2018/09/21 by Laurent Baratchart, Baratchart, Laurent, Cristóbal Villalobos Guillén +7
Engineering · Mathematics · #49 #Electrical and Bioimpedance Tomography #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1809.08334
openalex publication_date 2018/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study inverse problems for the Poisson equation with source term the\ndivergence of an \R3-valued measure, that is, the potential \Φ\nsatisfies \n
Delta
Phi=
textdiv
boldsymbol
mu,\n and boldsymbol\μ is to be reconstructed knowing (a component of) the\nfield grad \Φ on a set disjoint from the support of boldsymbol\μ.\nSuch problems arise in several electro-magnetic contexts in the quasi-static\nregime, for instance when recovering a remanent magnetization from measurements\nof its magnetic field. We develop methods for recovering boldsymbol\μ\nbased on total variation regularization. We provide sufficient conditions for\nthe unique recovery of boldsymbol\μ, asymptotically when the\nregularization parameter and the noise tend to zero in a combined fashion, when\nit is uni-directional or when the magnetization has a support which is sparse\nin the sense that it is purely 1-unrectifiable.\n Numerical examples are provided to illustrate the main theoretical results.\n