Inverse potential problems in divergence form for measures in the plane

Laurent Baratchart,Douglas P Hardin,Cristóbal Villalobos Guillén

doi:10.1051/cocv/2021082

Abstract

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 magnetizationμby 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 thatTV-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 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 recoveredvia 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.

Highlights

This paper considers inverse potential problems with source term in divergence form, in connection with the structure of 2-D divergence-free measures in the plane
A typical application, and the main motivation of the authors to carry out this research, is to inverse magnetization problems on thin plates
When magnetizations are modelled by R3-valued measures supported on a set S, the inverse magnetization problem consists in recovering such a measure, say μ, from knowledge of the magnetic field b(μ) that it generates, see Section 1.1 for details

Summary

Introduction

This paper considers inverse potential problems with source term in divergence form, in connection with the structure of 2-D divergence-free measures in the plane. A 2-D divergence-free measure ν in the plane can be decomposed as a superposition of elementary “loops”; i.e., contour integrations along rectifiable Jordan curves, in such a way that the Radon-Nykodim derivative dν/d|ν|(x) is essentially the unit tangent to any of these curves through x, (cf Thm. 2.8) This result, which is more precise (though limited to the planar case) than the general decomposition theorem for solenoids given in [28], gives us insight on the structure of the kernel of the forward operator, enabling us to give sufficient conditions for a magnetization to be T V -minimal on S; i.e., the magnetization has minimum total variation among those magnetizations supported on S that generate the same field. In the rest of this introductory section, we describe the inverse magnetization problem, explain our main results and set up notation

Background and overview of results

Notation

Curves and divergence-free measures in Rn

Applications to inverse magnetization problems

Regularization by penalizing the total variation