Tikhonov regularization in Hilbert scales under conditional stability assumptions

TL;DR

This paper investigates the use of Tikhonov regularization in Hilbert scales for solving nonlinear inverse problems with conditional stability, establishing optimal convergence rates and exploring the role of source conditions.

Contribution

It provides order optimal convergence rates for Tikhonov regularization under conditional stability in Hilbert scales, including analysis of source conditions and numerical validation.

Findings

Optimal convergence rates are achieved for a-priori and a-posteriori strategies.

The role of hidden source conditions is clarified.

Numerical tests confirm theoretical results.

Abstract

Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a-priori and a-posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.