[Sur les solutions en série des équations intégrales en diffusion]
Nous étudions la validité de l’approche série de Neumann ou de Born pour résoudre l’équation de Helmholtz, et pour l’identification de coefficients dans des problèmes inverses de diffusion. Plus précisément, nous obtenons une condition nécessaire et suffisante sous laquelle la série converge fortement. Cette condition est beaucoup plus faible que celle utilisée traditionnellement. Nous examinons également le taux de convergence de la série. Notre approche utilise des techniques d’espace de réduction proposées par Suzuki [21]. De plus, nous proposons une méthode d’interpolation qui permet l’utilisation de la série de Neumann dans tous les cas. Enfin, nous fournissons plusieurs tests numériques avec différentes fonctions de milieu et valeurs de fréquence pour valider nos résultats théoriques.
We study the validity of the Neumann or Born series approach in solving the Helmholtz equation and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the series is strongly convergent. We also investigate the rate of convergence of the series. The obtained condition is optimal and it can be much weaker than the traditional requirement for the convergence of the series. Our approach makes use of reduction space techniques proposed by Suzuki [21]. Furthermore we propose an interpolation method that allows the use of the Neumann series in all cases. Finally, we provide several numerical tests with different medium functions and frequency values to validate our theoretical results.
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Keywords: Helmholtz equation, Born series, scattering
Mot clés : Équation de Helmholtz, série de Born, diffusion
Faouzi Triki 1 ; Mirza Karamehmedović 2
CC-BY 4.0
@article{CRMATH_2024__362_G9_1023_0,
author = {Faouzi Triki and Mirza Karamehmedovi\'c},
title = {On the series solutions of integral equations in scattering},
journal = {Comptes Rendus. Math\'ematique},
pages = {1023--1035},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.621},
language = {en},
}
Faouzi Triki; Mirza Karamehmedović. On the series solutions of integral equations in scattering. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1023-1035. doi : 10.5802/crmath.621. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.621/
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