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 and 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. AtfehL. BaratchartJ. 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. BaratchartL. 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. LimaB. P. WeissL. BaratchartD. 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.

show all references

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. AtfehL. BaratchartJ. 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. BaratchartL. 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. LimaB. P. WeissL. BaratchartD. 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.

Figure 2.  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 $1$
Figure 3.  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 $\lambda \to +\infty$
Figure 1.  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
Figure 4.  "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
Figure 5.  $\phi_{\mathit{\boldsymbol{e}}_1}(\lambda)$ (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
Figure 6.  Synthetic magnetization (from left to right) $m_1$, $m_2$ and $m_3$ on $S$
Figure 7.  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
Table 1.  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$
[1]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations and Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005

[2]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

[3]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103

[4]

Anton Petrunin. Harmonic functions on Alexandrov spaces and their applications. Electronic Research Announcements, 2003, 9: 135-141.

[5]

Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems and Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058

[6]

Irene Benedetti, Nguyen Van Loi, Valentina Taddei. An approximation solvability method for nonlocal semilinear differential problems in Banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2977-2998. doi: 10.3934/dcds.2017128

[7]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[8]

Bo Jiang, Yongge Tian. On best linear unbiased estimation and prediction under a constrained linear random-effects model. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021209

[9]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025

[10]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems and Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229

[11]

Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092

[12]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

[13]

Kun-Peng Jin, Jin Liang, Ti-Jun Xiao. Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077

[14]

Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure and Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837

[15]

Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems and Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449

[16]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems and Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1

[17]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428

[18]

Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. Approximation of a nonlinear fractal energy functional on varying Hilbert spaces. Communications on Pure and Applied Analysis, 2018, 17 (2) : 647-669. doi: 10.3934/cpaa.2018035

[19]

Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems and Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77

[20]

Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899

[Back to Top]