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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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Contributor : Laurent Baratchart <>
Submitted on : Saturday, December 28, 2019 - 12:56:53 AM
Last modification on : Tuesday, January 19, 2021 - 6:16:01 PM

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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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