Recovering point sources for the inhomogeneous Helmholtz equation

TL;DR

This paper introduces a new stability estimate for recovering point sources within an inhomogeneous medium governed by the Helmholtz equation, using boundary measurements and assuming well-separated sources.

Contribution

It provides a novel Hölder stability estimate for inverse point source problems in inhomogeneous media, advancing the theoretical understanding of source recovery.

Findings

Established a Hölder stability estimate for source recovery

Proved uniqueness of the inverse problem under separation assumptions

Demonstrated the effectiveness of Carleman estimates in this context

Abstract

The paper is concerned with an inverse point source problem for the Helmholtz equation. It consists of recovering the locations and amplitudes of a finite number of radiative point sources inside a given inhomogeneous medium from the knowledge of a single boundary measurement. The main result of the paper is a new H\"{o}lder type stability estimate for the inversion under the assumption that the point sources are well separated. The proof of the stability is based on a combination of Carleman estimates and a technique for proving uniqueness of the Cauchy problem.