On a level-set method for ill-posed problems with piecewise non-constant coefficients

TL;DR

This paper develops a level-set regularization method for solving ill-posed inverse problems with piecewise non-constant solutions, proving convergence, stability, and applicability in elliptic PDE models.

Contribution

It introduces a novel Tikhonov-type regularization framework combined with level-set techniques for non-constant piecewise solutions in inverse problems.

Findings

Proves existence of generalized minimizers for the proposed functional.

Establishes convergence and stability of solutions with respect to noise.

Demonstrates applicability in elliptic PDE inverse problems.

Abstract

We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem we propose a Tikhonov-type regularization approach coupled with a level set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability of the regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level set method in some interesting inverse problems arising in elliptic PDE models. Keywords: Level Set Methods, Regularization, Ill-Posed Problems, Piecewise Non-Constant Coefficients