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 R-3-valued measure, that is, the potential Phi satisfies Delta Phi = del . mu, and mu is to be reconstructed knowing (a component of) the field grad Phi on a set disjoint from the support of 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 mu by penalizing the measure theoretic total variation norm parallel to mu parallel to(TV). We provide sufficient conditions for the unique recovery of 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.