An introduction to finite element methods for inverse coefficient problems in elliptic PDEs

TL;DR

This paper introduces finite element methods for solving inverse problems in elliptic PDEs, focusing on reconstructing spatial coefficients from limited measurements, and discusses implementation, convergence, and challenges such as non-uniqueness and non-linearity.

Contribution

It provides an efficient FEM-based approach for inverse coefficient problems, including implementation details, convergence proofs, and analysis of properties like monotonicity and convexity.

Findings

FEM implementations can accurately approximate the forward operator and its Jacobian.

Numerical examples highlight challenges like non-uniqueness and instability.

Monotonicity and convexity properties are identified for symmetric measurement settings.

Abstract

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 measurements can be made. This leads to the problem of inverting a finite-dimensional non-linear forward operator $\mathcal F:\ \mathcal D(\mathcal F)\subseteq \mathbb R^n\to \mathbb R^m$, where evaluating $\mathcal F$ 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.