Matrix Coefficient Identification in an Elliptic Equation with the Convex Energy Functional Method

TL;DR

This paper addresses the inverse problem of identifying the diffusion matrix in an elliptic PDE using a convex energy functional approach with Tikhonov regularization, discretization, and a gradient-projection algorithm, supported by convergence analysis and numerical experiments.

Contribution

It introduces a novel application of the convex energy functional method with Tikhonov regularization for matrix identification in elliptic PDEs, including convergence proofs and an effective numerical algorithm.

Findings

Proved convergence of discretized solutions.

Established error bounds under source conditions.

Demonstrated the method's effectiveness through numerical experiments.

Abstract

In this paper we study the inverse problem of identifying the diffusion matrix in an elliptic PDE from measurements. The convex energy functional method with Tikhonov regularization is applied to tackle this problem. For the discretization we use the variational discretization concept, where the PDE is discretized with piecewise linear, continuous finite elements. We show the convergence of approximations. Using a suitable source condition, we prove an error bound for discrete solutions. For the numerical solution we propose a gradient-projection algorithm and prove the strong convergence of its iterates to a solution of the identification problem. Finally, we present a numerical experiment which illustrates our theoretical results.