Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators

TL;DR

This paper introduces a novel level set regularization method for ill-posed operator equations involving discontinuous operators, capable of handling topology changes, with demonstrated effectiveness in inverse source problems.

Contribution

It develops a new regularization approach with a functional analytic framework for problems with discontinuous operators, including a novel concept of minimizers.

Findings

The method effectively handles changing topologies.

Numerical implementation demonstrates good performance.

Application to inverse source problems shows promising results.

Abstract

We consider a regularization concept for the solution of ill--posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a novel concept of minimizers. The proposed level set regularization is capable of handling changing topologies. A functional analytic framework explaining the splitting of topologies is given. The asymptotic limit of the level set regularization method is an evolution process, which is implemented numerically and the quality of the proposed algorithm is demonstrated by solving an inverse source problem.