Inverse Potential Problems for Divergence of Measures with Total Variation Regularization
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Inverse Potential Problems for Divergence of Measures with Total Variation Regularization L. Baratchart1 · C. Villalobos Guillén2 · D. P. Hardin2 · M. C. Northington3 · E. B. Saff2 Received: 14 December 2018 / Revised: 1 August 2019 / Accepted: 9 October 2019 © SFoCM 2019
Abstract We study inverse problems for the Poisson equation with source term the divergence of an R3 -valued measure, that is, the potential Φ satisfies Φ = ∇ · μ, and μ is to be reconstructed knowing (a component of) the field grad Φ on a set disjoint from the support of μ. 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 μ by penalizing the measure theoretic total variation norm μTV . 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. Keywords Divergence free · Distributions · Solenoidal · Total variation of measures · Magnetization · Inverse problems · Purely 1-unrectifiable · Sparse recovery · Total variation regularization Mathematics Subject Classification 31B20 · 49N45 · 49Q20 · 86A22
Communicated by Endre Süli. This research was supported, in part, by the U. S. National Science Foundation under Grant DMS-1521749, and by the INRIA grant to the associate team FACTAS.
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L. Baratchart [email protected]
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Foundations of Computational Mathematics
1 Introduction This work is concerned with inverse potential problems with source term in divergence form. That is, an R3 -valued measure on R3 has to be recovered knowing (one component of) the field of the Newton potential of its divergence on a piece of surface, away from the support. Such issues typically arise in source identification from field measurements for Maxwell’s equations, in the quasi-static regime. They occur for instance in electro-encephalography (EEG), magneto-encephalography (MEG), geomagnetism and paleomagnetism, as well as in several non-destructive testing problems, e.g., see [3,4,12,35,38] and their bibliographies. A model problem of our particular interest is inverse scanning magnetic microscopy, as considered for instance in [5,7,36] to recover magnetization distributions of thin rock samples, but the considerations below are of a more general and abstract nature. Our main objective is to introduce notions of sparsity that help recovery in this infinite-dimensional context, when regularization is performed by penalizing the total variation of the measure. Since the term “total variation regularization” also often refers to penalization of the L 1 -norm of the gradient of an unknown function, we emphasize tha
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