Impinge stands for “Inverse Magnetization Problems In Geosciences”. It is an associate team between Inria's project team Apics (France) and the Laboratory “Earth, Atmospheric and Planetary Sciences” at MIT (MA, USA). People of the Center for Constructive Approximation at Vanderbilt University (TN, USA) are also associated to the project.
| Inria | MIT | Vanderbilt U. |
|---|---|---|
| L. Baratchart | C. Borlina | D. P. Hardin |
| S. Chevillard | E. A. Lima | E. B. Saff |
| J. Leblond | B. P. Weiss | C. Villalobos |
| K. Mavreas |
Former members: M. Northington (VU) and D. Ponomarev (Inria) defended their PhD thesis in 2016 and are not members of Impinge any more. Nonetheless, they are still collaborators of us on topics related to inverse magnetization problems. I. Sanders (MIT) quitted the associate team in 2016.
In geosciences and paleomagnetism, estimating the remanent magnetization in old rocks is an important issue to study past evolution of the Earth and other planets or bodies. However, the magnetization cannot be directly measured and only the magnetic field that it produces can be recorded.
In this paper we consider the case of thin samples, to be modeled as a planar set S of R^2 x {0}, carrying a magnetization m (a 3-dimensional vector field supported on S). This setup is typical of scanning microscopy that was developed recently to measure a single component of a weak magnetic field, close to the sample. Specifically, one is given a record of b_3[m] (tiny: a few nano Teslas), the vertical component of the magnetic field produced by m, on a planar region Q of R^2 x {h} located at some fixed height h > 0 above the sample plane. We assume that both S and Q are Lipschitz-smooth bounded connected open sets in their respective planes, and that the magnetization m belongs to [L^2(S)]^3, whence b_3[m] belongs to L^2(Q). Such magnetizations possess net moments <m> (belonging to R^3) defined as their integral on S.
Recovering the magnetization m or its net moment <m> from available measurements of b_3[m] are inverse problems for the Poisson-Laplace equation in the upper half-space R^3_+ with right hand side in divergence form. Indeed, Maxwell's equations in the quasi-static approximation identify the divergence of m with the Laplacian of a scalar magnetic potential in R^3_+ whose normal derivative on Q coincides with b_3[m]. Hence Neumann data b_3[m] are available on Q (subset of R^3_+), and we aim at recovering m or <m> on S. We thus face recovery issues on the boundary of the harmonicity domain from (partial) data available inside.
Such inverse problems are typically ill-posed and call for regularization. Indeed, magnetization recovery is not even unique, due to the existence of silent sources m != 0 such that b_3[m] = 0. And though such sources have vanishing moment so that net moment recovery is unique, estimation of the latter turns out to be unstable with respect to measurements errors.
The present work investigates silent sources, equivalent magnetization of minimal L^2(S)-norm to some given m in [L^2(S)]^3 (two magnetizations are called equivalent if their difference is silent), as well as density / instability results.
Scanning magnetic microscopes typically measure the vertical component B_3 of the magnetic field on a horizontal rectangular grid at close proximity to the sample. This feature makes them valuable instruments for analyzing magnetized materials at fine spatial scales, e.g., in geosciences to access ancient magnetic records that might be preserved in rocks by mapping the external magnetic field generated by the magnetization within a rock sample. Recovering basic characteristics of the magnetization (such as its net moment, i.e., the integral of the magnetization over the sample's volume) is an important problem, specifically when the field is too weak or the magnetization too complex to be reliably measured by standard bulk moment magnetometers.
In this paper, we establish formulas, asymptotically exact when R goes large, linking the integral of x_1 B_3, x_2 B_3, and B_3 over a square region of size R to the first, second, and third component of the net moment (and higher moments), respectively, of the magnetization generating B_3. The considered square regions are centered at the origin and have sides either parallel to the axes or making a 45-degree angle with them. Differences between the exact integrals and their approximations by these asymptotic formulas are explicitly estimated, allowing one to establish rigorous bounds on the errors.
We show how the formulas can be used for numerically estimating the net moment, so as to effectively use scanning magnetic microscopes as moment magnetometers. Illustrations of the method are provided using synthetic examples.
We study inverse problems for the Poisson equation with source term the divergence of an R^3-valued measure, that is, the potential Phi satisfies Laplacian of Phi equals divergence of mu, and mu is to be reconstructed knowing (a component of) the field gradient of Phi on a set disjoint from the support of mu. 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 mu based on total variation regularization. We provide sufficient conditions for the unique recovery of mu, 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.
Here is a little movie showing how the solution φₑ₁ of the BEP problem evolves when λ ranges from 10⁻¹⁰ to 10⁻²⁶.