# American Institute of Mathematical Sciences

February 2019, 13(1): 39-67. doi: 10.3934/ipi.2019003

## Magnetic moment estimation and bounded extremal problems

 1 Team Factas, Inria, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France 2 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA 3 Department of Earth, Atmospheric and Planetary Sciences - MIT, Cambridge, MA 02139, USA 4 Center of Applied Mathematics, École des Mines ParisTech - CS 10 207, 06904 Sophia Antipolis Cedex, France

* Corresponding author: laurent.baratchart@inria.fr

Received  October 2017 Published  December 2018

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 L2-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.

Citation: Laurent Baratchart, Sylvain Chevillard, Douglas Hardin, Juliette Leblond, Eduardo Andrade Lima, Jean-Paul Marmorat. Magnetic moment estimation and bounded extremal problems. Inverse Problems & Imaging, 2019, 13 (1) : 39-67. doi: 10.3934/ipi.2019003
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975. [2] K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009. [3] B. Atfeh, L. Baratchart, J. Leblond and J. R. Partington, Bounded extremal and Cauchy-Laplace problems on the sphere and shell, Journal of Fourier Analysis and Applications, 16 (2010), 177-203. doi: 10.1007/s00041-009-9110-0. [4] L. Baratchart, L. Bourgeois and J. Leblond, Uniqueness results for inverse Robin problems with bounded coefficient, Journal of Functional Analysis, 270 (2016), 2508-2542. doi: 10.1016/j.jfa.2016.01.011. [5] 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 (eds. P. Jaming, A. Hartmann, K. Kellay, S. Kupin, G. Pisier and D. Timotin), Theta Series in Advanced Mathematics, The Theta Foundation, 19 (2017), 11–27. [6] 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, In preparation. Preprint available at https://hal.inria.fr/hal-01421157/. [7] 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, Inverse Problems, 29 (2013), 015004, 29pp. doi: 10.1088/0266-5611/29/1/015004. [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, 2011. doi: 10.1007/978-0-387-70914-7. [9] I. Chalendar and J. R. Partington, Constrained approximation and invariant subspaces, Journal of Mathematical Analysis and Applications, 280 (2003), 176-187. doi: 10.1016/S0022-247X(03)00099-4. [10] P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978. [11] B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Mathematica, 67 (1980), 297-314. doi: 10.4064/sm-67-3-297-314. [12] F. Demengel and G. Demengel, Espaces Fonctionnels, Utilisation Dans la Résolution des Équations Aux Dérivées Partielles, EDP Sciences/CNRS Editions, 2007. [13] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, 2004. doi: 10.1007/978-1-4757-4355-5. [14] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, 1998. doi: 10.1137/1.9780898719697. [15] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., New York-London-Sydney, 1975. [16] D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, Journal of Functional Analysis, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067. [17] T. Kato, Perturbation Theory for Linear Operators, vol. 132 of Grundlehren der mathematischen Wissenschaften, 2nd edition, Springer-Verlag, 1976. doi: 10.1007/978-3-642-66282-9. [18] J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Review, 26 (1984), 163-193. doi: 10.1137/1026033. [19] S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-0897-6. [20] 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, Journal of Geophysical Research: Solid Earth, 118 (2013), 2723-2752. doi: 10.1002/jgrb.50229. [21] J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, vol. 1, Dunod, 1968. [22] L. Schwartz, Théorie des Distributions, Hermann, 1966. [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [24] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univeristy Press, 1971. [25] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, 1986. [26] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975. [2] K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009. [3] B. Atfeh, L. Baratchart, J. Leblond and J. R. Partington, Bounded extremal and Cauchy-Laplace problems on the sphere and shell, Journal of Fourier Analysis and Applications, 16 (2010), 177-203. doi: 10.1007/s00041-009-9110-0. [4] L. Baratchart, L. Bourgeois and J. Leblond, Uniqueness results for inverse Robin problems with bounded coefficient, Journal of Functional Analysis, 270 (2016), 2508-2542. doi: 10.1016/j.jfa.2016.01.011. [5] 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 (eds. P. Jaming, A. Hartmann, K. Kellay, S. Kupin, G. Pisier and D. Timotin), Theta Series in Advanced Mathematics, The Theta Foundation, 19 (2017), 11–27. [6] 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, In preparation. Preprint available at https://hal.inria.fr/hal-01421157/. [7] 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, Inverse Problems, 29 (2013), 015004, 29pp. doi: 10.1088/0266-5611/29/1/015004. [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, 2011. doi: 10.1007/978-0-387-70914-7. [9] I. Chalendar and J. R. Partington, Constrained approximation and invariant subspaces, Journal of Mathematical Analysis and Applications, 280 (2003), 176-187. doi: 10.1016/S0022-247X(03)00099-4. [10] P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978. [11] B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Mathematica, 67 (1980), 297-314. doi: 10.4064/sm-67-3-297-314. [12] F. Demengel and G. Demengel, Espaces Fonctionnels, Utilisation Dans la Résolution des Équations Aux Dérivées Partielles, EDP Sciences/CNRS Editions, 2007. [13] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, 2004. doi: 10.1007/978-1-4757-4355-5. [14] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, 1998. doi: 10.1137/1.9780898719697. [15] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., New York-London-Sydney, 1975. [16] D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, Journal of Functional Analysis, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067. [17] T. Kato, Perturbation Theory for Linear Operators, vol. 132 of Grundlehren der mathematischen Wissenschaften, 2nd edition, Springer-Verlag, 1976. doi: 10.1007/978-3-642-66282-9. [18] J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Review, 26 (1984), 163-193. doi: 10.1137/1026033. [19] S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-0897-6. [20] 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, Journal of Geophysical Research: Solid Earth, 118 (2013), 2723-2752. doi: 10.1002/jgrb.50229. [21] J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, vol. 1, Dunod, 1968. [22] L. Schwartz, Théorie des Distributions, Hermann, 1966. [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [24] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univeristy Press, 1971. [25] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, 1986. [26] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.
Approximation error of $b_3^*[\phi_{\mathit{\boldsymbol{e}}_k}(\lambda)]$ with respect to $\mathit{\boldsymbol{e}}_k$ when $\lambda$ varies (the scale for $\lambda$ is logarithmic). The solid line corresponds to the case when $k = 1$, while the dashed line corresponds to the case when $k = 3$. As expected from Lemma 4.1, on the one hand, when $\lambda$ goes to $0$ (i.e., $\log_{10}(\lambda) \to -\infty$), the error tends to $0$. On the other hand, when $\lambda$ goes large, the constraint $M(\lambda)$ goes to $0$, meaning that ${\phi _{{\rm{opt}}}}$ is forced to go to $0$, whence the relative error tends to $1$
Constraint $M(\lambda) = \nabla \phi_{\mathit{\boldsymbol{e}}_k}(\lambda)$ as a function of $\lambda$ (the scale for $\lambda$ is logarithmic). The solid line corresponds to the case when $k = 1$, while the dashed line corresponds to the case when $k = 3$. As expected from Lemma 4.1, these are strictly decreasing smooth functions tending to $+\infty$ when $\lambda \to 0$ (i.e., $\log_{10}(\lambda) \to -\infty$) and tending to $0$ when $\lambda \to +\infty$
The mesh on $Q$, here with $P = 4$. The points of coordinates $(\kappa_p, \kappa_q)$ ($1 \le p, q \le P$) are represented by bullets. The elementary squares $Q_{p, q}$ overlap, as shown in the diagram
"L-curves" showing the approximation error $\|b_3^*[\phi_{\mathit{\boldsymbol{e}}_k}(\lambda)]-\mathit{\boldsymbol{e}}_k\|_{L^2(S, \mathbb{R}^3)}/\|\mathit{\boldsymbol{e}}_k\|_{L^2(S, \mathbb{R}^3)}$ as a function of the constraint $M = \|\nabla \phi_{\mathit{\boldsymbol{e}}_k}(\lambda)\|_{L^2(Q, \mathbb{R}^2)}$. The upper plot corresponds to the case when $k = 1$, and the lower one to the case when $k = 3$. As expected, the error decreases and tends to $0$ as the constraint is relaxed
$\phi_{\mathit{\boldsymbol{e}}_1}(\lambda)$ (top) and $\phi_{\mathit{\boldsymbol{e}}_3}(\lambda)$ (bottom) for $\lambda = 10^{-21}$. On both plots, the rectangles $Q$ (red) and $S$ (blue) are drawn together on the bottom layer to help visualize their respective positions
Synthetic magnetization (from left to right) $m_1$, $m_2$ and $m_3$ on $S$
Field $b_3[\mathit{\boldsymbol{m}}]$ corresponding to the synthetic magnetization shown on Figure 6 and an additive Gaussian white noise generated on the same $P \times P$ mesh on $Q$, with $P = 100$. The computed values are used to approximate the field and the noise as functions of $\text{Span}\{\psi_{p, q}\}_{1 \le p, q \le P}$. Note that the color scales are not the same on both pictures
The components $\langle m_k \rangle$ ($k = 1, 2, 3$) of the net moment $\langle \mathit{\boldsymbol{m}} \rangle$ are approximated thanks to the linear estimator as $\mu_k = \langle b, \, \phi_{\mathit{\boldsymbol{e}}_k}(\lambda) \rangle_{L^2(Q)}$ for several values of $\lambda$ and with $b$ being either the exact synthetic field $b_3[\mathit{\boldsymbol{m}}]$ or the exact field plus some noise.
The quantities $\delta_k$ are the relative errors $\delta_k = \big(\mu_k - \langle m_k \rangle\big)/\langle m_k \rangle$. The quantity $\delta_r$ is the relative error of the amplitude of $\mathit{\boldsymbol{\mu}} = (\mu_1, \mu_2, \mu_3)$ as a vector approximation of $\langle \mathit{\boldsymbol{m}}\rangle$, i.e., $\delta_r = \big(\|\mathit{\boldsymbol{\mu}}\|-\|\langle \mathit{\boldsymbol{m}} \rangle\|\big)/\|\langle \mathit{\boldsymbol{m}} \rangle\|$, where $\|\cdot\|$ denotes the Euclidean norm. Finally, $\theta$ is the angle between the vectors $\mathit{\boldsymbol{\mu}}$ and $\langle \mathit{\boldsymbol{m}} \rangle$, i.e., $\theta = \frac{360}{2\pi}\, \arccos\big(\frac{\mathit{\boldsymbol{\mu}}}{\|\mathit{\boldsymbol{\mu}}\|}.\frac{\langle \mathit{\boldsymbol{m}} \rangle}{\|\langle \mathit{\boldsymbol{m}} \rangle\|}\big)$
 No noise With noise $\lambda$ $\delta_1$ (%) $\delta_2$ (%) $\delta_3$ (%) $\delta_r$ (%) $\theta$ (°) $\delta_r$ (%) $\theta$ (°) $10^{-18}$ $12.56$ $14.77$ $3.02$ $-13.10$ $2.13$ $-12.93$ $2.28$ $10^{-19}$ $7.41$ $9.15$ $2.08$ $-8.05$ $1.23$ $-7.73$ $1.33$ $10^{-20}$ $4.91$ $5.52$ $1.61$ $-5.01$ $0.65$ $-4.44$ $0.70$ $10^{-21}$ $3.50$ $3.17$ $1.25$ $-3.10$ $0.34$ $-0.41$ $1.03$ $10^{-22}$ $2.50$ $1.71$ $0.95$ $-1.86$ $0.26$ $6.66$ $2.37$ $10^{-23}$ $1.71$ $0.86$ $0.73$ $-1.08$ $0.23$ $17.87$ $4.34$ $10^{-24}$ $1.11$ $0.38$ $0.53$ $-0.59$ $0.18$ $31.97$ $5.76$
 No noise With noise $\lambda$ $\delta_1$ (%) $\delta_2$ (%) $\delta_3$ (%) $\delta_r$ (%) $\theta$ (°) $\delta_r$ (%) $\theta$ (°) $10^{-18}$ $12.56$ $14.77$ $3.02$ $-13.10$ $2.13$ $-12.93$ $2.28$ $10^{-19}$ $7.41$ $9.15$ $2.08$ $-8.05$ $1.23$ $-7.73$ $1.33$ $10^{-20}$ $4.91$ $5.52$ $1.61$ $-5.01$ $0.65$ $-4.44$ $0.70$ $10^{-21}$ $3.50$ $3.17$ $1.25$ $-3.10$ $0.34$ $-0.41$ $1.03$ $10^{-22}$ $2.50$ $1.71$ $0.95$ $-1.86$ $0.26$ $6.66$ $2.37$ $10^{-23}$ $1.71$ $0.86$ $0.73$ $-1.08$ $0.23$ $17.87$ $4.34$ $10^{-24}$ $1.11$ $0.38$ $0.53$ $-0.59$ $0.18$ $31.97$ $5.76$
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