Modern Regularization Methods for Inverse Problems

TL;DR

This paper reviews the evolution of regularization methods for inverse problems, emphasizing modern nonlinear approaches like variational methods, and discusses their analysis, applications, and future research directions.

Contribution

It provides a comprehensive overview of recent developments in nonlinear regularization techniques, including variational methods and their connections to other fields.

Findings

Highlighting the shift from linear to nonlinear regularization methods.

Discussion of variational methods and their applications.

Identification of future research challenges in the field.

Abstract

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research. In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.