2018/09/21 by Laurent Baratchart, Cristóbal Villalobos Guillén, Cristobal Villalobos Guillen +9
Engineering · Mathematics · #49 #Electrical and Bioimpedance Tomography #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #math.OC #msc:49
paper · pdf · doi:10.48550/arxiv.1809.08334
arxiv created 2018/09/21 · openalex publication_date 2018/09/21 · arxiv updated 2018/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study inverse problems for the Poisson equation with source term the divergence of an R3-valued measure, that is, the potential Φ satisfies ΔΦ= div \boldsymbolμ, and \boldsymbolμ is to be reconstructed knowing (a component of) the field grad Φ on a set disjoint from the support of \boldsymbolμ. 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 develop methods for recovering \boldsymbolμ based on total variation regularization. We provide sufficient conditions for the unique recovery of \boldsymbolμ, 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.