Bregman Distances in Inverse Problems and Partial Differential Equation

TL;DR

This paper reviews recent advances in the application of Bregman distances to inverse problems, image processing, and nonlinear PDEs, highlighting their growing importance and potential for further research.

Contribution

It provides an overview of Bregman distances outside optimization, emphasizing their role in inverse problems, image processing, and PDE analysis, and discusses new mathematical questions.

Findings

Bregman distances are now standard in inverse problems and image processing.

They reveal hidden structures in nonlinear PDEs with variational formulations.

The paper identifies new mathematical questions related to Bregman distances.

Abstract

The aim of this paper is to provide an overview of recent development related to Bregman distances outside its native areas of optimization and statistics. We discuss approaches in inverse problems and image processing based on Bregman distances, which have evolved to a standard tool in these fields in the last decade. Moreover, we discuss related issues in the analysis and numerical analysis of nonlinear partial differential equations with a variational structure. For such problems Bregman distances appear to be of similar importance, but are currently used only in a quite hidden fashion. We try to work out explicitely the aspects related to Bregman distances, which also lead to novel mathematical questions and may also stimulate further research in these areas.