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.

- 01/28 to 02/01: L. Baratchart visits Vanderbilt University.
- 04/25 to 04/27: organisation of a 3-days workshop at MIT, with all participants of the associate team except D. Ponomarev.
- 04/28 to 04/30: L. Baratchart, S. Chevillard and J. Leblond stay further at MIT.

- 05/07 to 06/06: E. Saff visits Inria.
- 05/30 to 06/07: D. Hardin visits Inria.

- 03/03 to 03/07: E. Lima visits Inria.
- 05/26 to 06/06: L. Baratchart visits Vanderbilt University.
- 05/27 to 06/04: S. Chevillard visits Vanderbilt University.
- 07/25 to 07/31: D. Hardin and M. Northington visit Inria.
- 08/01 to 08/08: M. Northington stays further at Inria.

- 08/18 to 08/22: L. Baratchart visits MIT.
- 09/02 to 09/09: S. Chevillard, J. Leblond and D. Ponomarev visit MIT.
- 09/10 to 09/12: D. Ponomarev stays further at MIT.

- 03/30 to 04/05: L. Baratchart, S. Chevillard, J. Leblond and D. Ponomarev visit MIT.
- 04/06 to 04/12: J. Leblond and D. Ponomarev stay further at MIT.

- 07/21 to 07/30: M. Northington visits Inria.
- 09/06 to 09/12: E. Lima and I. Sanders visit Inria.

- 02/22 to 02/28: L. Baratchart, S. Chevillard (from 02/24 only), J. Leblond and D. Ponomarev visit MIT.
- 02/29 to 03/04: L. Baratchart, S. Chevillard, J. Leblond, E. Lima and D. Ponomarev attend the
*Shanks Workshop on Mathematical Methods for Inverse Magnetization Problems Arising in Geoscience*organized at Vanderbilt University.- 03/04 to 03/06: L. Baratchart, J. Leblond and D. Ponomarev (until 03/05 only) stay further at Vanderbilt University.

- 06/08 to 06/22: D. Hardin (from 06/11 to 06/21 only), M. Northington (from 06/11 only) and C. Villalobos (until 06/21 only) visit Inria.
- 06/13 to 06/17: E. Lima visits Inria.

- 04/23 to 04/27: C. Borlina, D. Hardin and E. Lima visit Inria.
- 11/01 to 11/10: S. Chevillard visits MIT and then Vanderbilt University.
- 11/05 to 11/12: J. Leblond visits Vanderbilt University. E. Lima also, but only from 11/06 to 11/08.
- August to December: L. Baratchart spends a sabbatical semester at Vanderbilt University.

- 05/15 to 05/21: E. Lima visits Inria.
- 08/07 to 08/12: L. Baratchart visits Vanderbilt University.
- 11/01 to 11/09: L. Baratchart, J. Leblond and K. Mavreas will visit MIT.
- 12/13 to 12/20: D. Hardin, E. Saff and C. Villalobos plan to visit Inria.

- L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff and B. P. Weiss.
*Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions.*In*Inverse Problems*, 29(1). 2013.#### Abstract

Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique inverse problem. This problem leads to an analysis of the kernel of the related magnetization operators, which also has relevance to the 'equivalent source problem' in the case of measurements taken from just one side of the magnetization. We characterize the kernel of the operator relating planar magnetization distributions to planar magnetic field maps in various function and distribution spaces (e.g., sums of derivatives of L^{p}(Lebesgue spaces) or bounded mean oscillation (BMO) functions). For this purpose, we present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce a magnetic field either above or below the sample, or that are magnetically silent (i.e. no magnetic field anywhere outside the sample). For example, we show that a thin-plate magnetization is silent (i.e. in the kernel) when its normal component is zero and its tangential component is divergence free. In addition, we show that compactly supported magnetizations (i.e. magnetizations that are zero outside of a bounded set in the source plane) that do not produce magnetic fields either above or below the sample are necessarily silent. In particular, neither a nontrivial planar magnetization with fixed direction (unidimensional) compact support nor a bidimensional planar magnetization (i.e. a sum of two unidimensional magnetizations) that is nontangential can be silent. We prove that any planar magnetization distribution is equivalent to a unidimensional one. We also discuss the advantages of mapping the field on both sides of a magnetization, whenever experimentally feasible. Examples of source recovery are given along with a brief discussion of the Fourier-based inversion techniques that are utilized. - E. A. Lima, B. P. Weiss, L. Baratchart, D. P. Hardin and E. B. Saff.
*Fast inversion of magnetic field maps of unidirectional planar geological magnetization.*In*Journal of Geophysical Research: Solid Earth*, 118(6), pp. 2723-2752. 2013.#### Abstract

Scanning magnetic microscopes are being increasingly utilized in paleomagnetic studies of geological samples. These instruments typically map a single component of the sample's magnetic field at close proximity with submillimeter horizontal spatial resolution. However, in most applications, an image of the magnetization distribution within the sample is desired rather than its external magnetic field. This requires carefully solving an ill-posed inverse problem to obtain solutions that are nearly free of artifacts and consistent with both natural and laboratory magnetization processes. We present a new, fast inversion technique based on classic methods developed for the Fourier domain that retrieves planar unidirectional magnetization distributions from magnetic field maps. Whereas our approach considers the subtle peculiarities of scanning magnetic microscopy which otherwise can complicate this technique, much of the formalism and algorithms described in this work can also be directly applied to province-scale magnetic field data from aeromagnetic surveys and may be used as an initial step in the modeling of magnetic sources with complex three-dimensional geometries. We discuss sources of inaccuracy observed in practical implementations of the technique and present strategies to improve the quality of inversions. Numerous examples of inversion of both synthetic and experimental data demonstrate the performance of the technique under different conditions. In particular, we retrieve magnetization distributions of a Hawaiian basalt and compare it to inversions calculated in a previous work. We conclude by showing a reconstructed magnetization for the eucrite meteorite ALHA81001 that displays in high resolution the spatial distribution of high-coercivity grains within the sample. - L. Baratchart, S. Chevillard and J. Leblond.
*Silent and equivalent magnetic distributions on thin plates.*In*Harmonic Analysis, Function Theory, Operator Theory, and their Applications*, pages 11-27,*Theta Series in Advanced Mathematics*, 2017, The Theta Foundation.#### Abstract

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.

- L. Baratchart, S. Chevillard, D. P. Hardin, J. Leblond, E. A. Lima and J.-P. Marmorat.
*Magnetic moment estimation and bounded extremal problems.*Submitted to*Inverse Problems and Imaging*.#### 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 L^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.

- L. Baratchart, S. Chevillard J. Leblond, E. A. Lima and D. Ponomarev.
*Asymptotic method for estimating magnetic moments from field measurements on a planar grid.*#### Abstract

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.

- L. Baratchart, C. Villalobos, D. P. Hardin, M. C. Northington and E. B. Saff.
*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 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.

- L. Baratchart presented a talk at IPNE in May 2013.
- S. Chevillard presented a talk at
*PICOF'14*in May 2014. - L. Baratchart presented a plenary talk at
*Constructive Functions 2014*in June 2014. - S. Chevillard presented a talk at the
*27th IFIP TC7 conference 2015*in July 2015. - L. Baratchart presented a talk at the 10th ISSAC conference in August 2015.
- L. Baratchart, S. Chevillard, D. Ponomarev, J. Leblond and M. Northington all gave talks at the Shanks Workshop on Mathematical Methods for Inverse Magnetization Problems Arising in Geoscience at Vanderbilt University (Nashville, USA), March 2016.
- S. Chevillard gave an invited talk at the conference 5th Approximation Day in Lille (France), May 2016.
- J. Leblond gave a talk at PICOF (Problèmes Inverses, Contrôle et Optimisation de Formes) in Autrans (France), June 2016
- L. Baratchart gave an invited talk at the 25th St.Petersburg Summer Meeting in Mathematical Analysis, in St Petersburg (Russia), June 2016.
- J. Leblond gave a plenary talk at WiSE (Waves in Science and Engineering) in Queretaro (Mexico), August 2016.
- L. Baratchart gave a plenary talk at the conference Quasilinear equations, inverse problems and their applications in Moscow (Russia), September 2016.
- L. Baratchart and J. Leblond both gave invited talks at the
*Workshop Sigma (Signal, Image, Geometry, Modelling, Approximation)*in Luminy (France), Oct-Nov 2016. - D. Hardin gave a talk at the Computational Mathematics Seminar, at Middle Tennessee State University, Murfreesboro (TN, USA), November 2016.
- L. Baratchart gave a talk at the conference AIP (Applied Inverse Problems) in Hangzhou (China), May-June 2017.
- L. Baratchart gave a talk in the session
*Harmonic Analysis and Inverse Problems*of the conference MCA (Mathematical Congress of the Americas) in Montréal (Canada), July 2017. - A poster (presented by J.-P. Marmorat, on the work we did together on the moment recovery problem, addressed by a bounded extremal problem) at the 18th ISEM (International Symposium on Applied Electromagnetic and Mechanics) in Chamonix (France), September 2017.
- J. Leblond gave a talk at the ERNSI workshop (European Research Network on Systems Identification) in Lyon (France), September 2017.
- J. Leblond gave a talk at the annual meeting of the GDR AFHP (Research Group on Functional and Harmonic Analysis, and Probabilities) in Bordeaux (France), October 2017.
- L. Baratchart and J. Leblond gave talks at the conference Mathematics, Signal Processing and Linear Systems: New Problems and Directions, Orange (California, USA), November 2017.
- L. Baratchart, E. Lima, and K. Mavreas all gave talks at the school/workshop Inverse problems and approximation techniques in planetary sciences that we organized at Inria, May 2018.
- L. Baratchart was invited to give a talk at the conference IPMS 2018 (Inverse Problems, Modeling & Simulation), Malta, May 2018.
- L. Baratchart is an invited speaker at the SMAI-SIGMA workshop, Paris, Novembre 2018.

- A small presentation of the subject of the associate team has been written by S. Chevillard for the blog Mαthématiques de la planète Terre 2013 (in French). Here is a translation of this text in English.
- L. Bartachart made a presentation of the subject of the associate team at Inria for a
*C@fé-In*(in French) on October 13, 2014. The summary and the slides of the presentation are available here.

- We organized in May 2018 at Inria the 3-days school/workshop Inverse problems and approximation techniques in planetary sciences. The event was sponsored by the Center for Planetary Origins (C4PO) from UCA Jedi Idex and gathered around 20 participants.
- L. Baratchart and J. Leblond organized a mini-symposium on
*Inverse Source Problems With Applications to Planetary Sciences and Medical Imaging*at the IPMS 2018 conference, on May 21-25 in Malta.

- [A] Construction of a synthetic example and results obtained on this example with prototype algorithms : current notes.
- [B] Study of a simplified version of the problem: working notes.
- [C] Notes about the spectrum of the truncated Poisson operator.

- [D] Notes on how to use Kelvin transform in order to compute the moment of a sample by integral methods using measurements of the field.
- [E] Notes on using dipolar asymptotics to extend the data known only on a finite rectangle to the whole plane, and how it can help to recover the moment of a sample (
**updated in 2015**).

- [F] Asymptotic recovery of the net moment from integrals of the vertical component of the field on a plane.
- [G] Net moments estimation and bounded extremal problem: technical notes on the definition of a bounded extremal problem to recover the net moment of a magnetization from data on a finite square, up to any given accuracy (
**updated in 2016**).

- [H] Moments estimation of magnetic source terms from partial data (this is a comprehensive version of [F], almost in final form before submission).

- [I] Technical notes reporting practical experimations using the asymptotic formulas to recover the moment of physical data.
- [J] Informal presentation of the multipole fitting approach to recover the net moment of samples with reduced support.

- [K] Slides of a presentation at MIT Paleomagnetism Meetings, about results obtained with the asymptotic method for synthetic and experimental sources.
- [L] Slides of the lecture by E. Lima at the school/workshop Inverse problems and approximation techniques in planetary sciences.
- [M] Notes on explicit formulas to compute the adjoint of the forward map operator.

Here is a little movie showing how the solution φₑ₁ of the BEP problem evolves when λ ranges from 10⁻¹⁰ to 10⁻²⁶.

- Master's thesis of Olga Permiakova (2014): how to recover the net moment of a magnetization whose thickness is small but not completely negligible, by looking for a thin-plate magnetization on a well-chosen plane.
- PhD dissertation of Dmitry Ponomarev (2016).

- Final report of the first three years of the associate team.
- Final report of the three years renewal of the associate team.

Web site of the Apics Team.

Web site of the Laboratory “Earth, Atmospheric and Planetary Sciences”.

Web site of the Math Department of Vanderbilt University.