Convergence of an inverse problem for discrete wave equations

TL;DR

This paper proves that discrete inverse problems for 1D wave equations converge to the continuous potential as the discretization becomes finer, using a novel uniform discrete Carleman estimate.

Contribution

It introduces a new uniform discrete Carleman estimate and applies a Lax-type argument to establish convergence of discrete inverse problems for wave equations.

Findings

Discrete fluxes converge to continuous flux in the inverse problem

Uniform stability is established via new discrete Carleman estimates

Convergence result proven for 1D semi-discrete wave equations

Abstract

It is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are interested in proving that trying to fit the discrete fluxes, given by discrete approximations of the wave equation, with the continuous one, one recovers, at the limit, the potential of the continuous model. In order to do that, we shall develop a Lax-type argument, usually used for convergence results of numerical schemes, which states that consistency and uniform stability imply convergence. In our case, the most difficult part of the analysis is the one corresponding to the uniform stability, that we shall prove using new uniform discrete Carleman estimates, where uniform means with respect to the discretization parameter. We shall then deduce a convergence result for the discrete inverse problems. Our analysis will be restricted to the 1-d case for space semi-discrete wave equations discretized on a uniform mesh using a finite differences approach.