Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

Abstract

We study inverse problems for the Poisson equation with source term the divergence of an $${{\mathbb {R}}}^3$$ -valued measure, that is, the potential $$\varPhi $$ satisfies $$\begin{aligned} \Delta \varPhi = \nabla \cdot {{\varvec{\mu }}}, \end{aligned}$$ and $${{\varvec{\mu }}}$$ is to be reconstructed knowing (a component of) the field $$\, {\mathrm{grad}}\,\varPhi $$ on a set disjoint from the support of $${{\varvec{\mu }}}$$ . 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 investigate methods for recovering $${{\varvec{\mu }}}$$ by penalizing the measure theoretic total variation norm $$\Vert {{\varvec{\mu }}}\Vert _{\mathrm{TV}}$$ . We provide sufficient conditions for the unique recovery of $${{\varvec{\mu }}}$$ , 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.