Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

TL;DR

This paper investigates the iterated quasi-reversibility method as a regularization technique for ill-posed elliptic and parabolic data completion problems, demonstrating its convergence and robustness through numerical experiments.

Contribution

It introduces an abstract framework for applying the method to both elliptic and parabolic problems and proposes a noise-handling strategy, showing effectiveness in practical scenarios.

Findings

Method converges to exact solution

Effective with highly corrupted data

Applicable to diverse elliptic and parabolic problems

Abstract

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data.