Lipschitz stability and reconstruction in inverse problems for semi-discrete parabolic operators
Rodrigo Lecaros, Juan López-Ríos, Ariel A. Pérez
TL;DR
The paper tackles inverse problems for spatially semi-discrete parabolic operators by developing a novel Carleman estimate whose parameter is constrained by the mesh size $h$, enabling Lipschitz stability results for inverse source and coefficient problems and a convergent Carleman-based reconstruction algorithm. A key contribution is a semi-discrete Carleman inequality that incorporates the second-order spatial term and yields an explicit discretization-related error term that vanishes as $h\to 0$, thus bridging to the continuous theory. The reconstruction framework minimizes a Carleman-weighted functional over a convex trajectory space, with provable existence, uniqueness, and convergence of the iterates for large the Carleman parameter $\tau$, and it explicitly accounts for discretization artifacts. These results provide rigorous foundations for stable identification in semi-discretized parabolic problems and point to natural extensions to boundary observations, discontinuous coefficients, and fully discrete schemes.
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.