Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation

TL;DR

This paper develops a numerical and theoretical reconstruction algorithm for identifying a potential in a wave equation using boundary measurements and Carleman estimates, addressing implementation challenges of the method.

Contribution

It introduces a practical, globally convergent algorithm based on Carleman estimates for the wave equation's inverse problem, with focus on numerical implementation.

Findings

Algorithm successfully reconstructs the potential from boundary data.

Addresses numerical challenges in implementing Carleman-based methods.

Provides theoretical guarantees of convergence.

Abstract

This article develops the numerical and theoretical study of a reconstruction algorithm of a potential in a wave equation from boundary measurements, using a cost functional built on weighted energy terms coming from a Carleman estimate. More precisely, this inverse problem for the wave equation consists in the determination of an unknown time-independent potential from a single measurement of the Neumann derivative of the solution on a part of the boundary. While its uniqueness and stability properties are already well known and studied, a constructive and globally convergent algorithm based on Carleman estimates for the wave operator was recently proposed in [L. Baudouin, M. de Buhan and S. Ervedoza, Global carleman estimates for waves and applications, Comm. Partial Differential Equations 38 (2013), no. 5]. However, the numerical implementation of this strategy still presents several challenges, that we propose to address here.