The "exterior approach" applied to the inverse obstacle problem for the heat equation

TL;DR

This paper introduces an iterative exterior approach combining quasi-reversibility and level set methods to solve the inverse obstacle problem for the heat equation, demonstrating its feasibility through numerical experiments.

Contribution

It develops a novel iterative method integrating quasi-reversibility and level set techniques for obstacle identification in heat equations.

Findings

Feasibility demonstrated through numerical experiments.

Effective handling of noisy Cauchy data.

Use of classical finite elements in the approach.

Abstract

In this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification.