Magnetic moment estimation and bounded extremal problems

Abstract

We consider the inverse problem in magnetostatics for recovering the moment of a planar magnetization from measurements of the normal component of the magnetic field at a distance from the support. Such issues arise in studies of magnetic material in general and in paleomagnetism in particular. Assuming the magnetization is a measure with 2 -density, we construct linear forms to be applied on the data in order to estimate the moment. These forms are obtained as solutions to certain extremal problems in Sobolev classes of functions, and their computation reduces to solving an elliptic differential-integral equation, for which synthetic numerical experiments are presented.2010 Mathematics Subject Classification. Primary: 35J15, 35R30, 35A35, 42B37, 86A22; Secondary: 45K05, 46F12, 47N20.* 3 ) −1 , ∇ ⟩ 2 ( ,R 2 ) > 0, ∀ ∈ 1,2 0 ( ), ̸ = 0. Now, the smoothness of the 1,2 0 ( )-valued function ↦ → opt ( ) entails that ↦ → ∇ opt ( ) is continuously differentiable with values in 2 ( , R 2 ), and that

Principles and techniques of the quantum diamond microscope

“…The sample magnetic field is generated by field sources, such as current densities or magnetic dipoles, with either known or unknown distributions. Measurement of the sample magnetic field can be used for the inverse problem of estimating an unknown source distribution under certain conditions [40][41][42]. The form of the sample magnetic field in terms of its sources is…”

Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation

“…There, the magnetization is assumed to be some R 3 -valued distribution (or integrable function) supported on a planar sample (square), of which we aim at estimating the net moment, given measurements of the normal component of the generated magnetic field on another planar measurements set taken to be a square parallel to the sample and located at some distance h to it. Moment and more general magnetization recovery issues are considered in [5,6,7]. More about the data acquisition process and the use of scanning SQUID (Superconducting Quantum Interference Device) microscopy devices for measuring a component of the magnetic field produced by weakly magnetized pieces of rocks can be found in the introductory sections of these references.…”

“…In 3D situation, these issues were efficiently analyzed with tools from harmonic analysis, specifically Poisson kernel and Riesz transforms [27], in [5,6,7], see also references therein, using their links with Hardy spaces of harmonic gradients. The existence of silent sources in 3D, elements of the non-zero kernel of the non injective magnetization-to-field operator (the operator that maps the magnetization to the measured component of the magnetic field) was established together with their characterization in [5].…”

“…Our purpose is to establish the existence and to build linear estimators for the mean values m i on S of the components m i of m for i = 1, 2, as was done in [6] for the 3D case. Indeed, though an academic version of the related physical issue, the present 2D situation possesses its own mathematical interest and specificity.…”

“…The results for moment estimation will be compared between them. Preliminary numerical computations in the 3D setting, with planar squared support and measurement set, were run on appropriate finite elements bases, see [6]; they are actually heavy. In the present 2D setting and on intervals, they are of course much less costly and we perform some of them using expansions on the Fourier bases, in Lebesgue and Sobolev spaces.…”