Linearizing non-linear inverse problems and an application to inverse backscattering

TL;DR

This paper introduces a linearization-based framework to establish local uniqueness and stability in non-linear inverse problems, demonstrated through an application to acoustic inverse backscattering.

Contribution

It presents a novel abstract approach that combines linearization with stability estimates, applicable to inverse problems like acoustic backscattering.

Findings

Proves local uniqueness and H"older stability for the inverse backscattering problem.

Shows that ellipticity of $A^*A$ ensures the stability condition.

Applies the method to acoustic equations near constant sound speed.

Abstract

We propose an abstract approach to prove local uniqueness and conditional H\"older stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization $A$, we need a stability estimate for $A$ as well. That condition is satisfied in particular, if $A^*A$ is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and H\"older stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.