Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

• Published in: Foundations of computational mathematics 2020-10, Vol.20 (5), p.1273-1307
• Main Authors: Baratchart, L., Villalobos Guillén, C., Hardin, D. P., Northington, M. C., Saff, E. B.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Computer Science; Divergence; Economics; Inverse problems; Linear and Multilinear Algebras; Magnetic fields; Magnetization; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Measurement; Numerical Analysis; Optimization and Control; Poisson equation; Regularization
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-019-09443-x

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 Δ 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 ‖ μ ‖ TV . 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.