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A partial inverse problem for the Sturm-Liouville operator on the lasso-graph
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Inverse problems for the heat equation with memory
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 |
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.
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/. Google Scholar |
[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. |
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. 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/. Google Scholar |
[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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