no. 19-20
Partial Differential Equations/Numerical Analysis
Time reversed absorbing conditions
[Conditions aux limites absorbantes retournées temporellement]

Présenté par : Olivier Pironneau

Assous, Franck1, 2 ; Kray, Marie3 ; Nataf, Frédéric3 ; Turkel, Eli4
1 Bar Ilan University, 52900 Ramat Gan, Israel
2 Ariel University Center, 40700 Ariel, Israel
3 UPMC Université Paris-06, UMR 7598, laboratoire J.L. Lions, 75005 Paris, France
4 Tel-Aviv University, 69978 Ramat Aviv, Israel
Comptes Rendus. Mathématique, Tome 348 (2010) no. 19-20, pp. 1063-1067.

Le but de cette note est d'introduire les conditions absorbantes retournées temporellement (TRAC) dans les méthodes de retournement temporel. Elles rendent possible la « reconstruction du passé » sans connaître la source émettrice des signaux enregistrés puis rétropropagés. Cette nouvelle méthode ne nécessite pas de connaissance a priori des propriétés physiques de l'inclusion. Nous démontrons une estimation d'énergie pour le problème non standard obtenu. Deux applications aux problèmes inverses sont proposées.

The aim of this note is to introduce the time reversed absorbing conditions (TRAC) in time reversal methods. These new boundary conditions enable one to “recreate the past” without knowing the source which has emitted the signals that are back-propagated. This new method does not rely on any a priori knowledge of the physical properties of the inclusion. We prove an energy estimate for the resulting non-standard boundary value problem. Two applications to inverse problems are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.09.014
Assous, Franck 1, 2 ; Kray, Marie 3 ; Nataf, Frédéric 3 ; Turkel, Eli 4

1 Bar Ilan University, 52900 Ramat Gan, Israel
2 Ariel University Center, 40700 Ariel, Israel
3 UPMC Université Paris-06, UMR 7598, laboratoire J.L. Lions, 75005 Paris, France
4 Tel-Aviv University, 69978 Ramat Aviv, Israel 
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Assous, Franck; Kray, Marie; Nataf, Frédéric; Turkel, Eli. Time reversed absorbing conditions. Comptes Rendus. Mathématique, Tome 348 (2010) no. 19-20, pp. 1063-1067. doi : 10.1016/j.crma.2010.09.014. http://www.numdam.org/articles/10.1016/j.crma.2010.09.014/

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[9] Hecht, F. FreeFem++. Numerical Mathematics and Scientific Computation, Laboratoire J.L. Lions, Université Pierre et Marie Curie, 2010 http://www.freefem.org/ff++/

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☆ This work was supported by the French–Israeli Grant FIST-MATHS2.