Stable determination of an anisotropic inclusion in the Schr\"odinger equation from local Cauchy data

TL;DR

This paper establishes a logarithmic stability estimate for identifying an anisotropic inclusion within a body governed by a Schrödinger equation using local Cauchy data, with implications for inverse problems in inhomogeneous media.

Contribution

It provides the first stability estimate for anisotropic inclusions in Schrödinger equations based on local Cauchy data, including practical misfit functional bounds.

Findings

Logarithmic stability estimate derived for the inverse problem.

Stability estimate expressed in terms of local Cauchy data.

Additional stability bounds provided using a misfit functional.

Abstract

We consider the inverse problem of determining an inclusion contained in a body for a Schr\"odinger type equation by means of local Cauchy data. Both the body and the inclusion are made by inhomogeneous and anisotropic materials. Under mild a priori assumptions on the unknown inclusion, we establish a logarithmic stability estimate in terms of the local Cauchy data. In view of possible applications, we also provide a stability estimate in terms of an ad-hoc misfit functional.