On Tikhonov functionals penalized by Bregman distances

TL;DR

This paper studies Tikhonov regularization with Bregman distance penalties for nonlinear ill-posed problems in Banach spaces, establishing convergence, stability, and convergence rates, and analyzing an iterated method with convex penalties.

Contribution

It introduces convergence and stability results for Tikhonov regularization with Bregman penalties and derives convergence rates under source conditions, also analyzing an iterated approach.

Findings

Proved convergence and stability of Tikhonov methods with Bregman penalties.

Derived convergence rates using source conditions.

Analyzed an iterated Tikhonov method with convex penalization.

Abstract

We investigate Tikhonov regularization methods for nonlinear ill-posed problems in Banach spaces, where the penalty term is described by Bregman distances. We prove convergence and stability results. Moreover, using appropriate source conditions, we are able to derive rates of convergence in terms of Bregman distances. We also analyze an iterated Tikhonov method for nonlinear problems, where the penalization is given by an appropriate convex functional.