Comptes Rendus
no. 6
Partial differential equations/Numerical analysis
Transmission eigenvalues with artificial background for explicit material index identification
[Une identification d'indice explicite au moyen de valeurs propres de transmission pour un milieu de référence artificiel]

Présenté par : the Editorial Board

Lorenzo Audibert1, 2  ; Lucas Chesnel2  ; Houssem Haddar2
1 Department STEP, EDF R&D, 6, quai Watier, 78401 Chatou cedex, France
2 INRIA/Centre de mathématiques appliquées, École polytechnique, Université Paris-Saclay, route de Saclay, 91128 Palaiseau, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 626-631.

Nous souhaitons retrouver l'indice n d'une inclusion pénétrable dans un milieu de référence connu à partir de la donnée de champs lointains associés à des ondes planes incidentes. Pour ce faire, nous utilisons les valeurs propres de transmission (VPT) qui dépendent de n et qui peuvent être déterminées à partir de l'opérateur de champ lointain F. Dans cette note, nous expliquons comment modifier F en un opérateur de champ lointain Fart=F−F˜, où F˜ est calculé numériquement, correspondant à un milieu de référence artificiel et pour lequel les VPT associées fournissent une information plus directe sur n.

We are interested in the problem of retrieving information on the refractive index n of a penetrable inclusion embedded in a reference medium from farfield data associated with incident plane waves. Our approach relies on the use of transmission eigenvalues (TEs) that carry information on n and that can be determined from the knowledge of the farfield operator F. In this note, we explain how to modify F into a farfield operator Fart=F−F˜, where F˜ is computed numerically, corresponding to well-chosen artificial background and for which the associated TEs provide more accessible information on n.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.015
Lorenzo Audibert 1, 2 ; Lucas Chesnel 2 ; Houssem Haddar 2

1 Department STEP, EDF R&D, 6, quai Watier, 78401 Chatou cedex, France
2 INRIA/Centre de mathématiques appliquées, École polytechnique, Université Paris-Saclay, route de Saclay, 91128 Palaiseau, France 
@article{CRMATH_2018__356_6_626_0,
     author = {Lorenzo Audibert and Lucas Chesnel and Houssem Haddar},
     title = {Transmission eigenvalues with artificial background for explicit material index identification},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {626--631},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.015},
     language = {en},
}
TY  - JOUR
AU  - Lorenzo Audibert
AU  - Lucas Chesnel
AU  - Houssem Haddar
TI  - Transmission eigenvalues with artificial background for explicit material index identification
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 626
EP  - 631
VL  - 356
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2018.04.015
LA  - en
ID  - CRMATH_2018__356_6_626_0
ER  - 
%0 Journal Article
%A Lorenzo Audibert
%A Lucas Chesnel
%A Houssem Haddar
%T Transmission eigenvalues with artificial background for explicit material index identification
%J Comptes Rendus. Mathématique
%D 2018
%P 626-631
%V 356
%N 6
%I Elsevier
%R 10.1016/j.crma.2018.04.015
%G en
%F CRMATH_2018__356_6_626_0
Lorenzo Audibert; Lucas Chesnel; Houssem Haddar. Transmission eigenvalues with artificial background for explicit material index identification. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 626-631. doi : 10.1016/j.crma.2018.04.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.015/

[1] L. Audibert Qualitative Methods for Heterogeneous Media, École polytechnique, Palaiseau, France, 2015 (PhD thesis)

[2] L. Audibert; H. Haddar A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements, Inverse Probl., Volume 30 (2014) no. 3

[3] L. Audibert; F. Cakoni; H. Haddar New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data, Inverse Probl., Volume 33 (2017) no. 12

[4] E. Blåsten; L. Päivärinta; J. Sylvester Corners always scatter, Comment. Math. Phys., Volume 331 (2014) no. 2, pp. 725-753

[5] F. Cakoni; D. Colton; H. Haddar On the determination of Dirichlet or transmission eigenvalues from farfield data, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010) no. 7–8, pp. 379-383

[6] F. Cakoni; D. Colton; H. Haddar Inverse Scattering Theory and Transmission Eigenvalues, CBMS Series, SIAM Publications, vol. 88, 2016

[7] F. Cakoni; H. Haddar Transmission Eigenvalues in Inverse Scattering Theory Inverse Problems and Applications, Inside Out 60, MSRI Publications, Berkeley, CA, USA, 2013

[8] F. Cakoni; D. Gintides; H. Haddar The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., Volume 42 (2010) no. 1, pp. 237-255

[9] F. Cakoni; D. Colton; S. Meng; P. Monk Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math., Volume 76 (2016) no. 4, pp. 1737-1763

[10] L. Chesnel Bilaplacian problems with a sign-changing coefficient, Math. Methods Appl. Sci., Volume 39 (2016) no. 17, pp. 4964-4979

[11] G. Giorgi; H. Haddar Computing estimates of material properties from transmission eigenvalues, Inverse Probl., Volume 28 (2012) no. 5

[12] F. Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3–4, pp. 251-265

[13] A. Kirsch; N. Grinberg The Factorization Method for Inverse Problems, vol. 36, Oxford University Press, 2008

[14] A. Kirsch; A. Lechleiter The inside–outside duality for scattering problems by inhomogeneous media, Inverse Probl., Volume 29 (2013) no. 10

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

An acoustic analogy applied to the laminar upstream flow over an open 2D cavity

Jonas Ask; Lars Davidson

C. R. Méca (2005)

Strictly stable high order difference approximations for computational aeroacoustics

Bernhard Müller; Stefan Johansson

C. R. Méca (2005)