New Stability Estimates for the Inverse Medium Problem with Internal Data

TL;DR

This paper develops new weighted stability estimates for an electro-acoustic inverse problem, addressing the challenge of critical points where data vanishes, and shows stability degrades near these points.

Contribution

It introduces stability estimates that do not rely on avoiding critical points, unlike previous results that required boundary conditions close to CGO solutions.

Findings

Stability deteriorates to logarithmic near critical points.

New estimates apply without assumptions on critical point locations.

Results improve understanding of inverse problem stability in complex scenarios.

Abstract

A major problem in solving multi-waves inverse problems is the presence of critical points where the collected data completely vanishes. The set of these critical points depend on the choice of the boundary conditions, and can be directly determined from the data itself. To our knowledge, in the most existing stability results, the boundary conditions are assumed to be close to a set of CGO solutions where the critical points can be avoided. We establish in the present work new weighted stability estimates for an electro-acoustic inverse problem without assumptions on the presence of critical points. These results show that the Lipschitz stability far from the critical points deteriorates near these points to a logarithmic stability.