There always is a variational source condition for nonlinear problems in Banach spaces

TL;DR

This paper demonstrates that variational source conditions are universally applicable to almost all ill-posed inverse problems across various spaces, providing a broad foundation for convergence rate analysis in regularization methods.

Contribution

It proves that variational source conditions are satisfied for nearly all inverse problems, regardless of linearity, space type, or solution multiplicity, establishing their universal applicability.

Findings

Variational source conditions hold for most ill-posed inverse problems.

They are valid in both Hilbert and Banach spaces.

Applicable to linear and nonlinear problems, with single or multiple solutions.

Abstract

Variational source conditions proved useful for deriving convergence rates for Tikhonov's regularization method and also for other methods. Up to now such conditions have been verified only for few examples or for situations which can be handled by classical techniques, too. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.