Lavrentiev's regularization method in Hilbert spaces revisited

TL;DR

This paper revisits Lavrentiev's regularization method for nonlinear ill-posed problems in Hilbert spaces, providing new convergence rate results based on variational and approximate source conditions.

Contribution

It introduces novel variational source conditions and explores their implications for convergence rates in Lavrentiev's regularization within Hilbert spaces.

Findings

Established new convergence rate results under variational source conditions.

Developed and analyzed approximate source conditions for Lavrentiev's method.

Provided interpretations of convergence results in specific problem settings.

Abstract

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.