Volume 27, 2021
|Number of page(s)||21|
|Published online||09 August 2021|
Inverse potential problems in divergence form for measures in the plane*,**
Projet FACTAS, INRIA, 2004 route des Lucioles, BP 93,
2 CMAP, Ecole Polytechnique, route de Saclay, 91128 Palaiseau Cedex, France.
3 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA.
*** Corresponding author: email@example.com
Accepted: 14 July 2021
We study inverse potential problems with source term the divergence of some unknown (ℝ3-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 ∥μ∥TV , 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 TV -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that TV -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 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.
Mathematics Subject Classification: 31B20 / 49N45 / 49Q20 / 86A22
Key words: Planar divergence free measures / purely 1-unrectifiable sets / inverse potential problems in divergence form / thin plate magnetizations / sparse recovery / total variation regularization
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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