{"title":"Topological Sensitivity Analysis for Reconstruction of the Inverse Source Problem from Boundary Measurement","authors":"Maatoug Hassine, Mourad Hrizi","volume":121,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":26,"pagesEnd":33,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10006334","abstract":"In this paper, we consider a geometric inverse source
\r\nproblem for the heat equation with Dirichlet and Neumann boundary
\r\ndata. We will reconstruct the exact form of the unknown source
\r\nterm from additional boundary conditions. Our motivation is to
\r\ndetect the location, the size and the shape of source support.
\r\nWe present a one-shot algorithm based on the Kohn-Vogelius
\r\nformulation and the topological gradient method. The geometric
\r\ninverse source problem is formulated as a topology optimization
\r\none. A topological sensitivity analysis is derived from a source
\r\nfunction. Then, we present a non-iterative numerical method for the
\r\ngeometric reconstruction of the source term with unknown support
\r\nusing a level curve of the topological gradient. Finally, we give
\r\nseveral examples to show the viability of our presented method.","references":" A. B. Abda, M. Hassine, M. Jaoua, and M. Masmoudi, Topological\r\nsensitivity analysis for the location of small cavities in stokes flow, SIAM\r\nJournal on Control and Optimization. 48(2009), 2871-2900.\r\n V. Akc\u00b8elik, G. Biros, O. Ghattas, K. R. Long, and B. van\r\nBloemenWaanders, A variational finite element method for source\r\ninversion for convectivediffusive transport, Finite Elements in Analysis\r\nand Design. 39(2003), 683-705.\r\n Y. Alber and I. Ryazantseva, Nonlinear ill-posed problems of monotone\r\ntype. Springer, 2006.\r\n G. Alessandrini and V. Isakov, Analicity and uniqueness for the inverse\r\nconductivity problem. 1996.\r\n M. A. Anastasio, J. Zhang, D. Modgil, and P. J. La Rivi`ere, Application\r\nof inverse source concepts to photoacoustic tomography, Inverse\r\nProblems. 23(2007), S21.\r\n O. Andreikiv, O. 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