Variational Inequalities and Improved Convergence Rates for Tikhonov Regularisation on Banach Spaces

TL;DR

This paper develops higher order convergence rates for Tikhonov regularisation in Banach spaces using variational inequalities on dual functionals, extending known results from Hilbert spaces to more general settings.

Contribution

It introduces a novel approach based on dual variational inequalities to achieve improved convergence rates for convex regularisation in Banach spaces.

Findings

Higher order convergence rates derived in terms of Bregman distance.

Approach extends convergence rate results from Hilbert to Banach spaces.

Method not limited to low-order rates, applicable to a broad range.

Abstract

In this paper we derive higher order convergence rates in terms of the Bregman distance for Tikhonov like convex regularisation for linear operator equations on Banach spaces. The approach is based on the idea of variational inequalities, which are, however, not imposed on the original Tikhonov functional, but rather on a dual functional. Because of that, the approach is not limited to convergence rates of lower order, but yields the same range of rates that is well known for quadratic regularisation on Hilbert spaces.