# 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 R 3-valued measure; that is, the potential $Φ$ satisfies $∆Φ = div µ$, 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 electromagnetic contexts in 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 total variation 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 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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Cited literature [42 references]

https://hal.inria.fr/hal-01880506
Contributor : Laurent Baratchart <>
Submitted on : Tuesday, September 25, 2018 - 1:41:30 AM
Last modification on : Thursday, August 22, 2019 - 2:44:01 PM
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• HAL Id : hal-01880506, version 1

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Laurent Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington, E. B. Saff. Inverse Potential Problems for Divergence of Measures with Total Variation Regularization. 2018. ⟨hal-01880506⟩

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