Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

Abstract

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.

Inverse potential problems in divergence form for measures in the plane

“…The magnetic field b(µ) generated by a magnetization µ ∈ M(S) 3 is defined, at a point x not in the support of µ, in terms of the scalar magnetic potential Φ(µ) by (see [13]):…”

“…where, for x, y ∈ R 3 , x • y and |x| denote the Euclidean scalar product and norm and ∇ y the gradient with respect to y. Clearly, Φ(µ) and the components of b(µ) are harmonic functions on R 3 \ S. Moreover, formula (2) defines Φ(µ) on the whole of R 3 as a member of L 2 (R 3 ) + L 1 (R 3 ) (see [3,Proposition 2.1]) so that b(µ), initially defined on R 3 \ S, extends to a R 3 -valued divergence-free distribution on R 3 . Indeed, we may write…”

“…For instance, it is usually so in Scanning Magnetic Microscopy experiments (SMM) where data consist of point-wise values of the normal component of the magnetic field on a planar region not intersecting S, see [14,15,21]. Geometric conditions on Q, S and v, ensuring that such measurements suffice to determine b(µ) in the entire region R 3 \S, are given in [3,Lemma 2.3] and recalled when S is planar in Section ? ?, for the convenience of the reader.…”

“…In short: regularizing schemes that penalize the total variation are consistent to recover strictly T V -minimal magnetizations, and thus, any assumption ensuring strict T V -minimality gives rise to a consistency result. For the larger class of magnetizations supported on a slender set S (see Section 1.1 for a definition), such a consistency result is obtained in [3,Theorem 2.6] by showing, using results from [19], that magnetizations supported on a purely 1-unrectifiable set are strictly T V -minimal. Specializing to the case of planar S and appealing to the above-mentioned loop decomposition, we shall obtain more general conditions: for instance magnetizations carried by the union of a purely 1-unrectifiable set and a collection of sufficiently separated line segments are strictly T V -minimal, see Corollary ??…”

“…[10] and [5]. In this connection, the gist of [3,Theorem 2.6] and Corollary ?? is to define notions of "sparsity" in the present, infinite-dimensional context.…”