diff --git "a/AtE1T4oBgHgl3EQfpAWX/content/tmp_files/load_file.txt" "b/AtE1T4oBgHgl3EQfpAWX/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/AtE1T4oBgHgl3EQfpAWX/content/tmp_files/load_file.txt" @@ -0,0 +1,533 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf,len=532 +page_content='A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure Marcello Longo∗, Christoph Schwab∗, Andreas Stein∗† January 10, 2023 Abstract We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The estimator’s reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 1 Introduction The efficient numerical approximation of high-dimensional, parametric partial differential equations (PDEs for short) received increasing attention during the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In this work, we address two classes of high-dimensional numerical integration problems which arise in connection with data assimilation and PDE constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We illustrate the abstract concepts for a parametric, linear elliptic PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The first class is the so-called Bayesian inverse problem (BIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' There, we are interested in the posterior expectation of a (linear) functional of the solution u of a parametric PDE, conditional on observation data subject to additive, centered Gaussian observation noise [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' See also [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' A second problem class is PDE-constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Specifically, the optimal control problem (OCP) of parametric PDEs under an entropic risk measure [7], where the state variable satisfies a parametric PDE constraint [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' When numerically approximating solutions of a BIP or an OCP, it is essential to quantify the error due to the numerical discretization in order to meet a prescribed numerical tolerance without wasting computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' As solving PDEs exactly is in general not possible, discretizations such as Finite Element Methods (FEM) must be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Additionally, the parametric ∗Seminar for Applied Mathematics, ETH Zürich †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Email: andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='stein@sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='ch 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='03327v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='NA] 9 Jan 2023 uncertainty in the forward PDE model is passed on to the solution, and this must be taken into account both in the computation and in the error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' This justifies the need for an a-posteriori error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assuming that the uncertain PDE coefficients can be described by means of a parameter vector y ∈ U := � − 1 2, 1 2 �s where s ∈ N, s ≫ 1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (6) below, we can employ suitable quasi-Monte Carlo (QMC) rules to approximate integrals over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Here, we select extrapolated polynomial lattice (EPL) rules as first introduced in [3,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' This choice is motivated by the deterministic nature or their quadrature nodes Pm, |Pm| = bm for some prime b [15], and good convergence properties under quantified parametric regularity of the integrand functions with respect to y ∈ U, uniformly in the dimension s [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Moreover, it was shown in [5,14] that under assumptions, EPL quadratures allow for computable a-posteriori quadrature error estimators that are asymptotically exact as m → ∞, with dimension robust ratio between estimated and actual quadrature error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Both, BIP and OCP problems for PDEs with parametric input take the form Z′ Z ∈ Y, where Z = � U Θ(y) dy, Z′ = � U Θ′(y) dy, (1) for some suitable integrable functions Θ: U → R and Θ′ : U → Y, where Y is a separable Hilbert space and Z, Z′ are Bochner integrals with respect to a product measure dy on the possibly high-dimensional parameter space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In particular, we have Y = R for the BIP case and Y ∈ L2(D) for the OCP case, with D being the physical domain of the considered PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Approximating the high-dimensional integrals with averages over polynomial lattices Pm ⊂ U yields a first approximation Z′ m Zm ∈ Y, where Zm = 1 bm � y∈Pm Θ(y), Z′ m = 1 bm � y∈Pm Θ′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (2) Then, since the integrands Θ, Θ′ depend on the solution of a y-parametric PDE, we discretize the parametric PDEs for y ∈ Pm, resulting in computable, parametric integrand functions Θh(y), Θ′ h(y) and in the computable estimates Z′ m,h Zm,h ∈ Y, where Zm,h = 1 bm � y∈Pm Θh(y), Z′ m,h = 1 bm � y∈Pm Θ′ h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (3) Here, the parameter h > 0 denotes the meshwidth of conforming Lagrangian Finite Element discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We present a computable a-posteriori estimator for the combined Finite Element discretization and quadrature error err = ���� Z′ Z − Z′ m,h Zm,h ���� Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (4) In the rest of this section we introduce the setting and we describe the two problems of interest, namely the BIP and the OCP with entropic risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Section 2 and Section 3 are devoted to the QMC and the FEM a-posteriori error analysis, respectively, and these results will be combined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We present numerical experiments in Section 5 and summary and conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1 Affine-Parametric Forward PDE For brevity of presentation, we consider a model, linear elliptic PDE with homogeneous Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Given a bounded polygon D ⊆ R2 and a parameter sequence y ∈ U, s ∈ N, consider the following parametric, linear, second order elliptic PDE in variational form: find u(·, y) ∈ X = H1 0(D) such that � D a(x, y)∇xu(x, y) · ∇xv(x) dx = � D f(x)v(x) dx ∀v ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (5) We assume that a is affine-parametric, namely that we are given a family of functions {ψj}j∈N0 ⊆ L∞(D) such that, with essinf ψ0 > κ > 0 and bj := 1 κ ∥ψj∥L∞(D), we have a(x, y) = ψ0(x) + s � j=1 yjψj, � j≥1 bj < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (6) Then essinf a(·, y) > essinf ψ0 − κ > 0 for all y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' By the Lax-Milgram lemma, the parametric weak solution u(·, y) ∈ X is well defined for any f ∈ X ∗ = H−1(D), where ∗ denotes the topological dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' To justify the a-posteriori QMC error analysis of Section 2, we will additionally require the summability b = (bj)j≥1 ∈ ℓp(N) for some p ∈ (0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (7) For the FEM approximation, we consider conforming subspaces1 Xh ⊆ X h ∈ H ⊆ (0, ∞), dim(Xh) < ∞, that are linked to shape-regular, simplicial partitions {Th}h∈H of D [6, Section 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assume that the resulting spaces are nested and conforming, that is Xh ⊆ Xh′ for any h, h′ ∈ H, h > h′ and that H accumulates at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We construct the Galerkin discretizations uh(y) ∈ Xh of (5), by solving � D a(x, y)∇xuh(x, y) · ∇xv(x) dx = � D f(x)v(x) dx ∀v ∈ Xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (8) To simplify notation, we write a(y) = a(·, y) and u(y) = u(·, y) and we omit the variable x for ∇x = ∇ and divx = div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='2 Bayesian inverse problem (BIP) Let X = {a ∈ L∞(D) : essinf a > 0} and fix f ∈ X ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, we can define the data-to-solution map S : X → X for the forward problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We also define the observation functional O ∈ (X ∗)K, with a finite number K ∈ N of observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' representing sensors), and a goal functional (also called quantity of interest) G ∈ X ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We define the prior measure π0 to be the uniform distribution on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The observations O(S(a)) are assumed to be additionally subject to additive observation noise η, which we assume to be centered Gaussian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=', η ∼ N(0, Γ) for some known, nondegenerate covariance matrix Γ ∈ RK×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In other words, we assume given noisy observation data δ ∈ RK modeled as δ = O(S(a)) + η ∈ L2 Γ(RK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (9) 1In practice, h either parametrizes the local mesh-size maxT ∈Th |T|1/2, for quasi-uniform collections of partitions, or it relates to the refinement level in case of adaptive refinement [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 3 We consider the Bayesian inverse problem of recovering the expected value of G(u), given observation data δ, that is Eπ0[G(u)|δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' By Bayes’ theorem [19], the posterior distribution πδ of y|δ is absolutely continuous with respect to π0 and its Radon-Nikodym derivative with respect to the prior π0 is dπδ dπ0 (y) = Θ(y) Z , (10) where Θ(y) := exp(− 1 2|δ − O(S(a(y)))|2 Γ) = exp(− 1 2|δ − O(u(y))|2 Γ) denotes the likelihood with the observation noise covariance-weighted, data-to-observation misfit, where |x|2 Γ := x⊤Γ−1x and Z is defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' As Θ(y) > 0 for all y ∈ U, Z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In the present setting, Bayesian inversion amounts to the numerical evaluation of the posterior mean Eπδ[G(u)] = 1 Z � U G(u(y))Θ(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' This is (1) upon setting Θ′(y) := G(u(y))Θ(y) and Y = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Define the FE solution operator Sh : X → Xh as the mapping Sha(y) = uh(y) via (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The FE approximations of Θ, Θ′ used in (3) are then Θh = exp(− 1 2|δ − O(uh(y))|2 Γ) and Θ′ h = G(uh(y))Θh(y), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='3 Optimal control with entropic risk measure (OCP) Let Y = L2(D), assume a parameter independent target state ˆu ∈ Y and a nonempty, closed and convex set X ⊆ Y of admissible controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Throughout the rest of the paper, we identify Y with its dual via Riesz representation and write ⟨·, ·⟩ for the inner product on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Once the affine parametric diffusion coefficient a(y) is fixed, (5) defines a linear solution operator Ly : Y → Y by Lyf = ι ◦ u(y) for all f ∈ Y, where ι denotes the continuous embedding X ⊂ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In particular, we view u(y) as a function of the right-hand side f of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For a function Φ: U → R and some θ ∈ (0, ∞), the entropic risk measure [12] is defined by R(Φ) = 1 θ log �� U exp(θΦ(y)) dy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (11) The entropic risk is especially relevant when favoring a risk averse behavior [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We consider the following minimization problem, for fixed constants α1, α2 > 0 f ∗ := argmin f∈X J(f), J(f) := R( α1 2 ∥Lyf − ˆu∥2 Y) + α2 2 ∥f∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (12) Due to convexity of R and α2 > 0, the functional J is strongly convex so that (12) is a well-posed minimization problem [9,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Define the shorthand notation Φf(y) = α1 2 ∥Lyf − ˆu∥2 Y and the adjoint state given by qf(y) = α1Ly(Lyf − ˆu) ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Under the above conditions on X, (12) is equivalent to the inequality ⟨J′(f ∗), f − f ∗⟩ ≥ 0 for all f ∈ X, where in analogy with [9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='6] the Fréchet derivative J′(f) ∈ Y of J at f ∈ X is J′(f) = 1 � U exp(θΦf(y)) dy � U exp(θΦf(y))qf(y) dy + α2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (13) 4 Next, we replace in (12) the integral over U by QMC rules and the exact solution operator Ly by the Galerkin solution Ly h : Y → Xh defined by Ly hf = uh(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, we obtain the discrete formulation f ∗ m,h := argmin f∈X Jm,h(f), Jm,h(f) := Rm( α1 2 ∥Ly hf − ˆu∥2 Y) + α2 2 ∥f∥2 Y , (14) where Rm(Φ) = 1 θ log � 1 bm � y∈Pm exp(θΦ(y)) � is again convex, due to positivity of the QMC quadrature weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The derivative J′ m,h(f) ∈ Y of Jm,h is analogous to (13), again replacing the integrals by sample averages over Pm, Φh,f(y) = α1 2 ∥Ly hf − ˆu∥2 Y and qh,f(y) = α1Ly h(Ly hf − ˆu) ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The next proposition recasts the error in the approximation f ∗ m,h ≈ f ∗ to the form (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Whenever it does not cause confusion, we will write q(y) = qf ∗ m,h(y) and qh(y) = qh,f ∗ m,h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For Y = L2(D), assume Z, Z′ in (1) are defined by Θ(y) = exp(θΦf ∗ m,h(y)) and Θ′(y) = q(y)Θ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Similarly, let Zm,h, Z′ m,h in (3) be defined by Θh(y) = exp(θΦh,f ∗ m,h(y)) and Θ′ h(y) = qh(y)Θh(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then ��f ∗ − f ∗ m,h �� Y ≤ 1 α2 ��J′(f ∗ m,h) − J′ m,h(f ∗ m,h) �� Y = 1 α2 ���� Z′ Z − Z′ m,h Zm,h ���� Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The inequalities ⟨J′(f ∗), f − f ∗⟩ ≥ 0 and ⟨J′ m,h(f ∗ m,h), f − f ∗ m,h⟩ ≥ 0 for all f ∈ X imply that ⟨J′ m,h(f ∗ m,h) − J′(f ∗), f ∗ − f ∗ m,h⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Moreover, strong convexity of J yields the relation ⟨J′(f ∗)−J′(f ∗ m,h)−α2(f ∗−f ∗ m,h), f ∗−f ∗ m,h⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Thus, we get α2 ��f ∗ − f ∗ m,h ��2 Y ≤ ⟨J′ m,h(f ∗ m,h) − J′(f ∗) + α2(f ∗ − f ∗ m,h), f ∗ − f ∗ m,h⟩ ≤ ⟨J′ m,h(f ∗ m,h) − J′(f ∗ m,h), f ∗ − f ∗ m,h⟩ ≤ ��J′ m,h(f ∗ m,h) − J′(f ∗ m,h) �� Y ��f ∗ − f ∗ m,h �� Y , which implies the inequality in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The equality follows by substituting J′ m,h(f ∗ m,h) = Z′ m,h Zm,h + α2f ∗ m,h and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Note that Θ, Θ′ as defined in Proposition 1, and hence Z, Z′, also implicitly depend on h via the discrete minimizer f ∗ m,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In particular, the exact minimizer f ∗ does not appear in the right hand side of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' This fact will be crucial for the ensuing a-posteriori error estimation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 2 A-posteriori QMC error estimation We develop computable a-posteriori error estimators for the PDE discretization error and for the QMC-quadrature error, the latter being reliable independent of the integration dimension s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1 Parametric regularity To leverage the results from [5, 14] and overcome the curse of dimensionality, we need to quantify the regularity with respect to y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' To this end, we 5 write |ν| = � j νj, supp(ν) = {j : νj ̸= 0} and, given smooth F : U → Y and ν ∈ F := {ν ∈ NN 0 : | supp(ν)| < ∞} we introduce the multi-index notation ∂ν yF(y) = � j∈supp(ν) ∂νj yj F(y) for the derivatives with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Moreover, given β = (βj)j∈N ∈ ℓp(N) for some p ∈ (0, 1), n ∈ N and c > 0, we define the SPOD weights γ = (γu)u⊆{1:s} via γu = � ν∈{1:α}|u| (|ν| + n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' � j∈u cβνj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (16) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let Y be a separable Hilbert space, α ∈ N, α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We define the weighted unanchored Sobolev space Ws,α,γ with dominating mixed smoothness as the completion of C∞(U, Y) with respect to the norm ∥F∥s,α,γ := max u⊆{1:s} γ−1 u � v⊆u � νu\\v∈{1:α}|u\\v| � [− 1 2 , 1 2 ]|v| ����� � [− 1 2 , 1 2 ]s−|v| ∂ (νu\\v,αv) y F(y) dy{1:s}\\v ����� Y dyv, where µ = (νu\\v, αv) ∈ F is such that µj = νj if j ∈ u \\ v, µj = α if j ∈ v and µj = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The inner integral is interpreted as a Bochner integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The relevance of the space Ws,α,γ in our context is justified by the following result, which will be the starting point of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' This is a consequence of the so-called component-by-component (CBC) construction as described in [5,14], which takes as input s, α, γ and m and returns a polynomial lattice Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let α ∈ N, α ≥ 2 and F : U → Y for some separable Hilbert space Y be such that F ∈ Ws,α,γ for some weights γ of the form (16) for β ∈ ℓp(N), p ∈ (0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, there exists a sequence (Pm)m∈N of polynomial lattice rules such that as m → ∞ Em(F) := � U F − 1 bm � y∈Pm F(y) = ∥F∥s,α,γ O(b−m), as well as for all ε > 0 Em(F) = 1 bm � y∈Pm F(y) − 1 bm−1 � y∈Pm−1 F(y) + ∥F∥s,α,γ O(b−2m+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Here the constants hidden in O(·) are independent of s, m ∈ N and F, but depend on ε and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In addition, the point sets (Pm)m∈N can be constructed explicitly by a CBC construction in O(smbm + s2bm) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For the case Y = R, this is proved in [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Otherwise, let v ∈ Y arbitrary such that ∥v∥Y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, ∂ν y⟨F(y), v⟩ = ⟨∂ν yF(y), v⟩ for all ν ∈ F implies ∥⟨F, v⟩∥s,α,γ ≤ ∥F∥s,α,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Hence, we reduced to the case Y = R and by linearity of the integral and the quadrature we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assume that the PDE (5) satisfies (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then we can construct polynomial lattice rules (Pm)m∈N (depending on b) in O(smbm + s2bm) operations such that for all ε > 0 it holds for the BIP Z − Zm = Zm − Zm−1 + O(b−2m+ε), Z′ − Z′ m = Z′ m − Z′ m−1 + O(b−2m+ε), as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The hidden constants in O(·) are independent of s, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' From [18, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1], [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1] and ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' := � j∈supp(ν) νj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' ≤ |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=', we have that sups ∥Θ∥s,α,γ + sups ∥Θ′∥s,α,γ < ∞ for α = 1 + ⌊1/p⌋ and some SPOD weights (16) defined by a sequence β ∼ b and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Hence we can apply Theorem 1 and conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' A similar result can be given for the OCP, based on the results in [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assume that the PDE (5) satisfies (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then we can construct polynomial lattice rules (Pm)m∈N (depending on b) in O(smbm + s2bm) operations such that for all ε > 0 it holds for the OCP Z − Zm = Zm − Zm−1 + O(b−2m+ε), Z′ − Z′ m = Z′ m − Z′ m−1 + O(b−2m+ε), as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The hidden constants in O(·) are independent of s, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Applying the result of [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='6] in [9, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='6], we conclude that sups ∥Θ∥s,α,γ + sups ∥Θ′∥s,α,γ < ∞ for α = 1 + ⌊1/p⌋ and for some SPOD weights (16) defined by a sequence β ∼ b and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='2 Ratio error estimator Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assume that the PDE (5) satisfies (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, for the BIP and the OCP, we can construct polynomial lattice rules (Pm)m∈N such that Z′ Z − Z′ m Zm = 1 bZm − Zm−1 �Zm−1Z′ m − ZmZ′ m−1 Zm � + O(b−2m+ε), (17) holds for all ε > 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The hidden constant in O(·) is independent of s, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' First, Z − Zm → 0 as m → ∞ implies that Zm > Z/2 > 0 for m sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Next, due to either Corollary 1 or 2, for all ε > 0 we have Z′ Z − Z′ m Zm = (Z′ − Z′ m)Zm − (Z − Zm)Z′ m (Z − Zm)Zm + Z2m = 1 b−1(Z′ m − Z′ m−1 + O(b−2m+ε))Zm − 1 b−1(Zm − Zm−1 + O(b−2m+ε))Z′ m 1 b−1(Zm − Zm−1 + O(b−2m+ε))Zm + Z2m = 1 b−1(Z′ m − Z′ m−1) 1 b−1(bZm − Zm−1) + O(b−2m+ε) − 1 b−1(Zm − Zm−1)Z′ m ( 1 b−1(bZm − Zm−1) + O(b−2m+ε))Zm + O(b−2m+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Clearly Am := 1 b−1(bZm − Zm−1) → Z as m → ∞, so that Am > Z/2 > 0 for m sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Hence, by a geometric sum argument, collecting in Bm all terms contained in O(b−2m+ε) in the denominator, we obtain for m → ∞ that 1 Am + Bm = 1 Am ∞ � k=0 (−1)k �Bm Am �k = 1 Am + O(b−2m+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 7 Therefore, as m → ∞ Z′ Z − Z′ m Zm = 1 b−1(Z′ m − Z′ m−1) Am − 1 b−1(Zm − Zm−1)Z′ m AmZm + O(b−2m+ε), (18) which, upon rearranging the terms, is the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Since Z′ Z − Z′ m Zm = O(b−m) ≫ O(b−2m+ε), Proposition 2 states that Ebm(Θ′, Θ) := 1 bZm − Zm−1 �Zm−1Z′ m − ZmZ′ m−1 Zm � (19) is a computable, asymptotically exact error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 3 A-posteriori FEM error estimation In practice the parametric solution u(y) ∈ X is not exactly available, and hence Zm, Z′ m are not computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For any y ∈ U, u(y) will be approximated by the corresponding Galerkin discretizations uh(y) ∈ Xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1 Ratio error estimator Let Θh, Θ′ h, Zh, Z′ h be defined replacing u and q by uh, qh in the definitions of Θ, Θ′, Z, Z′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Similarly, let Zm,h, Z′ m,h be defined replacing u and q by uh, qh in the definitions of Zm, Z′ m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assume there exist ζm,h, ζ′ m,h such that for some c > 0 inde- pendent of h ∈ H, |Zm − Zm,h| ≤ cζm,h and ���Z′ m − Z′ m,h ��� Y ≤ cζ′ m,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Assume that ζm,h → 0 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, there exists h0 ∈ H such that for all h ≤ h0 ���� Z′ m Zm − Z′ m,h Zm,h ���� Y ≤ c Zm,hζ′ m,h + ���Z′ m,h ��� Y ζm,h Z2 m,h − cζm,hZm,h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Due to the limit ζm,h → 0 there exists h1 ∈ H such that Zm,h ≥ Zm/2 > 0 for all h ≤ h1, h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Therefore, we can pick h0 ∈ H such that Zm,h > cζm,h for all h ≤ h0, h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Thus ���� Z′ m Zm − Z′ m,h Zm,h ���� Y = ����� (Z′ m − Z′ m,h)Zm,h − (Zm − Zm,h)Z′ m,h (Zm − Zm,h)Zm,h + Z2 m,h ����� Y ≤ ���Z′ m − Z′ m,h ��� Y Zm,h + |Zm − Zm,h| ���Z′ m,h ��� Y Z2 m,h − |Zm − Zm,h| Zm,h ≤ c Zm,hζ′ m,h + ���Z′ m,h ��� Y ζm,h Z2 m,h − cζm,hZm,h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 8 Due to this result, we are left with the task of finding computable ζm,h, ζ′ m,h satisfying the conditions |Zm − Zm,h| ≤ cζm,h and ���Z′ m − Z′ m,h ��� Y ≤ cζ′ m,h for some c > 0 independent of m, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Thus, in the following sections we will provide such error estimators for BIP and OCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='2 FEM error estimators for BIP For a finite collection of observation functionals O = (O1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' , OK) ∈ (X ∗)K, define ∥O∥X ∗ = ��K k=1 ∥Ok∥2 X ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The starting point to estimate the FEM error will be the following well-known result, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let {Th}h∈H be a family of shape-regular, simplicial meshes of D and let Pk(Th) be the set of piecewise polynomial functions on Th of degree at most k ∈ N0 in each T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let Eh be the set of interior edges of all elements T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We assume that Xh := P1(Th)∩X, that f ∈ L2(D) and that a(y) ∈ W 1,∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let hT = |T|1/2 for T ∈ Th and he the length of an edge e ∈ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then we define the a-posteriori error estimator η2 y,h := � T ∈Th � h2 T ∥f + div(a(y)∇uh(y))∥2 L2(T ) + 1 2 � e⊆∂T,e∈Eh he ∥�a(y)∇uh(y)�∥2 L2(e) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (20) By [16, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='3] there exists some c∗ > 0, only depending on D and the shape regularity constant of {Th}h∈H, and in particular independent of y ∈ U and h ∈ H, such that ∥u(y) − uh(y)∥X ≤ c∗ηy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (21) For the important special case O ∈ (L2(D))K and G ∈ L2(D) we may derive sharper estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' To simplify the presentation, we assume here that the physical domain D ⊆ R2 is a convex polygon (see also Remark 3 below), and introduce the L2(D)−residual estimator ˜η2 y,h := � T ∈Th � h4 T ��f ∗ m,h + div(a(y)∇uh(y)) ��2 L2(T ) + 1 2 � e⊆∂T,e∈Eh h3 e ∥�a(y)∇uh(y)�∥2 L2(e) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (22) The additional factors hT , he are derived from a standard duality argument, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' [20, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, there exists some c∗ > 0 depending only on D and the shape regularity constant of {Th}h∈H, and in particular independent of y ∈ U and h ∈ H, such that ∥u(y) − uh(y)∥L2(D) ≤ c∗˜ηy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (23) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Fix a regular mesh Th of simplices in D and a parameter vector y ∈ U and assume (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then |Θ(y) − Θh(y)| ≤ Θh(y)(eχy,h − 1) =: ζy,h 9 holds for χy,h := ���Γ−1/2O ��� X ∗ � |δ − O(uh(y))|Γ + 1 2 ���Γ−1/2O ��� X ∗ c∗ηy,h � c∗ηy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Furthermore, if O ∈ (L2(D))K and G ∈ L2(D), then |Θ(y) − Θh(y)| ≤ Θh(y)(e˜χy,h − 1) := ˜ζy,h holds for ˜χy,h := ���Γ−1/2O ��� L2(D) � |δ − O(uh(y))|Γ + 1 2 ���Γ−1/2O ��� L2(D) c∗˜ηy,h � c∗˜ηy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We define ∆h(y) := − 1 2 |δ − O(u(y))|2 Γ + 1 2 |δ − O(uh(y))|2 Γ to obtain |Θ(y) − Θh(y)| = |Θh(y)(e∆h(y) − 1)| ≤ Θh(y)(e|∆h(y)| − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The first part of the claim now follows with (21) and |∆h(y)| ≤ 1 2 ���Γ���1/2O ��� X ∗ |2δ − O(u(y) + uh(y))|Γ ∥u(y) − uh(y)∥X ≤ ���Γ−1/2O ��� X ∗ |δ − O(uh(y))|Γ ∥u(y) − uh(y)∥X + 1 2 ���Γ−1/2O ��� 2 X ∗ ∥u(y) − uh(y)∥2 X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The second part of the proof follows analogously by replacing X by L2(D) and using (23) instead of (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Fix a regular mesh of simplices Th and y ∈ U and assume (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then there exists a constant c∗ > 0 such that for all y ∈ U and h ∈ H |Θ′(y) − Θ′ h(y)| ≤ ∥G∥X ∗ (c∗ηy,hΘh(y)eχy,h + ζy,h ∥uh(y)∥X ) =: ζ′ y,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Furthermore, if O ∈ (L2(D))K and G ∈ L2(D), then |Θ′(y) − Θ′ h(y)| ≤ ∥G∥L2(D) � c∗˜ηy,hΘh(y)e˜χy,h + ˜ζy,h ∥uh(y)∥L2(D) � =: ˜ζ′ y,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Since Θ′(y) = G(u(y))Θ(y) = G(u(y)Θ(y)), we get |Θ′(y) − Θ′ h(y)| ≤ ∥G∥X ∗ ∥u(y)Θ(y) − uh(y)Θh(y)∥X ≤ ∥G∥X ∗ (∥u(y) − uh(y)∥X Θ(y) + ∥uh(y)∥X |Θ(y) − Θh(y)|), and hence the claim follows with Lemma 1 since Θ(y) ≤ Θh(y)eχy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The second part of the claim follows analogously by the second part of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We remark that the estimates in the proof of Lemma 2 are conservative: we used that K > 1 in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For K = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' for a single observation functional, goal-oriented AFEM results from [1] and the references there, can be used to obtain sharper a-posteriori error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 10 We now can define ζm,h by averaging ζy,h for y ∈ Pm, that is ζm,h := 1 bm � y∈Pm ζy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then we get from Lemma 1 |Zm − Zm,h| ≤ 1 bm � y∈Pm |Θ(y) − Θh(y)| ≤ ζm,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (24) Analogously, with ζ′ m,h = 1 bm � y∈Pm ζ′ y,h, Lemma 2 implies ��Z′ m − Z′ m,h �� ≤ 1 bm � y∈Pm |Θ′(y) − Θ′ h(y)| ≤ ζ′ m,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (25) In particular, if we construct Th such that it also holds ηy,h → 0 as h → 0 for all y ∈ U, (24) and (25) verify the hypotheses of Proposition 3 with c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='3 FEM error estimators for OCP with entropic risk In the case of OCP we require error estimates for the parametric state at the discrete optimal control f ∗ m,h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' u(y) = Lyf ∗ m,h and for the corresponding adjoint state q(y) = α1Ly(Lyf ∗ m,h − ˆu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The error will be measured in the L2(D)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Again we assume that Xh := P1(Th) ∩ X, that f ∈ L2(D) and a(y) ∈ W 1,∞(D), and that D ⊆ R2 is a convex polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Fix a mesh Th and y ∈ U and impose (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' With the notation of Proposition 1 and ˜ηy,h defined as in (22), we have |Θ(y) − Θh(y)| ≤ Θh(y)(eχy,h − 1) =: ζy,h, where χy,h := θc∗ �α1 2 ˜η2 y,h + α1 ��Ly hf ∗ m,h − ˆu �� L2(D) ˜ηy,h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' By twofold application of the triangle inequality we have ���Φf ∗ m,h(y) − Φh,f ∗ m,h(y) ��� ≤ α1 2 ��(Ly − Ly h)f ∗ m,h ��2 L2(D) + α1 ��Ly hf ∗ m,h − ˆu �� L2(D) ��(Ly − Ly h)f ∗ m,h �� L2(D) ≤ c∗ �α1 2 ˜η2 y,h + α1 ��Ly hf ∗ m,h − ˆu �� L2(D) ˜ηy,h � =: c∗ξy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Note that ξy,h is computable due to Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, it follows with ∆h(y) := θ(Φf ∗ m,h(y) − Φh,f ∗ m,h(y)) that |Θ(y) − Θh(y)| = ���Θh(y)(e∆h(y) − 1) ��� ≤ Θh(y)(e|∆h(y)| − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Using a residual estimator in the form (22) for the adjoint problem yields ∥q(y) − qh(y)∥L2(D) ≤ 2 max(c∗, 1)c∗˜˜ηy,h, (26) where ˜˜η2 y,h := α2 1 � T ∈Th � h4 T ∥uh(y) − ˆu + div(a(y)∇qh(y))∥2 L2(T ) (27) + 1 2 � e⊆∂T,e∈Eh h3 e ∥�a(y)∇qh(y)�∥2 L2(e) � + (max T ∈Th h4 T )˜η2 y,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 11 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let Y = L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Fix a mesh Th and y ∈ U and impose (23) and (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' With the notation of Proposition 1, we have ∥Θ′(y) − Θ′ h(y)∥Y ≤ ζy,h ∥qh(y)∥Y + 2c∗Θh(y)eχy,h ˜˜ηy,h =: ζ′ y,h Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' As in the proof of Lemma 2, we get by Θ(y) ≤ Θh(y)eχy,h that ∥q(y)Θ(y) − qh(y)Θh(y)∥Y ≤ |Θ(y) − Θh(y)| ∥qh(y)∥Y + Θ(y) ∥q(y) − qh(y)∥Y ≤ ζy,h ∥qh(y)∥Y + 2c∗Θh(y)eχy,h ˜˜ηy,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' If D ⊆ R2 is a non-convex polygon, the reliability assumption (23) and the corresponding definitions (22), (27) of ˜ηy,h, ˜˜ηy,h must be adapted by using weighted L2 norms, with weights near the re-entrant corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We refer to [21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1] for a precise result in the case of the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 4 Combined QMC-FEM estimator In view of Propositions 2 and 3 we employ the computable a-posteriori estimator ESTbm,h := ∥Ebm(Θ′ h, Θh)∥Y + Zm,hζ′ m,h + ���Z′ m,h ��� Y ζm,h Z2 m,h − ζm,hZm,h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (28) Note that the QMC error estimator ∥Ebm(Θ′, Θ)∥Y derived from Proposition 2 is itself approximated by the computable expression ∥Ebm(Θ′ h, Θh)∥Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In the next proposition we precise that the additional error committed due to this extra approximation is of higher asymptotic order, as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We equip the set C0(U, Y) with the norm ∥F∥∞ = supy∈U ∥F(y)∥Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Fix a family of regular meshes {Th}h∈H such that for some ˜C > 0 independent of s ∈ N, h ∈ H and some SPOD weights (16), it holds max(∥Θ∥s,α,γ , ∥Θ′∥s,α,γ , ∥Θh∥s,α,γ , ∥Θ′ h∥s,α,γ) ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' (29) Assume that the spaces {Xh}h are contained in X and that they are selected so that ∥Θ − Θh∥∞ → 0 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then we can construct a sequence of polynomial lattices (Pm)m∈N in O(smbm + s2bm) operations such that, for some h0 ∈ H and some constant C > 0 (independent of m, h, s) we have for any h < h0 that ∥Ebm(Θ′, Θ) − Ebm(Θ′ h, Θh)∥Y ≤ Cb−m(∥Θ − Θh∥∞ + ∥Θ′ − Θ′ h∥∞ + ∥Θ − Θh∥s,α,γ + ∥Θ′ − Θ′ h∥s,α,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Throughout the proof, C > 0 is a generic constant independent of m, h, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We compute the difference of the numerators ∆1 := Zm−1Z′ m − ZmZ′ m−1 − Zm−1,hZ′ m,h + Zm,hZ′ m−1,h = −(Zm − Zm−1)(Z′ m − Z′ m,h) − (Zm − Zm,h − Zm−1 + Zm−1,h)Z′ m,h + (Zm − Zm,h)(Z′ m − Z′ m−1) + Zm,h(Z′ m − Z′ m,h − Z′ m−1 + Z′ m−1,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 12 We have |Zm − Zm,h| ≤ ∥Θ − Θh∥∞ and ���Z′ m − Z′ m,h ��� Y ≤ ∥Θ′ − Θ′ h∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' From Theorem 1, we know that Zm = Zm−1 + ∥Θ∥s,α,γ O(b−m) and Z′ m = Z′ m−1 + ∥Θ′∥s,α,γ O(b−m) as m → ∞, with hidden constants in O(·) independent of s, m, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Furthermore Zm − Zm,h − Zm−1 + Zm−1,h = ∥Θ − Θh∥s,α,γ O(b−m), and Z′ m − Z′ m,h − Z′ m−1 + Z′ m−1,h = ∥Θ′ − Θ′ h∥s,α,γ O(b−m) also follow by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Therefore, we have ∥∆1∥Y ≤ Cb−m(∥Θ − Θh∥∞ + ∥Θ′ − Θ′ h∥∞ + ∥Θ − Θh∥s,α,γ + ∥Θ′ − Θ′ h∥s,α,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Next, we define T1 := Zm−1,hZ′ m,h − Zm,hZ′ m−1,h and obtain the estimate ∥T1∥Y ≤ C(∥Θh∥s,α,γ + ∥Θ′ h∥s,α,γ)b−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Moreover, ∆2 := (bZm − Zm−1)Zm − (bZm,h − Zm−1,h)Zm,h = (b(Zm − Zm,h) + (Zm−1,h − Zm−1))Zm + (bZm,h − Zm−1,h)(Zm − Zm,h) gives |∆2| ≤ C ∥Θ − Θh∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Next, we observe that T2 = (bZm,h − Zm−1,h)Zm,h is bounded from below away from 0, for h sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Therefore, for h sufficiently small we apply the elementary inequality ���� T1 + ∆1 T2 + ∆2 − T1 T2 ���� Y ≤ max(∥∆1∥Y , ∥T1∆2∥Y) 1 + |T2| |T2| (|T2| − |∆2|), valid for T1, T2, ∆1, ∆2 ∈ R with |∆2| < |T2|, which is satisfied since |∆2| → 0 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Combining all these observations we obtain the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For either the BIP or the OCP, assume that D ⊆ R2 is a con- vex polygon and that the PDE (5) satisfies (6), (7), f ∈ L2(D) and b′ ∈ ℓp(N), p ∈ (0, 1/2], with b′j = ∥ψj∥W 1,∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Let Xh = P1(Th) ∩ X for a family of shape-regular meshes Th such that h = maxT ∈Th hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then, we can construct polynomial lattices (Pm)m∈N such that the estimator ESTbm,h in (28) satisfies ���� Z′ Z − Z′ m,h Zm,h ���� Y ≤ ESTbm,h + O(b−2m+ε + b−mh), for any ε > 0 as m → ∞, h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The constant in O(·) is independent of s, m and h, but depends on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Since bj ≤ b′ j, from either Corollary 1 or 2, there exist SPOD weights γ′ as in (16) with β′ ∼ b′, such that sups ∥Θ∥s,α,γ′ + ∥Θ′∥s,α,γ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Combining (20) with Lemma 1 and h → 0 we have ζm,h → 0 for the BIP case for any m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Similarly, combining (22) with Lemma 3 yields the same observation for the OCP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Therefore we can apply Propositions 2 and 3 to get that we can construct polynomial lattices so that as m → ∞ ���� Z′ Z − Z′ m,h Zm,h ���� Y ≤ ∥Ebm(Θ′, Θ)∥Y + Zm,hζ′ m,h + ���Z′ m,h ��� Y ζm,h Z2 m,h − ζm,hZm,h + O(b−2m+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Next, we say that ρ ∈ (1, ∞)N is (b′, ε)-admissible if � j≥1(ρj − 1)b′ j ≤ ε, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Then we define Tb′,ε = � ρ,(b′,ε)-adm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='{y ∈ Cs : dist(yj, [− 1 2, 1 2]) ≤ ρj − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Fol- lowing the computations in [2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1], yield h0 ∈ H and ε > 0 sufficiently 13 small such that for all h ≤ h0, h ∈ H, we have supy∈Tb′,ε |Θ(y) − Θh(y)| + ∥Θ′(y) − Θ′ h(y)∥Y ≤ Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' By [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1], this implies that for a constant C independent of h, s, ∥Θ − Θh∥ ∞ + ∥Θ′ − Θ′ h∥∞ + ∥Θ − Θh∥s,α,γ′ + ∥Θ′ − Θ′ h∥s,α,γ′ ≤ Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Thus, (29) and ∥Θ − Θh∥∞ → 0 hold, and we apply Proposition 4 to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 5 Numerical experiment We consider the PDE (5) on the physical domain D := (0, 1)2, with f ≡ 10, and parametric diffusion coefficient given by a(x, y) = 1 2 + 16 � j=1 yj (k2 j,1 + k2 j,2)2 sin(kj,1x1) sin(kj,2x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The pairs (kj,1, kj,2) ∈ N2 are defined by the ordering of N2 such that for j ∈ N, k2 j,1 + k2 j,2 ≤ k2 j+1,1 + k2 j+1,2, and the ordering is arbitrary when equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We investigate a BIP with observation functional O = (O1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' , O4) ∈ (L2(D))4, given by Ok(v) := 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='01 � Ik vdx for v ∈ L2(D) and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' , 4, where I1 := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='2]2, I2 := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='2] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='9], I3 := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='9] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='2], I4 := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='9]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We draw a (random) sample of a to compute the "ground truth" of observa- tions O(S(a)) on a sequence of regular FE meshes of triangles obtained by uniform refinement, with 525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='313 degrees of freedom (dofs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We add ran- dom noise η ∼ N(0, σ2I4) to the observations, where σ is set as 10% of the average of O(S(a)) ∈ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The realized synthetic data is then given by δ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='5205, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='5037, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='5443, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='4609)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Our aim is to approximate Eπδ[G(u)] by the ratio estimator Z′ m,h Zm,h , where G ∈ L2(D) is given by G(v) := 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='5 � [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='25,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='75]2 vdx for v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The FE mesh and the polynomial lattice rule, that eventually determine h and m, are refined successively based on the combined estimator in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' For tolerances τFEM, τQMC > 0, we start from an initial FE mesh of D, that is uniformly refined until the stopping criterion Zm,hζ′ m,h+∥Z′ m,h∥Yζm,h Z2 m,h−ζm,hZm,h ≤ τFEM is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Thereafter, we increase the number bm of lattice points until there holds |Ebm(Θ′ h, Θh)| ≤ τQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We initialize by a FE mesh with 41 dofs and bm0 QMC points with base b = 2 and m0 = 2, and set the tolerances to τFEM = τQMC = 2−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' To assess the total realized error, we compute a reference solution Z′ ref Zref by a multilevel Monte Carlo ratio estimator, see [17], and report absolute error ��� Z′ m,h Zm,h − Z′ ref Zref ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The reference estimator uses 6 refinement levels with 545/525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='313 dofs on the coarsest/finest level, respectively, and uniform (pseudo-) random numbers y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The number of samples is adjusted to balance statistical error and discretization bias on each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The experiment has been implemented in MATLAB using the MooAFEM library [11] for the FE discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' All arising linear systems are solved directly by the \\-operator in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The estimated and realized errors vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' the number of iterations (in the sense of refinement steps) are depicted Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Here, the FE a-posteriori 14 estimator gives negative values on rather coarse meshes, where c∗˜ηy,h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Therefore, we discarded these "pre-asymptotic" values in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' We see that the FE a-posteriori estimator from Proposition 3 is rather conservative at first, but eventually approaches the actual error for finer meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The QMC estimator |Ebm(Θ′ h, Θ′ h)| is of the same magnitude as σ at first, and only two more refinement steps are needed once the FE mesh is sufficiently fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The combined error estimate ESTbm,h aligns well with the realized error, as expected from our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Figure 1: Results for the QMC-FEM ratio estimator with a-posterior ratio refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' First the FE mesh is refined until the tolerance τFEM is achieved (dashed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' circles), then the QMC a-posterior refinement takes place (dashed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The estimated error (solid w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' stars) is conservative for coarse meshes, but eventually approaches the realized error (solid w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 6 Conclusion In this paper we outlined the construction of an a-posteriori QMC-FEM estimator, that allows to quantify the approximation error to a) posterior expectation in Bayesian inverse problems and b) the optimal control under the entropic risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' The estimator is computable and viable for large number of parameters s and it is asymptotically an upper bound for the errors in a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Furthermore, the particular ratio structure Z′ Z of the sought quantities allows to tackle both the BIP and OCP, in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' In either case, we work under the assumption that the underlying model is a parametric elliptic PDE with affine-parametric diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Nevertheless, the present QMC methodology to high-dimensional integration is applicable to non-affine parametric PDEs with quantified, holomorphic-parametric dependence, see [4] and the references there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Since the error estimators we consider ηy,h, ˜ηy,h, ˜˜ηy,h are expressed as sums of local error contributions for T ∈ Th, a possible direction of research is to employ the presently proposed estimators ζy,h, ζ′ y,h to steer an adaptive QMC-FEM algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' 15 total error Z Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='h Total/estimated errors QMC-error est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' *-FEM+QMC-error est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' TFEM/TQMC LC 7 # IterationsReferences [1] R.' metadata={'source': 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Sparsity in Bayesian inversion of parametric operator equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Inverse Problems, 30(6):065007, 30, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Stuart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Inverse problems: a Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Acta Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=', 19:451–559, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Verfürth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' a-posteriori error estimation techniques for finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Numerical Mathematics and Scientific Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Oxford University Press, Oxford, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Wihler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Weighted L2-norm a-posteriori error estimation of FEM in polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfpAWX/content/2301.03327v1.pdf'} +page_content=' J.' metadata={'source': 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