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Foundations of Computational Mathematics

Erschienen in:

07.11.2019

Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

verfasst von: L. Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington, E. B. Saff

Erschienen in: Foundations of Computational Mathematics | Ausgabe 5/2020

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Abstract

We study inverse problems for the Poisson equation with source term the divergence of an ${{\mathbb {R}}}^3$-valued measure, that is, the potential $\varPhi $ satisfies

$$\begin{aligned} \Delta \varPhi = \nabla \cdot {{\varvec{\mu }}}, \end{aligned}$$

and ${{\varvec{\mu }}}$ is to be reconstructed knowing (a component of) the field $\, {\mathrm{grad}}\,\varPhi $ on a set disjoint from the support of ${{\varvec{\mu }}}$. 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 investigate methods for recovering ${{\varvec{\mu }}}$ by penalizing the measure theoretic total variation norm $\Vert {{\varvec{\mu }}}\Vert _{\mathrm{TV}}$. We provide sufficient conditions for the unique recovery of ${{\varvec{\mu }}}$, 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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Titel: Inverse Potential Problems for Divergence of Measures with Total Variation Regularization
verfasst von: L. Baratchart
C. Villalobos Guillén
D. P. Hardin
M. C. Northington
E. B. Saff
Publikationsdatum: 07.11.2019
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 5/2020
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-019-09443-x

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