Convergence of an Inverse Problem for a 1-D Discrete Wave Equation

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 when trying to fit the discrete flux, given by discrete approximations of the wave equation, with the continuous flux, 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 (one-dimensional) case for space semidiscrete wave equations discretized on a uniform mesh using a finite-difference approach.