diff --git "a/8NAyT4oBgHgl3EQf2_mV/content/tmp_files/load_file.txt" "b/8NAyT4oBgHgl3EQf2_mV/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/8NAyT4oBgHgl3EQf2_mV/content/tmp_files/load_file.txt" @@ -0,0 +1,1307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf,len=1306 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For highly complex parametric problems, these two approaches may become too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To reduce computational times, Proper Orthogonal Decomposition (POD) and Reduced Basis Methods (RBMs) have already been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The majority of these algorithms are however intrusive in the sense that the High-Fidelity (HF) code must be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To address this issue, non-intrusive strategies are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The NIRB two-grid method uses the HF code solely as a “black-box”, requiring no code modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Like other RBMs, it is based on an offline-online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The offline stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive than an HF evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' On a classical model problem, the heat equation, we prove that HF evaluations of sensitivities reach an optimal convergence rate in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)), and then establish that these rates are recovered by the proposed NIRB approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' These results are supported by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We then numerically demonstrate that a further deterministic post-treatment can be applied to the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This further reduces computational costs of the online step while only computing a coarse solution of the initial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 1 Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Sensitivity analysis is a critical step in optimizing the parameters of a parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The goal is to see how sensitive its results are to small changes of its input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It is especially useful in the biomedical field when experiments are extremely complex or prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, conducting several experiments to determine the impact of all parameters involved in biological processes may be difficult, if not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Several methods have been developed for computing sensitivities, see [4] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We focus here on two differential-based sensitivity analysis approaches in connection with models given as reaction-diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The sensitivities (of the solution or other outputs of interest) are computed directly from the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' One drawback is that it necessi- tates solving a new system for each parameter of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=', for P parameters of interest, P + 1 problems have to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “The adjoint state method”, also known as the ”backward method”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It may be a viable option [44] when the direct method becomes prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, the goal is to compute the sensitivities of an objective function that one aims at minimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The associated Lagrangian is formulated, and by choosing appropriate multipliers, a new system known as ”the adjoint” is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This approach is preferred in many situations since it avoids calculating the sensitivities with respect to the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='00761v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='NA] 2 Jan 2023 (which are frequently provided by multiple sensors) while combining it with a gradient-type optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As a result, we get the ”integrated effects” on the outputs over a time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The advantage is that it only requires two systems to solve regardless of the number of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For extremely complex simulations, both methods may still be impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Several reduction techniques have thus been investigated in order to reduce the complexity of the sensitivity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Among them, Reduced Basis Methods (RBMs) are a well-developed field [36, 40, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In what follows, we present a brief review of previous works on RBMs combined with both sensitivity methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us begin with the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It has been employed and studied with RB spaces in various applica- tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=', [39, 47, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The sensitivities may also be useful to enhance the reduced state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note also that variance-based sensitivity analysis has been investigated using RBMs [28] and non-intrusive RB [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The adjoint state formulation can be thought of as a PDE-constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' RBMs and POD have also been investigated in the context of optimal control under uncertainty [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In recent studies, the case of infinite-dimensional control function is considered with RB approximations on the state, adjoint, and control variables [29, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Even if the adjoint state method is frequently preferred, writing its associated reduced problem can be difficult when the adjoint formulation is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It may also be reformulated to take advantage of previously developed RB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For example, in [38], it is rewritten as a saddle-point problem for Stokes-type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Even though the Galerkin projection is prevalent in the literature, its main disadvantage lies in its intrusiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, in order to approximate the solution of a PDE, the matrices computed from its variational formulation must be changed in the HF code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Apparently, NIRB methods have not yet been used to approximate sensitivities except for statistical approaches such as variance-based sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The two-grid algorithm can be employed for a variety of PDEs and is simple to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It has been studied with FEM in the context of elliptic equations [7] and parabolic equations [19] (see also [17] for finite volume schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Furthermore, because it is non-intrusive, it is suitable for a wide range of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The 2 effectiveness of this method relies on its offline/online decomposition (as most RBMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The offline part is time- consuming but it is only performed once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It makes this portion of the algorithm much cheaper than a HF evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this paper, we combine the two-grids framework with both sensitivity analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This article is about extending the NIRB two-grid method to the computation of sensitiv- ities and performing the associated numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We present and illustrate the NIRB algorithms applied to both sensitivity analysis methods with several numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the direct method, we have carried out a thorough theoretical analysis of the heat equation as model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, we have optimal conver- gence rates in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)) for the spatial HF semi-discretized sensitivity solution and for its fully-discretized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It turns out that we obtain theoretically and numerically these optimal rates also for the NIRB sensitivity approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Our main theoretical result is given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The rest paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Section 2 describes both sensitivity methods along with established convergence results and the NIRB two-grid algorithm for parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Section 4 is devoted to the theoretical results on the rate of convergence for the NIRB sensitivity approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the last section 5, several numerical results are presented and illus- trate the theoretical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The implementation and the use of Automatic Differentiation (AD) is discussed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2 Mathematical Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we recall the NIRB algorithm in the context of parabolic equations, and finally, we detail the sensitivity problems for this model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 A model problem: The heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 The continuous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We use the conventional notations for space-time dependent Sobolev spaces [35] Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' V) := {u(x, t) | ∥u∥Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='V) := � � T 0 ��u(·, t) ��p V dt �1/p < ∞}, 1 ≤ p < ∞, L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' V) := {u(x, t) | ∥u∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='V) := ess sup 0≤t≤T ��u(·, t) �� V < ∞}, where V is a real Banach space with norm∥·∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The variational form of (3) is given by: � � � � � � � Find u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1 0(Ω)) with ut ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H−1(Ω)) such that (ut(t, ·), v) + a(u(t, ·), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � Ω A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ)∇w(x) · ∇v(x) dx, ∀w, v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that we will use the notation (·, ·) to denote the classical L2-inner product on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 The various discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the NIRB algorithm, we use the two spatial grids on the variational formulation (5) of our problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We employed P1 finite elements to discretize in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To avoid making notations more cumbersome, we will consider a uniform time step ∆tF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The time levels can be written tn = n ∆tF, where n ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Another time grid, denoted G, is used for coarse solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By analogy with the fine grid, we consider a uniform grid with time step ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Now, the time levels are written �tm = m ∆tG, where m ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the elliptic context [7], the NIRB algorithm is designed to recover the optimal estimate in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we consider a higher order time scheme for the coarse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To simplify the notations, we consider that both time grids end at time T here, T = NT ∆tF = MT ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (t), vh), ∀vh ∈ Vh and t ∈]0, T], (8) uh(·, 0) = u0 h(·) = P1 h(u0)(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT such that (∂un h, vh) + a(un h, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (tn), vh), ∀vh ∈ Vh and n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT, such that (∂um H, vH) + a( um H+um−1 H 2 , vH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (�tm− 1 2 ), vH), ∀vH ∈ VH and m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' MT, uH(·, 0) = u0 H(·), (11) where �tm− 1 2 = �tm+�tm−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the NIRB approximation, we will need to interpolate in space and in time the coarse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We denote by � uH n(µ) = I2 n[um H](µ) the quadratic interpolation of um H at a time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the next section, we recall the NIRB algorithm in the context of parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 Reminders on the Non-Intrusive Reduced Basis method (NIRB) in the context of parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since each snapshot is a HF finite element approximation in space at a time tn, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=', NT (NT being potentially very high), not all of the time steps may be required for the construction of the reduced space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Here, for each parameter µi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ, selected for the basis construction, the number of time steps employed (which depends on i) is denoted Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, the reduced basis is defined as XN h := Span{u (nj)i h (µi)| i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Ni, (nj)i ⊂ {1, · · · , NT}}, (13) with N := Nµ ∑ i=1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From training parameters (µi)i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Ntrain}, we define Gtrain = ∪ i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Ntrain}µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we employ a greedy procedure to adequately choose the parameters (µi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Nµ within Gtrain to construct the RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For this procedure, we refer to algorithm 1 (described for the setting Nµ = N in order to simplify notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that a POD-greedy algorithm may also be employed [19, 21, 20, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Algorithm 1 Greedy algorithm Input: tol, {un h(µ1), · · · , un h(µNtrain) with µi ∈ Gtrain, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Output: Reduced basis {Φh 1, · · · , Φh N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Choose µ1, n1 = arg max µ∈Gtrain, n∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',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}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' for k = 2 to N do: µk, nk = arg max (µ, n)∈(Gtrain×{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' end for The greedy algorithm is usually less expensive than the POD-greedy (thanks to a-posteriori error estimates for stationary problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The RB functions (time-independent), denoted (Φh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N, are generated at the end of this step, from fine fully-discretized solutions {un h(µi)}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Nµ}, n={0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT} (solving problem (10) with HF solver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that even if all the time steps are computed, only Ni are used for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ} in the RB construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Hence, XN h can be defined as XN h = Span{Φh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Φh N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In practice, the algorithm is halted with a stopping criterion such as an error threshold or a maximum number of basis functions to generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that with Gram-Schmidt procedure, we only obtain an L2-orthonormalized RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For any parameter µk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ, the classical NIRB approximation differs from the HF uh(µk) computed in the offline stage [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, as proposed in [7], to improve NIRB accuracy, we use a ”rectification post-processing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It comes from the fact that the coefficients of the matrix are directly derived from the snapshots in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online step” The online part of the algorithm is much faster than a HF evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution on the fine time grid with (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � uH n(µ) on the fine mesh in order to compute the L2-inner product with the RB functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The approximation used in the two-grid method is For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 7 In [19], we have proven the following estimate on the heat equation for n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Before adapting NIRB to the sensitivity analysis context, we first recall how to derive the sensitivities functions in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3 Sensitivity analysis: The direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this section, we recall the sensitivity systems (continuous and discretized versions) for P parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we prove the numerical results of the direct method on the model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To not make the notations too cumbersome, we will consider A(µ) = µ Id, with µ ∈ R+∗ for the analysis theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 The continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, we consider P parameters of interest, denoted µp = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , µP, and we want to approximate the exact derivatives Ψp(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) := ∂u ∂µp (t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) := −a(u(t), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) + ( f (t), v), ∀v ∈ V, t > 0, u(·, 0) = u0(·), where the bilinear form a is defined by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Using the chain rule and since the time and the parameter derivatives can commute, (Ψp,t(t), v) = ∂F ∂u (u(t), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) · Ψp(t) + ∂F ∂µ(u(t), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1 0(Ω)), and under the assumptions (4), the so-called ”parabolic regularity estimate” implies that u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1 0(Ω)) [14, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 The spatially semi-discretized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As previously for the state solution, we discretize in space and in time the sensitivity problems (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The corresponding spatially semi-discretized formulations (on Th) read � � � � � � � � � Find Ψp,h(t) ∈ Vh for t ∈ [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , T] such that (Ψp,h,t(t), vh) + a(Ψp,h(t), vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ)∇uh(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), ∇vh), for vh ∈ Vh, for t ∈]0, T], Ψ0 p,h(·) = P1 h(Ψ0 p)(·), (24) where P1 h is given by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Before proceeding with the proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1), we need several results that can be deduced from [45], but require some precisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, first, in [45], the estimates are proven on the heat equation with a non-varying diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' On the semi-discretized formulation, the following estimate holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψh be the corresponding sensitivities , respectively given by (23) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (30) – From the initial conditions, since u0 h = P1 hu0, T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' – 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (33) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In practice, this is not an issue since the effect of the initial data exponentially decreases [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore, from (32), we obtain T2 ≤ 2 √ T µ � � T 0 ��ρu,t ��2 L2(Ω) ds �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (36) – Finally, for T3, we only need to use (35) again, but with Ψ instead of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore T3 = � T 0 ��ρt �� L2(Ω) ds ≤ Ch2 � T 0 ∥Ψt∥H2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (37) Combining (27), (30), (36), and (37) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We can derive a similar result for the H1 0 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψh be the corresponding sensitivities , respectively given by (23) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∀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� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (39) For the estimate on θ(t), let us consider v ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (41) – From the initial conditions, T′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' – We can also write T′ 2 as T′ 2 = 1 µ3 � T 0 ��θu,t + ρu,t ��2 L2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (42) – Similarly, T′ 3 ≤ Ch4 µ � T 0 ∥Ψt∥2 H2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (43) Hence, combining (38) with (39), (41), (42) and (43) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3 The fully-discretized versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From (24), we can derive the fully-discretized systems for the fine and coarse grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂Ψn p,h, vh) + a(Ψn p,h, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ)∇un h(µ), ∇vh), for vh ∈ Vh, for n = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the fully-discretized version (44), the following estimate holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 12 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un h be the fully-discretized variational form (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψn h be the corresponding sensitivities , respectively given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Now, we define θn and ρn on the discretized time grid (tn)n=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, en := Ψn h − Ψ(tn) = (Ψn h − P1 hΨ(tn)) + (P1 hΨ(tn) − Ψ(tn)), = θn + ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (48) By definition of ∂ and by Cauchy-Schwarz inequality (and since the second term of the left-hand side of (47) is always positive),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ��θn��2 L2(Ω) ≤ ����θn−1��� L2(Ω) + ∆tF � 1 µ ���∂un h − ut(tn) ��� L2(Ω) + ��wn�� L2(Ω) ����θn�� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and by repeated application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and since ���θ0��� L2(Ω) = 0 (again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' the case where some θn are equal to 0 can be easily handled),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' it entails ��θn�� L2(Ω) ≤ ∆tF n ∑ j=1 1 µ ���∂uj h − ut(tj) ��� L2(Ω) � �� � T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n + ∆tF n ∑ j=1 ���wj��� L2(Ω) � �� � T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (49) – We first decompose T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where wj u := wj 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u with wj 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := (P1 h − I)∂u(tj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and wj 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := ∂u(tj) − ut(tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (51) 13 Let us begin by the estimate on Tθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (56) It remains to estimate Tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the fully-discretized version (44), the following estimate holds with H1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un h be the fully-discretized variational form (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψn h be the corresponding sensitivities , respectively given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' – To find a bound for T′ 3,n, we simply use (57) and (58) again but with the sensitivity function Ψ instead of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Combining the estimates on T′ 2,n and T′ 3,n with (62), and (60) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT} such that (∂Ψm p,H, vH) + a( Ψm p,H+Ψm−1 p,H 2 , vH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ) ∇um H(µ)+∇um−1 H (µ) 2 , ∇vH), for vH ∈ VH, for m = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT}, (63) Ψ0 p,H(·) = P1 HΨ0 p(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have the following result in the L2 norm with the Crank-Nicolson scheme on the coarse mesh TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ H3(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψm H be the corresponding sensitivities , respectively given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For ρm we have the same estimate as before (46) (but with the coarse size H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We introduce the following notation � um H = 1 2(um H + um−1 H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (∂θm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + µ(∇� θm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) = (∂Ψm H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) − (∂P1 H(Ψ(�tm)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + µ(∇� Ψm H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − µ 2 � (∇P1 HΨ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇P1 HΨ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' = −(∇ � um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − (∂P1 H(Ψ(�tm)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) − µ 2 � (∇Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇Ψ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' = −(∇ � um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) −(∂P1 H(Ψ(�tm)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + (∂Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) � �� � −wm I −(∂Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + (Ψt(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) � �� � −wm II − (Ψt(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) − µ 2 � (∇Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇Ψ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � = −(∇ � um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − wm I − wm II + (∇u(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + µ(∇Ψ(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − µ 2 � (∇Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇Ψ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � = (∇u(�tm− 1 2 ) − � ∇um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − (wm I + wm II + µwm III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where wm I ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II and wm III are defined by wm I := (P1 H − I)∂Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II := ∂Ψ(�tm) − Ψt(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and wm III := ∆ψ(�tm− 1 2 ) − 1 2(∆Ψ(�tm) + ∆Ψ(�tm−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By definition of ∂ (with the coarse time grid),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and since the second term in (66) is always positive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' we get (θm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' � θm) − (θm−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' � θm) ≤ ∆tG � 1 µ ���∂um H − ut(�tm− 1 2 ) ��� L2(Ω) + ��wm T �� L2(Ω) ����� θm ��� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 17 and by definition of � θm (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ��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) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' so that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' after cancellation of a common factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ��θm�� L2(Ω) − ���θm−1��� L2(Ω) ≤ ∆tG � 1 µ ���∂um H − ut(�tm− 1 2 ) ��� L2(Ω) + ��wm T �� L2(Ω) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and by recursive application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' it entails ��θm�� L2(Ω) ≤ ∆tG µ m ∑ j=1 ���∂uj H − ut(�tj− 1 2 ) ��� L2(Ω) � �� � T′′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n + ∆tG m ∑ j=1 ���wj T ��� L2(Ω) � �� � T′′ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (67) – To estimate T′′ 2,n, we use the same tricks as before (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' we can decompose T′′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n in 2 contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u+wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where we denote by wm I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u the same terms respectively defined by wm I ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II (65) but with u instead of Ψ wm I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := (P1 h − I)∂u(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := ∂u(�tm) − ut(�tm− 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (69) ≤ √�tm µ �� m ∑ j=1 ∆tG ���∂θj��� 2 L2(Ω) �1/2 + � m ∑ j=1 ∆tG ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + ���wj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� To estimate the first term of (69),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' we use v = ∂θm u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and we now have (from the Crank-Nicolson scheme on u (11)) ���∂θm u ��� 2 L2(Ω) + µ(∇� θm u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇∂θm u ) = −(wm I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wm II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + µwm III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∂θm u ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where wm III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := ∆u(�tm− 1 2 ) − 1 2(∆u(�tm) + ∆u(�tm−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (70) and θm u is the discrete version of (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) ���∂θm 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We find that (since the first term of (71) is positive) µ ∆tG ��∇θm u ��2 L2(Ω) ≤ m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + µwj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and thus (72) gives m ∑ j=1 ���∂θj u ��� 2 L2(Ω) ≤ 2 m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + µwj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) ≤ 4 m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + ���µwj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + m ∑ j=1 ∆tG ���wj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + 2 µ2 m ∑ j=1 ∆tG ���wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + m ∑ j=1 ∆tG ���wj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (74) Now, we can estimate the right-hand side terms of (74) as done in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy with the previous work on parabolic equations, we define � ΨH n = I2 n[Ψm H](µ), for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that, as before, we could have chosen another quadratic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='10 (of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Under the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9, let um H be the fully-discretized solution (11) on the coarse mesh TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψm H be the corresponding sensitivities, respectively given by (23) and by (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let � ΨH n be the quadratic interpolation of the coarse solution Ψm H given by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the next section, we proceed with the adjoint state formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4 Sensitivity analysis: The adjoint problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The adjoint method may be seen as an inverse method, where the goal is to retrieve the optimal parameter of an objective function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the second setting, it will be written F(µ) = NT ∑ n=1 gnun h(µ), (83) with gn some suitable weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that by differentiating F with respect to the parameters µp, p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (84) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 The continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us for instance consider the first case outlined above, given in the continuous version by F(µ) = 1 2 � T 0 ��err(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) ��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since here we consider homogeneous condition, we just impose ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Differentiating L with respect to the parameter µp, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), v) + (A(µ)∇χ(t), ∇v), ∀v ∈ V, t < T, χ(·, T) = 0, in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (88) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a stable implementation, one may have to add a regularization term depending of the parameter to the cost function F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 Discretized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The semi-discretized version on Th writes � � � � � � � Find χh(t) ∈ Vh for t ∈ [0, T] such that (χh,t(t), vh) − a(χh, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂errh ∂uh (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), vh), ∀vh ∈ Vh, t < T, χh(·, T) = 0, in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂χn h, vh) − a(χh, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −(un h − un, vh), ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT − 1, χNT h (·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (90) Note that to compute ∂errn h ∂uh , we need the fine solutions un h and the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT} such that (∂χm H, vH) − a( 1 2(χm H + χm−1 H ), vH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = − 1 2 � (um H − um, vH) + (um−1 H − um−1, vH) � ), ∀vH ∈ VH, ∀m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT − 1, (91) χMT H (·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The next section adapts the NIRB two-grid algorithm in the context of sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3 NIRB algorithms applied to sensitivity analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 On the direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 NIRB algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We consider P parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this context, we use the following offline/online decomposition for the NIRB procedure: “Offline part” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a set of training parameters (�µi)i=1,··· ,Np,train, we define Gp,train = ∪ i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Np,train}�µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, through a greedy algorithm 1, we adequately choose the parameters of the RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During this procedure, we compute fine fully-discretized solutions {Ψn p,h(�µi)}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='Nµ,p}, n={0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2) � � � � � � � � � Find Ψn p,h ∈ Vh for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂Ψn p,h, vh) + a(Ψn p,h, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' �µ) = −( ∂A ∂µp (µ)∇uN,n Hh (µ), ∇vh) for n = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}, (92) Ψ0 p,h(·) = P1 hΨ0 p(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that un h can directly be used (as in (44)) since this step belongs to the offline part of the NIRB algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Np, and the reduced spaces X Np p,h := Span{ζh p,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , ζh p,Np} for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (93) For each parameter p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the offline step 3 from section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2, we enhance the NIRB approximation with a rectification post- processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we introduce the rectification matrices, denoted Rp,n Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' They are associated to the sensitivities problem (44), defined for each p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P} and each ��ne time step n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}, and constructed from coarse snapshots, generated by solving (63) and whose parameters are the same as for the fine snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT and all p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In general, Np,train < Np and the parameter δp is required for the inversion of (Ap,n)TAp,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, we first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, for each p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we solve the coarse associated sensitivity problems (63) with the same parameter µ, at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution Ψm p,H on the fine time grid with (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � Ψp,H n(µ) on the fine mesh in order to compute the L2-inner product with the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The approximation used in the two-grid method is For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 23 In the next section, we propose an adaptation of this algorithm with a new post-treatment which reduces the online computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 New NIRB algorithm for the direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The online portion of the new algorithm described below only requires the resolution of two coarse problems, regardless the number of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We refer to the following offline/online decomposition: “Offline part” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a parameter training set Gtrain, we compute the RB functions of the initial problem, denoted (Φh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N and generates XN h by the steps 1-2 of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 (see algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As before, from the training sets Gp,train, we generate the reduced spaces X Np p,h, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P using steps 1 and 2 of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1, and at the end of this part, we obtain Np RB functions (time-independent), denoted (ζh p,i)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Np for each p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We introduce GT defined by GT := Gtrain ∩ Gp,train, (99) and Nµ,T the number of parameters in GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P} and each fine time step n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT and all p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that � uH n(�µk) is the quadratic interpolation given by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We highlight the fact that this step requires that GT ̸= ∅ (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online step” 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution um H on the fine time grid with (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � uH n(µ) on the fine mesh in order to compute the L2-inner product with the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 On the adjoint formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The adjoint formulation requires some modifications of the NIRB algorithm compared to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since in (88), for all n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As a matter of fact, in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1), the RB generation for u was optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' So let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and χ(µ) its adjoint given by (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, we use the following offline/online decomposition for the NIRB procedure: “Offline part” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During the offline stage, we first construct the reduced space XN h and the RB function (Φh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Φh N) with the steps 1-2 of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we use steps 1-2 of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for a set of training parameters (�µi)i=1,··· ,N1,train, we define G1,train = ∪ i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1,train}�µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, through a greedy procedure 1, we adequately choose the parameters of the RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During this proce- dure, we compute fine fully-discretized solutions {χn h(�µi)}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='Nµ,1}, n={0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2) � � � � � � � Find χn h ∈ Vh for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂χn h, vh) − a(χh, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −(uN,n Hh − un, vh), ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT − 1, χNT h (·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1, and the reduced space XN1 h := Span{ξh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , ξh N1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the offline step 3 from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1, we enhance the NIRB approximation with a rectifica- tion post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we introduce a rectification matrix, denoted Rn χ for each fine time step n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online part” 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we solve the coarse associated adjoint problem (92) with the same parameter µ, at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution χm H on the fine time grid with (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � χH n(µ) on the fine mesh in order to compute the L2-inner product with the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The approximation used for the adjoint in the two-grid method is For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we use the steps 5 and 6 of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Finally, the sensitivities NIRB approximations of F are given by for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (112) The next section gives our main result on the NIRB two-grid method error estimate in the context of sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4 NIRB error estimate on the sensitivities Main result Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (NIRB error estimate for the sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=') Let A(µ) = µ Id, with µ ∈ R+∗ , and let us consider the problem 3 with its exact solution u(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), and the full discretized solution un h(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) to the problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) and Ψn h(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) respectively by the corresponding sensitivities , given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let (ζh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1 be the L2-orthonormalized and H1-orthogonalized RB generated with the greedy algorithm 1 through the NIRB algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us consider the NIRB approximation, For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ΨN,n Hh (µ) := N1 ∑ i=1 (� ΨH n(µ), ζh i ) ζh i , (113) where � ΨH n(µ) is given by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, the following estimate holds ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The term ε depends on the Kolmogorov N-width and measures the error given by (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 then states that we recover optimal error estimates in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We now go on with the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The NIRB approximation at time step n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, for a new parameter µ ∈ G is defined by (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The first term T1 may be estimated using the inequality given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='8, such that ��Ψ(tn)(µ) − Ψn h(µ) �� H1(Ω) ≤ C(µ) (h + ∆tF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (116) We then denote by S′ h = {Ψn h(µ, t), µ ∈ G, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' NT} the set of all the sensitivities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For our model problem, this manifold has a low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, T2 is bounded by ε which depends on the Kolmogorov N-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' T2 = ������ Ψn h(µ) − N1 ∑ i=1 (Ψn h(µ), ζh i ) ζh i ������ H1(Ω) ≤ ε(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (117) Since (ζh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, defined by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (120) Now by definition of � ΨH n(µ) and by corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='10 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Combining these estimates (116), (117) and (122) concludes the proof and yields the estimate (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 27 Figure 1: H1 0 NIRB errors 5 Numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this section, we have applied the NIRB algorithms on several numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have implemented both schemes (Euler and RK2) using FreeFem++ (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9) [26] to compute the fine and coarse snapshots, and the solutions have been stored in VTK format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' After saving the NIRB approximations with Paraview module on Python, the errors have been computed with FreeFem++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 On the heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have solved (3) and (23) on the parameter set G = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5], with u0 solution of Poisson’s equation −∆u0 = f and Φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have taken 18 parameters in G for the RB construction such that µi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' A Derivation of the adjoint for the heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this appendix, we recall the main steps to derive the adjoint of our model problem, in order to compute ( ∂F ∂µk )k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a more general problem, we refer to [44] in case of FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 28 FEM H relative errors NIRB H relative error with H = V h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='80- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='80- h h FEM coarse error NiRB+rectification 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='50 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='50- FEM fine error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='20 Error (log Error (log 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='05- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Differentiating L with respect to the parameter µp, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we obtain dL dµp (u, χ, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � T 0 � � Ω derr dµp (µ) dx + � χ, d[∇ · (A(µ)∇u) + f − ut] dµp �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (123) In our setting, the objective does not depend directly on the parameter µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore, using the chain rule, it may be rewritten dL dµp (u, χ, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) := ∂u ∂µp (t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As we saw before, a classical forward sensitivity computation would require P + 1 systems of PDEs to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Here, we want to avoid calculating the sensitivities of the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � T 0 ��∂err ∂u , Ψp � + � χ, ∇ · ( ∂A ∂µp (µ)∇u) � + � χ, ∂[∇ · (A(µ)∇u)] ∂u Ψp � + � χt, Ψp �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 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[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (126) 29 Acknowledgment This work is supported by the SPP2311 program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We would like to give special thanks to Ole Burghardt for his precious help on Automatic Differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Bader, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' K¨archer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' A Grepl, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Veroy.' metadata={'source': 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