Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

TL;DR

This paper addresses inverse problems involving the Poisson equation with measure-valued sources, proposing total variation regularization methods for unique recovery under specific conditions, with theoretical analysis and numerical validation.

Contribution

It introduces novel total variation regularization techniques for reconstructing divergence of measures in inverse Poisson problems, with conditions for uniqueness and stability.

Findings

Unique recovery conditions for divergence of measures.

Asymptotic stability when regularization and noise vanish.

Numerical examples confirming theoretical results.

Abstract

We study inverse problems for the Poisson equation with source term the divergence of an $\mathbf{R}^3$-valued measure, that is, the potential $\Phi$ satisfies $$ \Delta \Phi= \text{div} \boldsymbol{\mu}, $$ and $\boldsymbol{\mu}$ is to be reconstructed knowing (a component of) the field grad $\Phi$ on a set disjoint from the support of $\boldsymbol{\mu}$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering $\boldsymbol{\mu}$ based on total variation regularization. We provide sufficient conditions for the unique recovery of $\boldsymbol{\mu}$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.