Converse results, saturation and quasi-optimality for Lavrentiev regularization of accretive problems

TL;DR

This paper investigates Lavrentiev regularization for linear ill-posed problems involving accretive operators, providing new theoretical insights into converse, saturation, and quasi-optimality results, mainly in Banach and Hilbert spaces.

Contribution

It introduces novel converse and saturation results for Lavrentiev regularization and establishes a new quasi-optimality result for a posteriori parameter choices.

Findings

Converse results clarify limits of regularization effectiveness.

Saturation results identify conditions where regularization cannot improve accuracy.

A new quasi-optimality theorem supports better parameter selection methods.

Abstract

This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible.