2020/06/16 by Laurent Baratchart, L. Baratchart, D. Hardin +4
Engineering · Mathematics · #46 G (Primary) #46 N #49 Q (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #Statistical and numerical algorithms #math.FA #msc:46 #msc:49
paper · pdf · doi:10.48550/arxiv.2006.09072
40 pages
openalex publication_date 2020/06/16 · openalex created_date 2020/06/19 · arxiv created 2021/01/11 · arxiv updated 2021/01/12 · openalex updated_date 2026/07/28
We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in elementary solenoids given by S.K. Smirnov, when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by R3-valued Borel measures. We investigate methods for recovering a magnetization μ by penalizing its measure theoretic total variation norm (TV). In particular, we prove that if a magnetization is supported in a plane, then TV-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTV-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV-regularization schemes in the zero noise limit, by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing