Convergence results and low order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

TL;DR

This paper investigates convergence and low order rates for nonlinear Tikhonov regularization with oversmoothing penalties, providing new theoretical results without requiring additional smoothness assumptions on solutions.

Contribution

It extends the theory of nonlinear Tikhonov regularization to include oversmoothing penalties, establishing convergence and low order rates without source conditions.

Findings

Provided sufficient conditions for convergence with various regularization parameter strategies.

Proved low order convergence rates under logarithmic source conditions.

Included numerical illustrations demonstrating theoretical results.

Abstract

For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where H\"older-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies, without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.