Variational source conditions in Lp-spaces

TL;DR

This paper develops and validates variational source conditions for Tikhonov regularization with Lp-norm penalties in Banach spaces, providing convergence rates and applying the theory to inverse elliptic problems with measure data.

Contribution

It introduces a new analytical framework using Littlewood-Paley theory and R-boundedness for VSC in Lp-spaces, extending regularization theory to inverse problems with measure data.

Findings

Validated VSC under stability and regularity conditions.

Derived convergence rates for Tikhonov regularization in Lp-spaces.

Applied theory to inverse elliptic problems with measure data.

Abstract

We propose and analyze variational source conditions (VSC) for the Tikhonov regularization method with Lp-norm penalties for a general ill-posed operator equation in a Banach space. Our analysis is based on the use of the celebrated Littlewood-Paley theory and the concept of (Rademacher) R-boundedness. On the basis of these two analytical tools, we validate the proposed VSC under a conditional stability estimate and a regularity requirement of the true solution in terms of Triebel-Lizorkin-type spaces. In the final part of the paper, the developed theory is applied to an inverse elliptic problem with measure data for the reconstruction of possibly unbounded diffusion coefficients in the Lp-setting. By means of VSC, convergence rates for the associated Tikhonov regularization with Lp-norm penalties are obtained.