Variational Regularization Theory Based on Image Space Approximation Rates

TL;DR

This paper introduces a new convergence rate analysis for variational regularization using image space approximation rates, avoiding Bregman distances, and applies it to Besov space regularization with broad penalty parameters.

Contribution

It develops a nearly minimax theorem linking the modulus of continuity to reconstruction error, and establishes equivalence of source conditions and approximation rates.

Findings

Derived convergence rates for Besov space regularization with various penalties.

Proved the equivalence of H"older-type source conditions and approximation rates.

Established bounds on the Tikhonov functional defect.

Abstract

We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of continuity is an upper bound on the reconstruction error up to a constant. Applied to Besov space regularization we obtain convergence rate results for $0,2,q$- and $0,p,p$-penalties without restrictions on $p,q\in (1,\infty).$ Finally we prove equivalence of H\"older-type variational source conditions, bounds on the defect of the Tikhonov functional, and image space approximation rates.