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首页> 外文期刊>Jahresbericht der Deutschen Mathematiker-Vereinigung >An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs
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An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs

机译：椭圆PDE中逆系数问题有限元方法的介绍

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Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the unknown coefficient is often assumed to be piecewise constant on a given pixel partition (corresponding to the desired resolution), and only finitely many measurement can be made. This leads to the problem of inverting a finite-dimensional non-linear forward operator F:D(F)⊆Rn→Rmdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathcal{F}: mathcal{D}(mathcal{F})subseteq mathbb{R}^{n}o mathbb{R}^{m}$end{document}, where evaluating ℱ requires one or several PDE solutions.Numerical inversion methods require the implementation of this forward operator and its Jacobian. We show how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their true-solution counterparts. We present simple example codes for Comsol with the Matlab Livelink package, and numerically demonstrate the challenges that arise from non-uniqueness, non-linearity and instability issues. We also discuss monotonicity and convexity properties of the forward operator that arise for symmetric measurement settings.This text assumes the reader to have a basic knowledge on Finite Element Methods, including the variational formulation of elliptic PDEs, the Lax-Milgram-theorem, and the Céa-Lemma. Section 3 also assumes that the reader is familiar with the concept of Fréchet differentiability.
机译：几种新型成像和非破坏性测试技术基于从其解决方案的测量重建椭圆局部微分方程中的空间相关系数。在实际应用中，通常假设在给定的像素分区（对应于期望的分辨率）上是分段常数的未知系数，并且仅可以进行有限许多测量。这导致反转有限维非线性前向操作员F：D（f）⊆rn→rm documentclass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsideDemargin} {-69pt} begin {document} $ mathcal {f}： mathcal {d}（ mathcal {f}） subseteq mathbb {r} ^ {n} to mathbb {r} ^ $ end {document}，其中评估ℱ需要一个或多个PDE解决方案.Numerical反演方法需要实现这个前进的运营商及其雅各比亚。我们展示了如何使用标准的FEM包来有效地实现，并证明对其真实解决方案对应的有效近似的收敛。我们向Matlab Livelink包提供了COMSOL的简单示例代码，数值上展示了来自非唯一性，非线性和不稳定问题的挑战。我们还讨论出现对称测量设置的前向操作员的单调性和凸性属性。该文本假定读者对有限元方法具有基本知识，包括椭圆PDE，LAX-MILGRAM-定理的变分制剂和Céa-lemma。第3节还假设读者熟悉Fréchet可怜的概念。

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