Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

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.