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 $Φ$ satisfies $$ ΔΦ= \text{div} \boldsymbolμ, $$ and $\boldsymbolμ$ is to be reconstructed knowing (a component of) the field grad $Φ$ on a set disjoint from the support of $\boldsymbolμ$. 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 $\boldsymbolμ$ based on total variation regularization. We provide sufficient conditions for the unique recovery of $\boldsymbolμ$, 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.