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

Abstract

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.

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

“…Recently, the authors have developed a method for the verification of variational source conditions in Hilbert spaces under standard smoothness conditions. This method has been successfully applied to a number of interesting inverse problems, both linear and nonlinear [25,26,35,40].…”

Convergence analysis of (statistical) inverse problems under conditional stability estimates

“…as for all f ∈ D (F ) the problem (56a)-(56c) admits a unique solution. For this problem, it has been shown in [23] holds true for any 0 < θ < 1. This also implies the conditional stability estimate (7) with −a = m and ϕ as in the above formula.…”

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

“…Variational source conditions can be obtained from classical and general source conditions for linear problems in Hilbert spaces (see [4, Section 3.2], [19], [5,Chapter 13]), as well as from problem specific calculations for certain nonlinear problems and Banach space settings (see [3,7,11,14,18]). Details on variational source conditions can be found in, e. g., [2,5,9,16].…”