A new approach to source conditions in regularization with general residual term

TL;DR

This paper introduces a novel approach combining approximate source conditions and variational inequalities to establish convergence rates for Tikhonov-like regularization methods in Banach and topological spaces, accommodating general residuals and nonsmooth operators.

Contribution

It presents a new unified framework for proving convergence rates in regularization that handles non-convex residuals and nonsmooth operators, extending existing results.

Findings

Provides a wide range of convergence rates.

Handles general, not necessarily convex residual terms.

Applicable to Banach and Hilbert space settings.

Abstract

This paper addresses Tikhonov like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates which combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results.