On the series solutions of integral equations in scattering
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...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.621/ |
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| author | Triki, Faouzi Karamehmedović, Mirza |
| author_facet | Triki, Faouzi Karamehmedović, Mirza |
| author_sort | Triki, Faouzi |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-59f180ba2d3a4f808f9f5026fbc6b244 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-59f180ba2d3a4f808f9f5026fbc6b2442025-02-07T11:23:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G91023103510.5802/crmath.62110.5802/crmath.621On the series solutions of integral equations in scatteringTriki, Faouzi0Karamehmedović, Mirza1Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, Grenoble, FranceDepartment of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, DenmarkWe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.621/Helmholtz equationBorn seriesscattering |
| spellingShingle | Triki, Faouzi Karamehmedović, Mirza On the series solutions of integral equations in scattering Comptes Rendus. Mathématique Helmholtz equation Born series scattering |
| title | On the series solutions of integral equations in scattering |
| title_full | On the series solutions of integral equations in scattering |
| title_fullStr | On the series solutions of integral equations in scattering |
| title_full_unstemmed | On the series solutions of integral equations in scattering |
| title_short | On the series solutions of integral equations in scattering |
| title_sort | on the series solutions of integral equations in scattering |
| topic | Helmholtz equation Born series scattering |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.621/ |
| work_keys_str_mv | AT trikifaouzi ontheseriessolutionsofintegralequationsinscattering AT karamehmedovicmirza ontheseriessolutionsofintegralequationsinscattering |