no. 15-16
Partial Differential Equations/Mathematical Physics
An inverse source problem with multiple frequency data
[Un problème inverse de source avec des données multi-fréquentielles]

Présenté par : Olivier Pironneau

Bao, Gang1, 2 ; Lin, Junshan2 ; Triki, Faouzi3
1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3 Université Joseph-Fourier, LJK, 38041 Grenoble cedex 9, France
Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 855-859.

Dans cette Note on considère un problème inverse de source pour lʼéquation de Helmholtz. Il consiste à déterminer la fonction source à partir du champ radié loin de la source, et à des multiples fréquences. On donne une nouvelle estimation de stabilité qui montre que la résolution dans la reconstruction de la source sʼaméliore avec lʼaugmentation de la fréquence. Ensuite, on propose une méthode de continuation pour résoudre numériquement le problème inverse. Cette méthode permet de capturer à la fois les détails fins et grossiers de la source. Un résultat numérique est présenté afin de montrer lʼefficacité de la méthode.

The Note is concerned with an inverse source problem for the Helmholtz equation, which determines the source from measurements of the radiated field away at multiple frequencies. Our main result is a novel stability estimate for the inverse source problem. Our result indicates that the ill-posedness of the inverse problem decreases as the frequency increases. Computationally, a continuation method is introduced to solve the inverse problem by capturing both the macro and the small scales of the source function. A numerical example is presented to demonstrate the efficiency of the method.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.07.009
Bao, Gang 1, 2 ; Lin, Junshan 2 ; Triki, Faouzi 3

1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3 Université Joseph-Fourier, LJK, 38041 Grenoble cedex 9, France 
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Bao, Gang; Lin, Junshan; Triki, Faouzi. An inverse source problem with multiple frequency data. Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 855-859. doi : 10.1016/j.crma.2011.07.009. http://www.numdam.org/articles/10.1016/j.crma.2011.07.009/

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