Lipschitz stability and reconstruction in inverse problems for semi-discrete parabolic operators

Abstract

This work addresses an inverse problem for a semi-discrete parabolic equation, consisting of identifying the right-hand side of the equation from solution measurements at an intermediate time and within a spatial subdomain. We apply this result to establish a stability estimate for a coefficient inverse problem involving the recovery of a spatially dependent potential function. Furthermore, we present a reconstruction algorithm for recovering this coefficient and provide a proof of its convergence. Our approach relies on a novel semi-discrete Carleman estimate in which the parameter is constrained by the mesh size. Due to the discrete terms arising in the Carleman inequality, this method naturally introduces an error term associated with the solution's initial condition.