# Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

Abstract : We study inverse problems for the Poisson equation with source term thedivergence of an $R3-valued$ measure; that is, the potential $Φ$ satisfies $∆Φ = divμ$,and $μ$ is to be reconstructed knowing (a component of) the field grad $Φ$ on a set dis-joint from the support of $μ$. Such problems arise in several electro-magnetic contextsin the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering $μ$ based on totalvariation regularization. We provide sufficient conditions for the unique recovery of $μ$, 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 whichis sparse in the sense that it is purely 1-unrectifiable.Numerical examples are provided to illustrate the main theoretical results.
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Journal articles

https://hal.inria.fr/hal-02424672
Contributor : Laurent Baratchart <>
Submitted on : Saturday, December 28, 2019 - 12:56:53 AM
Last modification on : Monday, January 13, 2020 - 1:12:32 AM

### Citation

Laurent Baratchart, Cristobal Villalobos Guillén, Doug Hardin, Michael Northington, Edward Saff. Inverse Potential Problems for Divergence of Measures with Total Variation Regularization. Foundations of Computational Mathematics, Springer Verlag, 2019, ⟨10.1007/s10208-019-09443-x⟩. ⟨hal-02424672⟩

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