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 ∆Φ = div 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 develop methods for recovering µ based on total variation regularization. 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.
We study inverse potential problems with source term the divergence of some unknown (R 3 -valued) measure supported in a plane; e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetization μ by penalizing the measure-theoretic total variation norm kμk T V , and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane that T V -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree- like. We note that such magnetizations can be recovered via T V -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.
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 terms of 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 R 3 -valued Borel measures. We investigate methods for recovering a magnetization µ by penalizing the measure theoretic total variation norm µ T V . In particular, we show that if a magnetization is supported in a plane, then T V -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V -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 T V -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.
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