family symmetry, the GJ model hypothesizes the following textures of the
+Yukawa coupling matrices,
+Yd =
+�
+�
+�
+0
+B
+0
+A C
+0
+0
+0
+D
+�
+�
+� ,
+Y T
+e =
+�
+�
+�
+0
+B
+0
+A −3C
+0
+0
+0
+D
+�
+�
+� ,
+(1.3)
+implying the GUT scale relations yτ/yb = 1, yµ/ys = −3, ye/yd = −1/3 which were at that
+time compatible with the experimental data. However, the current data suggests (taking
+– 2 –
+
+only the known SM particles into account in the renormalization group (RG) evolution)
+that other ratios such as yτ/yb = 3/2, yµ/ys = 9/2 are better suited (see e.g. [27]).
+Interestingly, these latter ratios can be obtained from higher dimensional operators
+[28, 29]. With these higher dimensional operators at hand, models similar to the GJ model
+can be build if the following two conditions are satisfied: (i) the Yukawa matrices should
+be hierarchical, (ii) the 22- and 33- entry should be dominated by a single GUT operator,
+a concept which is referred to as single operator dominance [28–30].1
+Following this approach, non-SUSY GUT scenarios in which neutrino masses are gen-
+erated by a type I or a type II seesaw have been investigated in [27], respectively [45]. For
+GUT scenarios with a type I seesaw it was shown that the GUT scale ratios yτ/yb = 3/2
+and yµ/ys = 9/2 are compatible with the experimental data. Moreover, for GUT scenarios
+in which neutrino masses are generated by a type II seesaw it was found, that two combi-
+nations of GUT scale relations are viable, namely (i) yτ/yb = 3/2 and yµ/ys = 9/2 and (ii)
+yτ/yb = 2 and yµ/ys = 6.
+In this paper we will investigate the viability of such GUT scale ratios for the case
+that neutrino masses stem from a combination of a type I [46–50] and a type III [51]
+seesaw mechanism. In this regard, we will consider a GUT scenario in which the particle
+content of the GG model is extended by a fermionic adjoint representation as well as by
+a 45-dimensional Higgs field.2 The former representation is needed to generate neutrino
+masses, while the latter gives rise to operators yielding potentially viable GUT scale Yukawa
+ratios. Moreover, both of these representations help to allow for gauge coupling unification.
+Using the Mathematica package ProtonDecay [52] and extending the above scenario to “toy
+models” we also compute the nucleon decay widths for various decay channels. Finally, we
+compute the masses of the added fermion and scalar fields.
+The paper is organized as follows: While the GUT scenario as well as the toy models
+are introduced in Section 2, the procedure for the numerical analysis is explained in Section
+3. In Section 4 the results are presented and discussed, before concluding in Section 5.
+In Appendix A, definitions of the newly introduced Yukawa couplings are given, while all
+relevant RGEs that we have derived are listed in Appendix B.
+2
+GUT scenario
+2.1
+Particle content
+The SM fermions are embedded as usual into three generations of 5F i and 10F i
+5F i = dc
+i(3, 1, 1
+3) ⊕ ℓi(1, 2, −1
+2),
+(2.1)
+10F i = qi(3, 2, 1
+6) ⊕ uc
+i(3, 1, −2
+3) ⊕ ec
+i(1, 1, 1).
+(2.2)
+In the considered scenario, neutrino masses are generated via a combination of a type I and
+a type III seesaw mechanism. The corresponding fermionic singlet Σc and triplet Σb (under
+1For models in which the concept of single operator dominance has been applied, see e.g. [31–44].
+2A non-supersymmetric SU(5) GUT with this particle content was first considered in [13]. However, so
+far it has not been studied under the assumption of single operator dominance.
+– 3 –
+
+SU(2)L) are contained in an adjoint fermionic representation
+24F = Σa(8, 1, 0) ⊕ Σb(1, 3, 0) ⊕ Σc(1, 1, 0) ⊕ Σd(3, 2, −5
+6) ⊕ Σe(3, 2, 5
+6).
+(2.3)
+Moreover, the GUT Higgs fields decompose under the SM gauge group as
+24H = Φa(8, 1, 0) ⊕ Φb(1, 3, 0) ⊕ Φc(1, 1, 0) ⊕ Φd(3, 2, −5
+6) ⊕ Φe(3, 2, 5
+6),
+(2.4)
+5H = Ta(3, 1, −1
+3) ⊕ Ha(1, 2, 1
+2),
+(2.5)
+45H = φa(8, 2, 1
+2) ⊕ φb(6, 1, −1
+3) ⊕ φc(3, 3, −1
+3) ⊕ φd(3, 2, −7
+6) ⊕ φe(3, 1, −4
+3)
+⊕ Tb(3, 1, −1
+3) ⊕ Hb(1, 2, 1
+2).
+(2.6)
+After the SU(5) breaking, the color triplets Ta and Tb mix to yield the mass eigenstates
+t1 = cos(α)Ta +sin(α)Tb and t2 = − sin(α)Ta +cos(α)Tb. Similarly, Ha and Hb mix to form
+the mass eigenstates h1 = cos(β)Ha + sin(β)Hb and h⊥
+2 = − sin(β)Ha + cos(β)Hb, where
+h1 is the SM Higgs doublet.
+2.2
+Neutrino masses
+At tree-level the relevant GUT operators for neutrino mass generation read3
+L ⊃ YA 5F 24F 5H + YB 5F 24F 45H.
+(2.7)
+After the GUT symmetry breaking the following relevant terms emerge
+L ⊃ −Y2ℓΣbHa − Y8ℓΣbHb − Y4ℓΣcHa − Y13ℓΣcHb − mΣbΣbΣb − mΣcΣcΣc,
+(2.8)
+where mΣb and mΣb are the respective masses of Σb and Σc, and where the GUT scale
+relations
+Y2 = −
+�
+3
+10 YA,
+Y4 = YA,
+Y8 =
+√
+5
+4 YB,
+and
+Y13 =
+√
+3
+4 YB
+(2.9)
+hold. After the SU(2) triplet Σb and SU(2) singlet Σc have been integrated out and the
+two Higgs fields Ha and Hb have taken their vacuum expectation values (vevs) va and vb,
+where v2
+a + v2
+b = v2 = (246 GeV)2, and where va = v cos(β) and vb = v sin(β), the neutrino
+mass matrix mν reads
+mij
+ν = −(Y i
+2 va + Y i
+8 vb)(Y j
+2 va + Y j
+8 vb)
+4mΣb
+− (Y i
+4 va + Y i
+13 vb)(Y j
+4 va + Y j
+13 vb)
+4mΣc
+.
+(2.10)
+Since the neutrino mass matrix mν is of rank two, two massive and one massless neutrino
+are predicted.
+3After the GUT symmetry breaking these two GUT operators decompose into 19 SM Yukawa interac-
+tions. For details see Appendix A.
+– 4 –
+
+2.3
+Quark-lepton Yukawa ratios
+With X and Y representing one or multiple Higgs fields, the charged fermion masses stem
+from GUT operators of the form
+Y ij
+5
+:
+10F i5F jX
+⊃
+Y ij
+d , Y ij
+e
+(2.11)
+Y ij
+10 :
+10F i10F jY
+⊃
+Y ij
+u ,
+(2.12)
+where Yu, Yd and Ye denote the usual SM charged fermion Yukawa matrices. Assuming
+in the charged fermion Yukawa sector the concept of single operator dominance, i.e. that
+each Yukawa entry is dominated by a singlet GUT operator, allows to connect the down-
+type with the charged lepton Yukawa matrix via group theoretical Clebsch-Gordan (CG)
+factors cij. In SU(5) GUTs, and considering up to dimension five operators, the potentially
+viable CG factors are |cij| ∈ {1/6, 1/2, 2/3, 1, 3/2, 2, 3, 9/2, 6, 9, 18}. The possible GUT
+operators yielding these ratios are given in [28, 29]. Moreover, if the matrix Y5 is assumed
+to be of hierarchical nature and dominated by its diagonal entries, then the second and
+third family down-type quark and charged lepton masses stem dominantly from the GUT
+operators O2 and O3 dominating the 22 and 33 positions in Y5.
+Depending on which
+operators are chosen for O2 and O3, different GUT scale Yukawa ratios yτ/yb and yµ/ys
+are predicted. Our numerical analysis (cf. Section 4) shows that there are only two possible
+choices for the GUT scale ratio yτ/yb, namely 3/2 or 2. The former CG factor can be
+complemented by a factor 9/2 for the second family, while for the latter CG factor two
+different completions, yµ/ys = 6 or yµ/ys = 9/2, are possible.
+2.4
+Toy models
+We now extend the above motivated scenarios to three toy models which also include the first
+family. For simplicity, we chose the matrix Y5 to be of diagonal nature. The double ratio
+(yµyd)/(yeys) = 10.7+1.6
+−0.9, which is nearly constant under renormalization group running
+(see e.g. [53]), suggests, that the the ratio yµ/ys = 9/2 is best complemented by a ratio
+ye/yd = 4/9, while the best completion of the ratio yµ/ys = 6 is given by ye/yd = 1/2.
+Utilizing these ratios, our three toy models relate the down-type with the charged lepton
+Yukawa matrix via
+Model 1:
+Ye = diag
+�4
+9, 9
+2, 3
+2
+�
+· Y T
+d ,
+(2.13)
+Model 2:
+Ye = diag
+�4
+9, 9
+2, 2
+�
+· Y T
+d ,
+(2.14)
+Model 3:
+Ye = diag
+�1
+2, 6, 2
+�
+· Y T
+d .
+(2.15)
+Moreover, for simplicity4 we assume in each toy model that Y10 is dominated by the
+operator 10F 10F 5H in all entries, yielding a symmetric up-type Yukawa matrix, i.e. Yu =
+4We might consider higher-dimensional operators also for Y10, for example to explain the mass hierarchy,
+however since no Yukawa ratio predictions arise from this sector, we stick to the simplest case in our toy
+models.
+– 5 –
+
+Y T
+u .
+Finally, in all toy models neutrino masses stem from a linear combination of the
+operators 5F 24F 5H and 5F 24F 45H.
+3
+Numerical procedure
+3.1
+Implementation of the charged fermion Yukawa sector
+We implement all three toy models at the GUT scale as described in Section 2.4. In all
+three models the down-type Yukawa matrix Yd is simply implemented as
+Yd = diag(yd
+1, yd
+2, yd
+3),
+(3.1)
+while the charged lepton Yukawa matrix Ye is implemented according to Eq. (2.13), (2.14),
+and (2.15), respectively. Since Yu is symmetric we use a Takagi decomposition and imple-
+ment it as
+Yu = U †
+uY diag
+u
+U ∗
+u,
+(3.2)
+where5
+Uu =
+�
+�
+�
+1
+0
+0
+0 cu
+23
+su
+23
+0 −su
+23 cu
+23
+�
+�
+�
+�
+�
+�
+cu
+13
+0 su
+13e−iδu
+0
+1
+0
+−su
+13eiδu 0
+cu
+13
+�
+�
+�
+�
+�
+�
+cu
+12
+su
+12 0
+−su
+12 cu
+12 0
+0
+0 1
+�
+�
+�
+�
+�
+�
+eiβu
+1
+0
+0
+0
+eiβu
+2 0
+0
+0
+1
+�
+�
+� ,
+(3.3)
+and where Y diag
+u
+= diag(yu
+1, yu
+2, yu
+3).
+3.2
+Implementation of the neutrino sector
+In order to simplify the analysis we assume in the neutrino sector that the Yukawa matrices
+Y5 and Y6 (for the definitions of these couplings, see Appendix A) are of the form
+Y5 = z1
+�
+�
+�
+0
+1
+1
+�
+�
+� ,
+Y6 = z2
+�
+�
+�
+1
+1
+3
+�
+�
+� ,
+(3.4)
+where z1 and z2 are real parameters. Furthermore, we denote the relative phase difference
+between mΣb and mΣc by γ (i.e. γ = arg(mΣb/mΣc)). This structure is motivated by CSD3
+[56] which in the case of type I seesaw has been shown to correctly describe the low-scale
+neutrino observables together with a normal neutrino mass hierarchy (see e.g. [27] for a
+recent work).
+3.3
+GUT scale parameters and low energy observables
+Each toy model contains 33 input parameters which decompose into the GUT scale MGUT,
+the SU(5) gauge coupling gGUT, the masses of the added particles,6 mΦa, mΦb, mφa, mφb,
+5Here we have dropped three unphysical parameters but kept the GUT phases βu
+1 and βu
+2 which effect
+the nucleon decay widths [54, 55].
+6Note that mΣd = mΣe.
+– 6 –
+
+mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mt1, mt2, mh2, the singular values yu
+1, yu
+2, yu
+3, yd
+1,
+yd
+2, yd
+3 and angles θu
+12, θu
+13, θu
+23 as well as phases δu, βu
+1 , βu
+2 of the charged fermion Yukawa
+matrices, the parameters of the neutrino Yukawa couplings z1, z2, and γ, and the eigenstate
+mixing angles α and β. The respective ranges of these input parameters are given by7
+MGUT < MPl,
+mΦa, mΦb, mφa, mφb, mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mh2 ∈ [1 TeV, MGUT],
+mt1, mt2 ∈ [1011 GeV, MGUT],
+gGUT, yu
+1, yu
+2, yu
+3, yd
+1, yd
+2, yd
+3 ∈ [0, 1],
+(3.5)
+θu
+12, θu
+13, θu
+23, α, β ∈ [0, π/2],
+δu, βu
+1 , βu
+2 , γ ∈ [−π, π),
+z1, z2 > 0.
+These input parameters are fitted to the 22 low-scale observables (listed in Eq. (3.6)) and
+the nucleon decay widths of thirteen decay channels (listed in Table I).
+g1, g2, g3,
+yu, yc, yt, yd, ys, yb, θCKM
+12
+, θCKM
+13
+, θCKM
+23
+, δCKM, ye, yµ, yτ,
+(3.6)
+∆m2
+21, ∆m2
+31, θPMNS
+12
+, θPMNS
+13
+, θPMNS
+23
+, δPMNS.
+For the SM gauge couplings and Yukawa observables we take the experimental values from
+[53], while the values for the neutrino sector are taken from NuFIT 5.1 [57].
+3.4
+Fitting procedure
+After implementing the input parameters given in Eq. (3.5) at the GUT scale we compute
+the RG evolution to the Z scale. For the gauge couplings we use a 2-loop running, while
+we compute the running of the Yukawa matrices and the effective neutrino mass operator
+at 1-loop. The nucleon decay widths are computed using the Mathematica package Proton
+Decay [52] (for a description of the calculation see e.g. [27]). Taking into account all observ-
+ables we compute at the low scale the χ2-function which we minimize using a differential
+evolution algorithm giving us a benchmark point. With a flat prior distribution we calcu-
+late 4 × 106 data points performing a Markov-chain-Monte-Carlo (MCMC) analysis using
+an adaptive Metropolis-Hastings algorithm [65] which we start from this benchmark point.
+These data points are finally used to compute the highest posterior density (HPD) ranges
+of various quantities.
+4
+Results
+The results of our numerical analysis are presented in this section. We are in particular
+interested in the nucleon decay predictions, the intermediate-scale particle masses as well
+7Note that although we do not put any perturbativity constraints on the neutrino Yukawa couplings z1
+and z2 the fit automatically choses them to be below 1 (cf. Section 4).
+– 7 –
+
+decay channel
+τ/B [year]
+Γpartial [GeV]
+Reference
+Proton:
+p → π0 e+
+> 2.4 · 1034
+< 8.7 · 10−67
+[58]
+p → π0 µ+
+> 1.6 · 1034
+< 1.3 · 10−66
+[58]
+p → η0 e+
+> 1.0 · 1034
+< 2.0 · 10−66
+[59]
+p → η0 µ+
+> 4.7 · 1033
+< 4.4 · 10−66
+[59]
+p → K0 e+
+> 1.1 · 1033
+< 1.9 · 10−65
+[60]
+p → K0 µ+
+> 3.6 · 1033
+< 5.8 · 10−66
+[61]
+p → π+ ν
+> 3.9 · 1032
+< 5.3 · 10−65
+[62]
+p → K+ ν
+> 6.6 · 1033
+< 3.2 · 10−66
+[63]
+Neutron:
+n → π− e+
+> 5.3 · 1033
+< 3.9 · 10−66
+[59]
+n → π− µ+
+> 3.5 · 1033
+< 5.9 · 10−66
+[59]
+n → π0 ν
+> 1.1 · 1033
+< 1.9 · 10−65
+[62]
+n → η0 ν
+> 5.6 · 1032
+< 3.7 · 10−65
+[60]
+n → K0 ν
+> 1.2 · 1032
+< 1.7 · 10−64
+[60]
+Table I: Current experimental bounds on the decay widths Γpartial, respectively lifetime
+τ/B at 90 % confidence level, where B is the branching ratio for the decay channel. See
+also [64] for future projections and sensitivities of various upcoming detectors.
+as the low scale predictions for the charged lepton and down-type quark mass ratios. In
+Section 4.1 we show the results of our minimization procedure. Starting an MCMC analysis
+from these benchmark points allows us to obtain the HPD ranges of various quantities. The
+results of this analysis is presented in Section 4.2.
+4.1
+Benchmark points
+We obtain for all three models benchmark points through a minimization of the χ2-function
+as described in Section 3. In Table II the input parameters for the respective benchmark
+points are listed. Moreover, the dominant pulls χ2
+i are presented in Table III. All three
+models can be very well fitted to the data. The strongest (though quite small) pull is given
+by the first and second family down-type quark masses. The biggest difference between the
+three models is the respectively favored GUT scale. For Models 2 and 3 a GUT scale above
+1017 GeV is favored, while for the benchmark point of Model 1 a GUT scale below 1016 GeV
+is obtained. This also results in different results for the predicted nucleon decay rates (cf.
+Section 4.2). Another difference is the preferred choice of some of the intermediate-scale
+particle masses. In the presented benchmark point the mass of the fermionic field Σa is
+obtained to be at the GUT scale for Model 1, at the intermediate scale for Model 2 and at
+the relatively low scale (23 TeV) for Model 3. Moreover, a mass of the leptoquark φc of 1
+– 8 –
+
+TeV, respectively 4 TeV is obtained for Model 3, respectively Model 2, whereas for Model 1
+the mass of this field is above 106 TeV. For the HPD results of these particle masses confer
+the subsequent section.
+Model 1
+Model 2
+Model 3
+gGUT / 10−1
+5.94
+6.17
+6.33
+log10(MGUT / GeV)
+15.6
+17.2
+17.3
+log10(mφa / GeV)
+9.43
+14.0
+16.7
+log10(mφc / GeV)
+9.02
+3.63
+3.00
+log10(mΣa / GeV)
+15.6
+7.53
+4.36
+log10(mΣb / GeV)
+14.2
+14.9
+14.7
+log10(mΣc / GeV)
+13.8
+12.8
+13.2
+log10(mΣd / GeV)
+14.2
+15.9
+14.1
+yu
+1 / 10−6
+2.63
+2.11
+1.99
+yu
+2 / 10−3
+1.46
+1.37
+1.18
+yu
+3 / 10−1
+4.54
+4.26
+3.65
+yd
+1 / 10−6
+6.21
+6.30
+5.46
+yd
+2 / 10−4
+1.31
+1.21
+0.99
+yd
+3 / 10−3
+6.64
+6.01
+5.36
+z1 / 10−1
+3.50
+9.42
+6.86
+z2 / 10−1
+1.12
+0.32
+0.50
+γ
+1.85
+1.48
+1.68
+α
+0.50
+1.00
+0.50
+Table II: The GUT scale input parameters of the benchmark points for all three models.
+χ2
+χ2
+yd
+χ2
+ys
+χ2
+yb
+χ2
+yµ
+χ2
+yτ
+χ2
+Γ(p→π0e+)
+Model 1
+1.36
+0.27
+0.41
+0.06
+0.04
+0.14
+0.44
+Model 2
+0.31
+0.23
+0.02
+0.01
+0.00
+0.05
+0.00
+Model 3
+0.33
+0.17
+0.03
+0.00
+0.06
+0.07
+0.00
+Table III: The total χ2 as well as the dominant pulls χ2
+i for the benchmark points of all
+three models.
+4.2
+Highest posterior densities
+As described in Section 3.4 we vary the input parameters listed in Eq. 3.5 around their
+benchmark points (cf. Table II) using an MCMC analysis. From these generated points we
+then compute the HPD intervals of various parameters and observables.
+– 9 –
+
+Model 1
+Model 2
+Model 3
+0.14
+0.15
+0.16
+0.17
+0.18
+0.19
+ye/yd
+HPD intervals for ye/yd
+Model 1
+Model 2
+Model 3
+1.7
+1.8
+1.9
+2.0
+2.1
+yμ/ys
+HPD intervals for yμ/ys
+Model 1
+Model 2
+Model 3
+0.59
+0.60
+0.61
+0.62
+0.63
+yτ/yb
+HPD intervals for yτ/yb
+Figure 1: Low scale (MZ) HPD intervals for charged lepton and down-type quark Yukawa
+ratios of all three families. The 1σ (2σ) HDP intervals are colored dark (light).
+In Figures 1 – 4 we use the following color coding: For Model 1, 2, and 3 the HPD
+intervals of various quantities are colored red, green, and blue, respectively, while the 1σ
+(2σ) HPD intervals are colored dark (light).
+4.2.1
+Quark-lepton mass ratios
+The HPD results for the low scale charged lepton and down-type quark mass ratios are
+presented in Figure 1.
+The horizontal dashed line represents the current experimental
+central value, whereas the white region shows the current experimental 1σ range. Clearly,
+all three models are capable of reproducing viable mass ratios. This strengthens the results
+of the benchmark points in the previous subsection (cf. Tables II and III). Compared to
+Model 2 and 3, Model 1 gives a bit smaller predictions for the mass ratios for all three
+generations.
+4.2.2
+Intermediate-scale particle masses
+Figure 2 shows the predicted HPD intervals of the intermediate-scale particle masses. Most
+of the masses are predicted to be out of the reach of current and future colliders, because
+they would either produce too much proton decay, spoil gauge coupling unification or be-
+cause of the fit of the fermion masses. But interestingly, the fields Φb, φc and Σa are not
+only potentially within the reach of future searches, but can also be used to distinguish be-
+tween the different models: An observation of the one of the fields Φb or Σa would strongly
+hint towards Model 3, while an observation of the field φc would disfavor Model 1. In fact,
+the most promising lookout could be for the leptoquark φc. The upper bound of the HPD
+1σ range is predicted to be 23 TeV (2.8 TeV) in Model 2 (3), whereas the upper bound of
+the 2σ intervals is 175 TeV (17 TeV). In the following, we briefly state the current collider
+bounds on these particles.
+The scalar triplet, Φb, with zero hypercharge, residing in the 24H multiplet is expected
+to be light in Model 3. Note that Φb contains a neutral Φ0
+b and a pair of singly charged
+Φ±
+b states. In the low-energy effective theory, a term of the form h†
+1Φ2
+bh1 is allowed, where
+– 10 –
+
+mΦa
+mΦb
+mTa
+mTb
+m ϕa
+m ϕb
+m ϕc
+m ϕd
+m ϕe
+mΣa
+mΣb
+mΣc
+mΣd
+Model 1
+Model 2
+Model 3
+2
+4
+6
+8
+10
+12
+14
+16
+18
+log10(μ/GeV)
+HPD intervals for intermediate-scale particle masses
+Figure 2: HPD intervals of the intermediate-scale particle masses. The 1σ (2σ) HDP
+intervals are colored dark (light).
+h1 is the SM Higgs doublet. As a result of this coupling, the SM Higgs can decay into two
+photons h0 → γγ via a one-loop diagram mediated by the Φ±
+b states. Consistency with the
+LHC data requires these charged states to have masses above 250 GeV [66].
+The scalar leptoquark φc, which is a triplet of SU(2)L, resides around the TeV scale
+in Models 2 and 3.
+In both models, its coupling to the SM fermions is dominated by
+the third-generation quark and lepton. Hence, within our scenarios, its decay branching
+fraction is dominated by a bτ final state. Since leptoquarks carry color, they are efficiently
+produced at the LHC through gluon-initiated as well as quark-initiated processes [67]. LHC
+searches of pp → bbττ from pair-produced leptoquarks rule out leptoquark masses below
+1400 GeV [68, 69].
+As can be seen from Eq. (A.2), the color octet fermion Σa, which is expected to be
+light in Model 3, couples, for example, to a singlet down-quark (lepton doublet) and a
+super-heavy colored triplet (octet) scalar. Consequently, the lifetime of a TeV scale Σa is
+expected to be large, and it behaves like a long-lived gluino that typically arises in split-
+supersymmetric scenarios [70, 71]. Long-lived colored particles would hadronize, forming
+so-called R-hadrons [72]. These bound states are comprised of the long-lived state and
+light SM quarks or gluons, and interact with the detector material, typically inside the
+calorimeters, via hadronic interactions of the light-quark constituents. Motivated by split-
+supersymmetric models, R-hadrons are extensively searched for at the LHC [73, 74]. Non-
+observation of any deviations of the signal from the expected background puts to a lower
+– 11 –
+
+limit on the mass of the long-lived Σa fermion of 2000 GeV [73].
+4.2.3
+Nucleon decay width and GUT scale
+Figure 3 shows the predictions for the HPD intervals of the GUT scale MGUT. Moreover,
+the predicted HPD ranges for the nucleon decay widths of the various decay channels
+are presented in Figure 4. The blue line segments in the latter picture indicate the current
+experimental bounds at 90 % confidence level (cf. Table I). Moreover, the future constraints
+on the decay widths for the decay channels p → π0e+ and n → π−e+ which will be provided
+by Hyper-Kamiokande [75] are presented by orange line segments.
+In Figure 3 it can be seen that Model 1 clearly predicts the GUT scale to be below 1016
+GeV. On the other hand, a much larger GUT scale is preferred by the Models 2 and 3. Since
+the nucleon decay width is inversely proportional to the forth power of the GUT scale in the
+case of gauge boson mediated nucleon decay, this also results in strongly different prediction
+for the nucleon decay widths of the various channels as it can be seen in Figure 4. The
+nucleon decay predictions for Model 1 are very close to the current bounds, the 1σ HPD
+interval of the proton decay channel p → π0e+ will be fully probed by Hyper-Kamiokande.
+Moreover, Hyper-Kamiokande will probe most of the 1σ HPD interval of the neutron decay
+channel n → π−e+. On the other hand, the gauge boson mediated nucleon decay is highly
+suppressed in Models 2 and 3 and cannot be probed by any planed experiments. Therefore,
+observation of nucleon decay in the decay channels p → π0e+ and n → π−e+ would clearly
+favour Model 1 over the Models 2 and 3.
+5
+Conclusion
+In this paper we considered an extension of the Georgi-Glashow SU(5) GUT scenario by
+a 45-dimensional scalar and a 24-dimensional fermionic representation. Neutrino masses
+in this scenario are generated by a combination of a type I and a type III seesaw mech-
+anism. Assuming the concept of single operator dominance we investigated which GUT
+scale charged lepton and down-type quark Yukawa ratios can be viable for the second and
+third family and found that three combinations work: (i) yτ/yb = 3/2, yµ/ys = 9/2, (ii)
+yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6. Also taking into account the origin
+of the first family masses we extended these possibilities to three toy models and analyzed
+various of their predictions. We showed that experimental discrimination between these
+models could be possible since they predict different nucleon decay rates as well as distinct
+light relics.
+Appendices
+A
+Definition of new Yukawa couplings
+The Lagrangian density contains the two terms
+L ⊃ YA 5i
+F 24F 5H + YB 5i
+F 24F 45H.
+(A.1)
+– 12 –
+
+Model 1
+Model 2
+Model 3
+15.5
+16.0
+16.5
+17.0
+17.5
+18.0
+18.5
+MGUT
+HPD intervals for MGUT
+Figure 3: Predicted HPD intervals of the GUT scale. The 1σ (2σ) HDP intervals are
+colored dark (light).
+p → π0e+
+p → π0 μ+
+p → η0e+
+p → η0 μ+
+p → K0e+
+p → K0 μ+ p → π+ν
+p → K+ν
+n → π-e+
+n → π- μ+
+n → π0ν
+n → η0ν
+n → K0ν
+Model 1
+Model 2
+Model 3
+-85
+-80
+-75
+-70
+-65
+log10(Γ/GeV)
+HPD intervals for decay widths Γ of different nucleon decay channels
+Figure 4:
+Predicted HPD intervals of the nucleon decay widths.
+The 1σ (2σ) HDP
+intervals are colored dark (light). For each decay channel the blue line segments represent
+the current experimental constraints. The future Hyper-Kamiokande constraints for the
+decay channels p → π0e+ and n → π−e+ are indicated by orange line segments.
+– 13 –
+
+After the GUT symmetry breaking they decompose into
+L =
+�
+2
+15YA dcΣcTa −
+�
+3
+10YA ℓΣcHa + YA dcΣaTa + YA ℓΣbHa+
+YA dcΣdHa + YA ℓΣeTa +
+�
+5
+12YB dcΣcTb +
+√
+5
+4 YB ℓΣcHb+
+1
+2
+√
+2YB dcΣaTb + 1
+√
+2YB dcΣaφb + 1
+√
+2YB ℓΣaφa + 1
+√
+2YB dcΣbφc+
+√
+3
+4 YB ℓΣbHb − 1
+√
+2YB dcΣdφa −
+1
+2
+√
+6YB dcΣdHb + 1
+√
+2YB ℓΣdφe−
+1
+√
+2YB dcΣeφd +
+1
+2
+√
+2YB ℓΣeTb − 1
+√
+2YB ℓΣeφc
+≡ Y1 dcΣcTa + Y2 ℓΣcHa + Y3 dcΣaTa + Y4 ℓΣbHa+
+Y5 dcΣdHa + Y6 ℓΣeTa + Y7 dcΣcTb + Y8 ℓΣcHb+
+Y9 dcΣaTb + Y10 dcΣaφb + Y11 ℓΣaφa + Y12 dcΣbφc+
+Y13 ℓΣbHb + Y14 dcΣdφa + Y15 dcΣdHb + Y16 ℓΣdφe+
+Y17 dcΣeφd + Y18 ℓΣeTb + Y19 ℓΣeφc ,
+(A.2)
+where we defined the Yukawa matrices YN, with N = 1, . . . , 19.
+B
+Renormalization group equations
+Here the RGEs for the gauge and Yukawa couplings as well as for the effective neutrino
+mass operator are listed. We have used the Mathematica package SARAH [76, 77] to obtain
+the RGEs for the gauge and Yukawa couplings. The SM contribution for the RGE of the
+effective neutrino mass operator is taken from [78]. In order to compute the new contribution
+for this RGE we have used the method described therein. We use the following definition
+for the Heaviside-Theta function
+H(µ, m) =
+�
+1, for µ ≥ m,
+0, for µ < m.
+(B.1)
+B.1
+Gauge couplings
+The RGEs for gauge couplings (i, k = 1 − 3) are given by
+µdgi
+dµ =
+βgi
+1−loop
+16π2
++
+βgi
+2−loop
+(16π2)2 ,
+(B.2)
+where βgi
+1−loop is the 1-loop and βgi
+2−loop is the 2-loop contribution given by
+βgi
+1−loop =
+�
+aSM
+i
++ H(µ, m)∆ai
+�
+g3
+i
+(B.3)
+βgi
+2−loop =
+�
+k
+bSM
+ik g2
+k +
+�
+k
+∆bikg2
+k H(µ, m) + βY,SM
+i
++ ∆βY
+i .
+(B.4)
+– 14 –
+
+Here, aSM
+i
+, bSM
+ik
+and βY,SM
+i
+are the well known SM 1-loop and 2-loop coefficients as well as
+Yukawa contributions [79, 80]. Moreover, the ∆βY
+i are given by
+∆βY
+i = g3
+i
+�
+k
+cikY T
+k Y ∗
+k H2
+k,
+(B.5)
+where we introduced the abbreviation H2
+k = H(µ, mF )H(µ, mH) associated to each of the
+Yukawa interactions, where, F and H refer to the BSM fermion and scalar appearing in
+that interaction, respectively, and where the cik are given by
+c1k = −
+�1
+5, 3
+10, 8
+15, 9
+20, 29
+10, 17
+5 , 1
+5, 3
+10, 8
+15, 8
+15, 12
+5 , 3
+5, 9
+20, 116
+15 , 29
+10, 17
+5 , 29
+5 , 17
+5 , 51
+10
+�
+,
+(B.6)
+c2k = −
+�
+0, 1
+2, 0, 11
+4 , 3
+2, 3, 0, 1
+2, 0, 0, 4, 6, 11
+4 , 4, 3
+2, 3, 3, 3, 9
+2
+�
+,
+(B.7)
+c3k = −
+�1
+2, 0, 13
+3 , 0, 2, 1, 1
+2, 0, 13
+3 , 13
+3 , 6, 3
+2, 0, 16
+3 , 2, 1, 4, 1, 3
+2
+�
+.
+(B.8)
+Finally, the ∆ai and ∆bi are given as a sum over the 1-loop and 2-loop coefficients of the
+BSM fermions and scalars, i.e.
+∆ai =
+�
+I
+∆aI
+i ,
+∆bi =
+�
+I
+∆bI
+i ,
+(B.9)
+where I runs over all BSM particles. The 1-loop coefficients are then given by
+∆aφa
+i
+=
+�4
+5, 4
+3, 2
+�
+,
+∆aφb
+i
+=
+� 2
+15, 0, 5
+6
+�
+,
+∆aφc
+i
+=
+�1
+5, 2, 1
+2
+�
+,
+∆aφd
+i
+=
+�49
+30, 1
+2, 1
+3
+�
+,
+∆aφe
+i
+=
+�16
+15, 0, 1
+6
+�
+,
+∆aΦa
+i
+= {0, 0, 1
+2},
+∆aΦb
+i
+=
+�
+0, 1
+3, 0
+�
+,
+∆aΣa
+i
+= {0, 0, 2},
+∆aΣb
+i
+=
+�
+0, 4
+3, 0
+�
+,
+∆aΣd,e
+i
+=
+�5
+3, 1, 2
+3
+�
+,
+∆ah⊥
+i
+=
+� 1
+10, 1
+6, 0
+�
+,
+∆at,t⊥
+i
+=
+� 1
+15, 0, 1
+6
+�
+,
+(B.10)
+whereas the 2-loop coefficients read
+∆bφa
+ik =
+�
+�
+�
+36
+25
+36
+5
+144
+5
+12
+5
+52
+3
+48
+18
+5 18 84
+�
+�
+� ,
+∆bφb
+ik =
+�
+�
+�
+8
+75 0
+16
+3
+0 0
+0
+2
+3 0 115
+3
+�
+�
+� ,
+∆bφc
+ik =
+�
+�
+�
+4
+25
+24
+5
+16
+5
+8
+5 56 32
+2
+5 12 11
+�
+�
+� ,
+∆bφd
+ik =
+�
+�
+�
+2401
+150
+147
+10
+392
+15
+49
+10
+13
+2
+8
+49
+15
+3
+22
+3
+�
+�
+� ,
+∆bφe
+ik =
+�
+�
+�
+1024
+75
+0 256
+15
+0
+0
+0
+32
+15
+0
+11
+3
+�
+�
+� ,
+∆bΦa
+ik =
+�
+�
+�
+0 0 0
+0 0 0
+0 0 21
+�
+�
+� ,
+∆bΦb
+ik =
+�
+�
+�
+0 0 0
+0 28
+3 0
+0 0 0
+�
+�
+� ,
+∆bΣa
+ik =
+�
+�
+�
+0 0 0
+0 0 0
+0 0 48
+�
+�
+� ,
+∆bΣb
+ik =
+�
+�
+�
+0 0 0
+0 64
+3 0
+0 0 0
+�
+�
+� ,
+∆bΣd,e
+ik
+=
+�
+�
+�
+25
+12
+15
+4
+20
+3
+5
+4
+49
+4
+4
+5
+6
+3
+2
+38
+3
+�
+�
+� ,
+∆bh⊥
+ik =
+�
+�
+�
+9
+50
+9
+10 0
+3
+10
+13
+6 0
+0
+0 0
+�
+�
+� ,
+∆bt,t⊥
+ik
+=
+�
+�
+�
+4
+75 0 16
+15
+0 0 0
+2
+15 0 11
+3
+�
+�
+� .
+(B.11)
+– 15 –
+
+B.2
+Yukawa matrices
+The RGEs of the Yukawa matrices read
+µdYf
+dµ =
+βf
+16π2 ,
+(B.12)
+where f = {u, d, e, k} and (k = 1, . . . , 19). For the SM Yukawa matrices Yu, Yd and Ye (i.e.
+f = u, d, e) the beta functions are given by
+βf = βSM
+f
++ δβf,
+(B.13)
+where βSM
+f
+is the SM beta function [79, 80], and where
+δβf = YfT1 +
+�
+k
+af
+k(Yk)j(Y T
+d Y ∗
+k )i H2
+k .
+(B.14)
+Here, we have defined T1 as
+T1 = Y T
+2 Y ∗
+2 H2
+2 + 3
+2Y T
+4 Y ∗
+4 H2
+4 + 3Y T
+5 Y ∗
+5 H2
+5.
+(B.15)
+while the af
+k are given by
+au
+k =
+�
+0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
+�
+,
+(B.16)
+ad
+k =
+�1
+2, 0, 4
+3, 0, 3, 0, 1
+2, 0, 4
+3, 4
+3, 0, 3
+2, 0, 8
+3, 1, 0, 2, 0, 0
+�
+,
+(B.17)
+ae
+k =
+�
+0, −3
+2, 0, 15
+4 , 0, 3
+2, 0, 1
+2, 0, 0, 4, 0, 3
+4, 0, 0, 3
+2, 0, 3
+2, 9
+4
+�
+.
+(B.18)
+In order to simplify the notation, from hereon, associated to each Yukawa Yi → Yi H2
+i must
+be understood. The beta function of the Yukawa matrices Y1, . . . , Y19 then read
+β1 = Y1
+�
+−1
+5g2
+1 − 4g2
+3 +
+�
+k
+a1
+kY T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y1 +
+�
+w
+b1
+w
+�
+Y T
+1 Y ∗
+w
+�
+Yw
++ 8
+3
+�
+Y T
+3 Y ∗
+9
+�
+Y7 + 2
+�
+Y T
+6 Y ∗
+18
+�
+Y7,
+(B.19)
+β2 = Y2
+�
+− 9
+20g2
+1 − 9
+4g2
+2 +
+�
+k
+a2
+kY T
+k Y ∗
+k + T
+�
++
+�
+−3
+2Y T
+e Y ∗
+e
+�
+Y2 +
+�
+w
+b2
+w
+�
+Y T
+2 Y ∗
+w
+�
+Yw
++ 3
+2
+�
+Y T
+4 Y ∗
+13
+�
+Y8 + 3
+�
+Y T
+5 Y ∗
+15
+�
+Y8,
+(B.20)
+β3 = Y3
+�
+−1
+5g2
+1 − 13g2
+3 +
+�
+k
+a3
+kY T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y3 +
+�
+w
+b3
+w
+�
+Y T
+3 Y ∗
+w
+�
+Yw
++
+�
+Y T
+1 Y ∗
+7
+�
+Y9 + 2
+�
+Y T
+6 Y ∗
+18
+�
+Y9,
+(B.21)
+β4 = Y4
+�
+− 9
+20g2
+1 − 33
+4 g2
+2 +
+�
+k
+a4
+kY T
+k Y ∗
+k + T
+�
++
+�5
+2Y T
+e Y ∗
+e
+�
+Y4 +
+�
+w
+b4
+w
+�
+Y T
+4 Y ∗
+w
+�
+Yw
+– 16 –
+
++
+�
+Y T
+2 Y ∗
+8
+�
+Y13 + 3
+�
+Y T
+5 Y ∗
+15
+�
+Y13,
+(B.22)
+β5 = Y5
+�
+−29
+20g2
+1 − 9
+4g2
+2 − 8g2
+3 +
+�
+k
+a5
+kY T
+k Y ∗
+k + T
+�
++
+�
+3YdY †
+d
+�
+Y5 +
+�
+w
+b5
+w
+�
+Y T
+5 Y ∗
+w
+�
+Yw
++
+�
+Y T
+2 Y ∗
+8
+�
+Y15 + 3
+2
+�
+Y T
+4 Y ∗
+13
+�
+Y15,
+(B.23)
+β6 = Y6
+�
+−17
+10g2
+1 − 9
+2g2
+2 − 4g2
+3 +
+�
+k
+a6
+kY T
+k Y ∗
+k
+�
++
+�1
+2Y T
+e Y ∗
+e
+�
+Y6 +
+�
+w
+b6
+w
+�
+Y T
+6 Y ∗
+w
+�
+Yw
++
+�
+Y T
+1 Y ∗
+7
+�
+Y18 + 8
+3
+�
+Y T
+3 Y ∗
+9
+�
+Y18,
+(B.24)
+β7 = Y7
+�
+−1
+5g2
+1 − 4g2
+3 +
+�
+k
+a7
+kY T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y7 +
+�
+w
+b7
+w
+�
+Y T
+7 Y ∗
+w
+�
+Yw
++
+�
+2Y T
+18Y ∗
+6
+�
+Y1 + 8
+3
+�
+Y T
+9 Y ∗
+3
+�
+Y1,
+(B.25)
+β8 = Y8
+�
+− 9
+20g2
+1 − 9
+4g2
+2 +
+�
+k
+a8
+kY T
+k Y ∗
+k
+�
++
+�
+Y T
+e Y ∗
+e
+�
+Y8 +
+�
+w
+b8
+w
+�
+Y T
+8 Y ∗
+w
+�
+Yw
++ 3
+2
+�
+Y T
+13Y ∗
+4
+�
+Y2 + 3
+�
+Y T
+15Y ∗
+5
+�
+Y2,
+(B.26)
+β9 = Y9
+�
+−1
+5g2
+1 − 13g2
+3 +
+�
+k
+a9
+kY T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y9 +
+�
+w
+b9
+w
+�
+Y T
+9 Y ∗
+w
+�
+Yw
++ 2
+�
+Y T
+18Y ∗
+6
+�
+Y3 +
+�
+Y T
+7 Y ∗
+1
+�
+Y3,
+(B.27)
+β10 = Y10
+�
+−1
+5g2
+1 − 13g2
+3 +
+�
+k
+a10
+k Y T
+k10Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y10 +
+�
+w
+b10
+w
+�
+Y T
+10Y ∗
+w
+�
+Yw,
+(B.28)
+β11 = Y11
+�
+− 9
+20g2
+1 − 9
+4g2
+2 − 9g2
+3 +
+�
+k
+a11
+k Y T
+k Y ∗
+k
+�
++
+�
+Y T
+e Y ∗
+e
+�
+Y11 +
+�
+w
+b11
+w
+�
+Y T
+11Y ∗
+w
+�
+Yw,
+(B.29)
+β12 = Y12
+�
+−1
+5g2
+1 − 6g2
+2 − 4g2
+3 +
+�
+k
+a12
+k Y T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y12 +
+�
+w
+b12
+w
+�
+Y T
+12Y ∗
+w
+�
+Yw,
+(B.30)
+β13 = Y13
+�
+− 9
+20g2
+1 − 33
+4 g2
+2 +
+�
+k
+a13
+k Y T
+k Y ∗
+k
+�
++
+�1
+2Y T
+e Y ∗
+e
+�
+Y13 +
+�
+w
+b13
+w
+�
+Y T
+13Y ∗
+w
+�
+Yw
++ 3
+�
+Y T
+15Y ∗
+5
+�
+Y4 +
+�
+Y T
+8 Y ∗
+2
+�
+Y4,
+(B.31)
+β14 = Y14
+�
+−29
+20g2
+1 − 9
+4g2
+2 − 8g2
+3 +
+�
+k
+a14
+k Y T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y14 +
+�
+w
+b14
+w
+�
+Y T
+14Y ∗
+w
+�
+Yw,
+(B.32)
+β15 = Y15
+�
+−29
+20g2
+1 − 9
+4g2
+2 − 8g2
+3 +
+�
+k
+a15
+k Y T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y15 +
+�
+w
+b15
+w
+�
+Y T
+15Y ∗
+w
+�
+Yw
+– 17 –
+
++ 3
+2
+�
+Y T
+13Y ∗
+4
+�
+Y5 +
+�
+Y T
+8 Y ∗
+2
+�
+Y5,
+(B.33)
+β16 = Y16
+�
+−17
+10g2
+1 − 9
+2g2
+2 − 4g2
+3 +
+�
+k
+a16
+k Y T
+k Y ∗
+k
+�
++
+�1
+2Y T
+e Y ∗
+e
+�
+Y16 +
+�
+w
+b16
+w
+�
+Y T
+16Y ∗
+w
+�
+Yw,
+(B.34)
+β17 = Y17
+�
+−29
+20g2
+1 − 9
+4g2
+2 − 8g2
+3 +
+�
+k
+a17
+k Y T
+k Y ∗
+k
+�
++
+�
+YdY †
+d
+�
+Y17 +
+�
+w
+b17
+w
+�
+Y T
+17Y ∗
+w
+�
+Yw,
+(B.35)
+β18 = Y18
+�
+−17
+10g2
+1 − 9
+2g2
+2 − 4g2
+3 +
+�
+k
+a18
+k Y T
+k Y ∗
+k
+�
++
+�1
+2Y T
+e Y ∗
+e
+�
+Y18 +
+�
+w
+b18
+w
+�
+Y T
+18Y ∗
+w
+�
+Yw
++
+�
+Y T
+7 Y ∗
+1
+�
+Y6 + 8
+3
+�
+Y T
+9 Y ∗
+3
+�
+Y6,
+(B.36)
+β19 = Y19
+�
+−17
+20g2
+1 − 9
+2g2
+2 − 4g2
+3 +
+�
+k
+a19
+k Y T
+k Y ∗
+k
+�
++
+�1
+2Y T
+e Y ∗
+e
+�
+Y19 +
+�
+w
+b19
+w
+�
+Y T
+19Y ∗
+w
+�
+Yw,
+(B.37)
+where the coefficients af
+k are given by
+a1
+k = {3, 1, 8
+3, 0, 0, 2, 3
+2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
+(B.38)
+a2
+k = {3
+2, 5
+2, 0, 3
+2, 3, 0, 3
+2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
+(B.39)
+a3
+k = {1, 0, 9
+2, 0, 0, 2, 0, 0, 1
+2, 1
+2, 1, 0, 0, 0, 0, 0, 0, 0, 0},
+(B.40)
+a4
+k = {0, 1, 0, 11
+4 , 3, 0, 0, 0, 0, 0, 0, 3
+2, 1
+2, 0, 0, 0, 0, 0, 0},
+(B.41)
+a5
+k = {0, 1, 0, 3
+2, 9
+2, 0, 0, 0, 0, 0, 0, 0, 0, 4
+3, 1
+2, 1
+2, 0, 0, 0},
+(B.42)
+a6
+k = {1, 0, 8
+3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1
+2, 3
+4},
+(B.43)
+a7
+k = {3
+2, 1, 0, 0, 0, 0, 3, 1, 8
+3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0},
+(B.44)
+a8
+k = {3
+2, 1, 0, 0, 0, 0, 3
+2, 5
+2, 0, 0, 0, 0, 3
+2, 0, 3, 0, 0, 0, 0},
+(B.45)
+a9
+k = {0, 0, 1
+2, 0, 0, 0, 1, 0, 9
+2, 1
+2, 1, 0, 0, 0, 0, 0, 0, 2, 0},
+(B.46)
+a10
+k = {0, 0, 1
+2, 0, 0, 0, 0, 0, 1
+2, 19
+6 , 1, 0, 0, 0, 0, 0, 0, 0, 0},
+(B.47)
+a11
+k = {0, 0, 1
+2, 0, 0, 0, 0, 0, 1
+2, 1
+2, 6, 0, 0, 1, 0, 0, 0, 0, 0},
+(B.48)
+a12
+k = {0, 0, 0, 1
+2, 0, 0, 0, 0, 0, 0, 0, 4, 1
+2, 0, 0, 0, 0, 0, 1},
+(B.49)
+a13
+k = {0, 0, 0, 1
+2, 0, 0, 0, 1, 0, 0, 0, 3
+2, 11
+4 , 0, 3, 0, 0, 0, 0},
+(B.50)
+a14
+k = {0, 0, 0, 0, 1
+2, 0, 0, 0, 0, 0, 1, 0, 0, 5, 1
+2, 1
+2, 0, 0, 0},
+(B.51)
+– 18 –
+
+a15
+k = {0, 0, 0, 0, 1
+2, 0, 0, 1, 0, 0, 0, 0, 3
+2, 4
+3, 9
+2, 1
+2, 0, 0, 0},
+(B.52)
+a16
+k = {0, 0, 0, 0, 1
+2, 0, 0, 0, 0, 0, 0, 0, 0, 4
+3, 1
+2, 4, 0, 0, 0},
+(B.53)
+a17
+k = {0, 0, 0, 0, 0, 1
+2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1
+2, 3
+4},
+(B.54)
+a18
+k = {0, 0, 0, 0, 0, 1
+2, 1, 0, 8
+3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3
+4},
+(B.55)
+a19
+k = {0, 0, 0, 0, 0, 1
+2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1
+2, 4},
+(B.56)
+while the coefficients bf
+k read
+b1
+w = {0, 0, 4
+3, 0, 1, 0, 3
+2, 0, 0, 4
+3, 0, 3
+2, 0, 8
+3, 1, 0, 2, 0, 0},
+(B.57)
+b2
+w = {0, 0, 0, 3
+4, 0, 3
+2, 0, 3
+2, 0, 0, 4, 0, 3
+4, 0, 0, 3
+2, 0, 3
+2, 9
+4},
+(B.58)
+b3
+w = {1
+2, 0, 0, 0, 1, 0, 1
+2, 0, 0, 4
+3, 0, 3
+2, 0, 8
+3, 1, 0, 2, 0},
+(B.59)
+b4
+w = {0, 1
+2, 0, 0, 0, 3
+2, 0, 1
+2, 0, 0, 4, 0, 9
+4, 0, 0, 3
+2, 0, 3
+2, 9
+4},
+(B.60)
+b5
+w = {1
+2, 0, 4
+3, 0, 0, 0, 1
+2, 0, 4
+3, 4
+3, 0, 3
+2, 0, 8
+3, 4, 0, 2, 0, 0},
+(B.61)
+b6
+w = {0, 1
+2, 0, 3
+4, 0, 0, 0, 1
+2, 0, 0, 4, 0, 3
+4, 0, 0, 3
+2, 0, 7
+2, 9
+4},
+(B.62)
+b7
+w = {3
+2, 0, 4
+3, 0, 1, 0, 0, 0, 4
+3, 4
+3, 0, 3
+2, 0, 8
+3, 1, 0, 2, 0, 0},
+(B.63)
+b8
+w = {0, 3
+2, 0, 3
+4, 0, 3
+2, 0, 0, 0, 0, 4, 0, 3
+4, 0, 0, 3
+2, 0, 3
+2, 9
+4},
+(B.64)
+b9
+w = {1
+2, 0, 4, 0, 1, 0, 1
+2, 0, 0, 4
+3, 0, 3
+2, 0, 8
+3, 1, 0, 2, 0, 0},
+(B.65)
+b10
+w = {1
+2, 0, 4
+3, 0, 1, 0, 1
+2, 0, 4
+3, 0, 0, 3
+2, 0, 8
+3, 1, 0, 2, 0, 0},
+(B.66)
+b11
+w = {0, 1
+2, 0, 3
+4, 0, 3
+2, 0, 1
+2, 0, 0, 0, 0, 3
+4, 0, 0, 3
+2, 0, 3
+2, 9
+4},
+(B.67)
+b12
+w = {1
+2, 0, 4
+3, 0, 1, 0, 1
+2, 0, 4
+3, 4
+3, 0, 0, 0, 8
+3, 1, 0, 2, 0, 0},
+(B.68)
+b13
+w = {0, 1
+2, 0, 9
+4, 0, 3
+2, 0, 1
+2, 0, 0, 4, 0, 0, 0, 0, 3
+2, 0, 3
+2, 9
+4},
+(B.69)
+b14
+w = {1
+2, 0, 4
+3, 0, 1, 0, 1
+2, 0, 4
+3, 4
+3, 0, 3
+2, 0, 0, 1, 0, 2, 0, 0},
+(B.70)
+b15
+w = {1
+2, 0, 4
+3, 0, 4, 0, 1
+2, 0, 4
+3, 4
+3, 0, 3
+2, 0, 8
+3, 0, 0, 2, 0, 0},
+(B.71)
+b16
+w = {0, 1
+2, 0, 3
+4, 0, 3
+2, 0, 1
+2, 0, 0, 4, 0, 3
+4, 0, 0, 0, 0, 3
+2, 9
+4},
+(B.72)
+b17
+w = {1
+2, 0, 4
+3, 0, 1, 0, 1
+2, 0, 4
+3, 4
+3, 0, 3
+2, 0, 8
+3, 1, 0, 0, 0, 0},
+(B.73)
+b18
+w = {0, 1
+2, 0, 3
+4, 0, 7
+2, 0, 1
+2, 0, 0, 4, 0, 3
+4, 0, 0, 3
+2, 0, 0, 9
+4},
+(B.74)
+b19
+w = {0, 1
+2, 0, 3
+4, 0, 3
+2, 0, 1
+2, 0, 0, 4, 0, 3
+4, 0, 0, 3
+2, 0, 3
+2, 0}.
+(B.75)
+– 19 –
+
+B.3
+Effective neutrino mass operator
+The RGE for the effective neutrino mass operator reads
+16π2µdκ
+dµ = βSM
+κ
++ ∆βκ,
+(B.76)
+where βSM
+κ
+is the SM contribution as given in [78] and ∆βκ is the correction due to the
+added BSM particles. For ∆βκ we find8
+∆βκ = κ
+�1
+2Y ∗
+2 Y T
+2 + 3
+4Y ∗
+4 Y T
+4 + 3
+2Y ∗
+6 Y T
+6 + 1
+2Y ∗
+8 Y T
+8 + 4Y ∗
+11Y T
+11 + 3
+4Y ∗
+13Y T
+13 + 3
+2Y ∗
+16Y T
+16
++ 3
+2Y ∗
+18Y T
+18 + 9
+4Y ∗
+19Y T
+19
+�
++
+�1
+2Y ∗
+2 Y T
+2 + 3
+4Y ∗
+4 Y T
+4 + 3
+2Y ∗
+6 Y T
+6 + 1
+2Y ∗
+8 Y T
+8
++ 4Y ∗
+11Y T
+11 + 3
+4Y ∗
+13Y T
+13 + 3
+2Y ∗
+16Y T
+16 + 3
+2Y ∗
+18Y T
+18 + 9
+4Y ∗
+19Y T
+19
+�T
+κ
++
+�
+2Y T
+2 Y ∗
+2 + 3Y T
+4 Y ∗
+4 + 6Y T
+5 Y ∗
+5 + 2Y T
+8 Y ∗
+8 + 3Y T
+13Y ∗
+13 + 6Y T
+15Y ∗
+15
+�
+κ .
+(B.77)
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+

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