Penalty-based smoothness conditions in convex variational regularization

TL;DR

This paper introduces a new framework for analyzing Tikhonov regularization in Banach spaces using penalty-based inequalities, providing error bounds that separate smoothness and noise effects, applicable to various convex penalties.

Contribution

It develops a novel penalty-based smoothness condition framework for convex Tikhonov regularization, extending error analysis beyond traditional smoothness assumptions.

Findings

Error bounds split into noise-independent and noise-dependent terms

Variational inequalities imply the proposed smoothness conditions

Applicable to specific examples in convex regularization

Abstract

The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.