Internship 2023: Field preprocessing in inverse magnetisation problems

Expected duration: 4-6 months (with regular remuneration).
Level: Second year of Master degree or Engineers School (PFE).
Location: Factas team, Centre Inria d’Université Côte d’Azur (Sophia Antipolis)

Advisors: Dmitry Ponomarev (contact:, Juliette Leblond
Inverse magnetisation problem consists in inferring information about static magnetic source from partial knowledge of the produced field. Over the last several years, Factas team has accumulated expertise in study of certain instances of the inverse magnetisation problem. One such instance is relevant to a concrete experimental set-up in paleomagnetism lab at EAPS (Earth, Atmospheric & Planetary Sciences) Department, MIT (Massachusetts Institute of Technology, USA) [1]. The issue consists in reconstruction of the net magnetisation of a nearly planar magnetised rock sample (typically, extraterrestial) from measurements of only one component of the field available, in vicinity of the sample, on a portion of the plane orthogonal to that component. In addition to implicit characterisation [2] of this desired quantity (net magnetisation vector), one can explicitly obtain, in multiple ways, its asymptotic estimates [3]. Both approaches rely on precise measurements of the field available on the area sufficiently large compared to the source size. In practice, however, the measurements are not only spatially limited but also heavily polluted by noise already at rather small distances away from the source. The second aspect makes it especially difficult to deal with rocks whose magnetisation is spreaded over the entire sample.
The goal of the internship is to explore different ways of preprocessing the field which would make the
previous results applicable. This would, on the one hand, require choosing an appropriate strategy for the field extrapolation based on its analytic structure. On the other hand, study of the sensitivity of the procedure to the noise and performing the denoising would be of the equal importance. The denoising issue may also include incorporating different stochastic and/or deterministic models of the noise. Even though preliminary results in this direction look promising, a number of theoretical and practical aspects have yet to be better understood.
This would require the candidate to have basic numerical skills in addition to strong interest in physics and engineering applications.
As a follow-up work, a Ph.D. thesis could potentially be offered to the successful intern, extending the
obtained results to treat other issues. Reconstruction of the whole magnetisation distribution could be
attempted under appropriate assumptions. Moreover, adapting results to different experimental set-ups and problems of magnetometry would also be a lucrative possibility.
[1] Baratchart, L., Hardin, D. P., Lima, E. A., Saff, E. B., Weiss, B. P., “Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions”, Inverse Problems, 29 (1), 2013.
[2] Baratchart, L., Chevillard, S., Hardin, D., Leblond, J., Lima, E. A., Marmorat, J.-P., “Magnetic moment estimation and bounded extremal problems”, Inverse Problems and Imaging, 13 (1), 2019.
[3] Ponomarev, D., “Magnetisation moment of a bounded 3D sample: Asymptotic recovery from planar measurements on a large disk using Fourier analysis”, arXiv:2205.14776, 2022.

About Juliette LEBLOND

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