Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

TL;DR

This paper establishes optimal convergence rates for Tikhonov regularization in Besov spaces, applicable to linear and nonlinear ill-posed problems, including the backward heat equation and noisy models, using variational source conditions.

Contribution

It provides the first comprehensive analysis of convergence rates for Tikhonov regularization in Besov spaces, extending results to nonlinear problems and noisy data.

Findings

Order optimal convergence rates for finitely smoothing operators.

Optimal rates for the backward heat equation in Besov spaces.

Convergence results for white noise models using variational source conditions.

Abstract

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.