Conditional Stability and Numerical Reconstruction of a Parabolic Inverse Source Problem Using Carleman Estimates

TL;DR

This paper introduces a new numerical method for reconstructing spatially dependent sources in parabolic equations, leveraging Carleman estimates to establish stability and provide error bounds, validated through numerical experiments.

Contribution

The paper develops a novel numerical approach for inverse source problems in parabolic equations, using Carleman estimates to achieve stability and error analysis.

Findings

Established conditional Lipschitz and Hölder stability estimates.

Proposed finite element-based numerical reconstruction method.

Validated effectiveness through numerical experiments.

Abstract

In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder stability for the inverse problem with and without boundary conditions, respectively, using suitable Carleman estimates. Then we propose a numerical approach for solving the inverse problem using conforming finite element approximations in both time and space. Moreover, by utilizing the conditional stability estimates, we prove rigorous error bounds on the discrete approximation. We present several numerical experiments to illustrate the effectiveness of the approach.