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arXiv · 1809.08334

Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

Laurent Baratchart·Cristobal Villalobos Guillen·Douglas P. Hardin·Michael C. Northington·Edward B. Saff
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Abstract

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.

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Laurent Baratchart, Cristobal Villalobos Guillen, Douglas P. Hardin, Michael C. Northington, Edward B. Saff. 2018-09-21. Inverse Potential Problems for Divergence of Measures with Total Variation Regularization. https://arxiv.org/abs/1809.08334

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