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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 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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Contributor : Laurent Baratchart <>
Submitted on : Tuesday, September 25, 2018 - 1:41:30 AM
Last modification on : Thursday, January 21, 2021 - 2:44:02 PM
Long-term archiving on: : Wednesday, December 26, 2018 - 12:31:05 PM


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  • HAL Id : hal-01880506, version 1



Laurent Baratchart, Cristobal 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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