A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

TL;DR

This paper establishes optimal convergence rate bounds for Tikhonov regularization in Banach spaces, extending previous Hilbert space results to more general settings without spectral theory.

Contribution

It provides a converse result showing that the derived error bounds are not only upper bounds but also asymptotic lower bounds in Banach spaces.

Findings

Optimal bounds for Bregman distances in Banach spaces.

Extension of converse results from Hilbert to Banach spaces.

Error bounds are tight and asymptotically optimal.

Abstract

We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.