Verification of a variational source condition for acoustic inverse medium scattering problems

TL;DR

This paper provides the first rigorous proof of convergence rates for Tikhonov regularization in acoustic inverse medium scattering, linking stability estimates with regularization theory under smoothness assumptions.

Contribution

It introduces a novel proof of convergence rates for Tikhonov regularization in inverse scattering, combining stability estimates and variational regularization techniques.

Findings

Established logarithmic convergence rates for Tikhonov regularization.

Connected stability estimates with regularization theory for acoustic inverse problems.

Provided a rigorous mathematical foundation for regularization in medium scattering.

Abstract

This paper is concerned with the classical inverse scattering problem to recover the refractive index of a medium given near or far field measurements of scattered time-harmonic acoustic waves. It contains the first rigorous proof of (logarithmic) rates of convergence for Tikhonov regularization under Sobolev smoothness assumptions for the refractive index. This is achieved by combining two lines of research, conditional stability estimates via geometrical optics solutions and variational regularization theory.