Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces

TL;DR

This paper establishes optimal convergence rates for wavelet-based sparsity-promoting regularization methods in Besov spaces, applicable to various inverse problems including Radon transform and nonlinear differential equations.

Contribution

It introduces a framework demonstrating minimax-optimal convergence rates for wavelet Besov norm penalties in both deterministic and statistical noise settings.

Findings

Achieves minimax-optimal convergence rates for finitely smoothing operators.

Applicable to linear and nonlinear inverse problems with wavelet Besov penalties.

Includes analysis for Radon transform and differential equation inverse problems.

Abstract

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We focus on $B^0_{p,1}$ penalty terms which yield estimators that are sparse with respect to a wavelet frame. Our framework includes among others, the Radon transform and some nonlinear inverse problems in differential equations with distributed measurements. Using variational source conditions it is shown that such estimators achieve minimax-optimal rates of convergence for finitely smoothing operators in certain Besov balls both for deterministic and for statistical noise models.