Convergent Algorithm Based on Carleman Estimates for the Recovery of a Potential in the Wave Equation

Abstract

This article develops the numerical and theoretical study of the 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 of 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, Comm. Partial Differential Equations, 38 (2013), pp. 823--859]. However, the numerical implementation of this strategy still presents several challenges, which we propose to address here.