An inverse source problem with multiple frequency data

Gang Bao; Junshan Lin; Faouzi Triki

doi:10.1016/j.crma.2011.07.009

Partial Differential Equations/Mathematical Physics

An inverse source problem with multiple frequency data

Presented by: Olivier Pironneau

Gang Bao ^1,²; Junshan Lin ²; Faouzi Triki 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

Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 855-859.

Abstract
Résumé

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.

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.

Received: 2010-09-01
Accepted: 2011-07-07
Published online: 2011-08-01

DOI: 10.1016/j.crma.2011.07.009

Author's affiliations:

Gang Bao ^{1, 2}; Junshan Lin ²; Faouzi Triki 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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     author = {Gang Bao and Junshan Lin and Faouzi Triki},
     title = {An inverse source problem with multiple frequency data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {855--859},
     publisher = {Elsevier},
     volume = {349},
     number = {15-16},
     year = {2011},
     doi = {10.1016/j.crma.2011.07.009},
     language = {en},
}

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Gang Bao; Junshan Lin; Faouzi Triki. An inverse source problem with multiple frequency data. Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 855-859. doi : 10.1016/j.crma.2011.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.07.009/

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