Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

TL;DR

This paper establishes stability estimates for reconstructing point sources in parabolic equations from boundary data, using advanced analytical techniques and providing numerical validation.

Contribution

It introduces a novel approach combining Carleman estimates and regularity analysis to derive stability estimates for source reconstruction in parabolic problems.

Findings

Derived stability estimates for source location and amplitude reconstruction.

Applied Carleman estimates and regularity results in the analysis.

Provided numerical reconstructions supporting theoretical results.

Abstract

In this work, we investigate the stability issue of the inverse problem of determining the locations and time-dependent amplitudes of point sources in a parabolic equation with a non-self adjoint elliptic operator from boundary observations. We derive different stability estimates for determining the locations and the amplitudes of the sources in the space, the plane as well as in dimension one. The analysis employs a novel approach that combines several different arguments, including the improved regularity of the solutions, the application of Carleman estimates, time extension of solutions, and construction of explicit solutions to the adjoint equations. Further we provide numerical reconstructions to complement the theoretical findings.