Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

TL;DR

This paper establishes optimal convergence rates for regularisation methods solving linear ill-posed problems in Hilbert spaces, extending existing results to broader source conditions including logarithmic and variational types.

Contribution

It generalizes convergence rate optimality to encompass a wider class of source conditions, linking variational and approximative source conditions.

Findings

Proves optimal convergence rates for logarithmic source conditions

Extends optimality results to variational source conditions

Connects variational and approximative source conditions

Abstract

In this paper, we prove optimal convergence rates results for regularisation methods for solving linear ill-posed operator equations in Hilbert spaces. The result generalises existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.