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Topological Sensitivity Analysis for Reconstruction of the Inverse Source Problem from Boundary Measurement
Abstract:In this paper, we consider a geometric inverse source problem for the heat equation with Dirichlet and Neumann boundary data. We will reconstruct the exact form of the unknown source term from additional boundary conditions. Our motivation is to detect the location, the size and the shape of source support. We present a one-shot algorithm based on the Kohn-Vogelius formulation and the topological gradient method. The geometric inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from a source function. Then, we present a non-iterative numerical method for the geometric reconstruction of the source term with unknown support using a level curve of the topological gradient. Finally, we give several examples to show the viability of our presented method.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1339982Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 761
 A. B. Abda, M. Hassine, M. Jaoua, and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in stokes flow, SIAM Journal on Control and Optimization. 48(2009), 2871-2900.
 V. Akc¸elik, G. Biros, O. Ghattas, K. R. Long, and B. van BloemenWaanders, A variational finite element method for source inversion for convectivediffusive transport, Finite Elements in Analysis and Design. 39(2003), 683-705.
 Y. Alber and I. Ryazantseva, Nonlinear ill-posed problems of monotone type. Springer, 2006.
 G. Alessandrini and V. Isakov, Analicity and uniqueness for the inverse conductivity problem. 1996.
 M. A. Anastasio, J. Zhang, D. Modgil, and P. J. La Rivi`ere, Application of inverse source concepts to photoacoustic tomography, Inverse Problems. 23(2007), S21.
 O. Andreikiv, O. Serhienko, et al, Acoustic-emission criteria for rapid analysis of internal defects in composite materials, Materials Science. 37(2001), 106-117.
 S. R. Arridge, Optical tomography in medical imaging, Inverse problems. 15(1999), R41.
 H. Bellout, A. Friedman, and V. Isakov, Stability for an inverse problem in potential theory, Transactions of the American Mathematical Society. 332(1992), 271-296.
 J. Benoit, C. Chauvi`ere, and P. Bonnet, Source identification in time domain electromagnetics, Journal of Computational Physics. 231(2012), 3446-3456.
 A. Canelas, A. Laurain, and A. A. Novotny, A new reconstruction method for the inverse potential problem, Journal of Computational Physics. 268(2012), 417-431.
 A. Canelas, A. Laurain, and A. A. Novotny, A new reconstruction method for the inverse source problem from partial boundary measurements, Inverse Problems. 31(2015), 075009.
 X. Cheng, R. Gong, and W. Han, A new general mathematical framework for bioluminescence tomography, Computer Methods in Applied Mechanics and Engineering. 197(2008), 524-535.
 M. Choulli, Local stability estimate for an inverse conductivity problem, Inverse problems. 19(2003), 895.
 M. Choulli, On the determination of an inhomogeneity in an elliptic equation, Applicable Analysis. 85(2006), 693-699.
 M. Choulli, Une introduction aux probl‘emes inverses elliptiques et paraboliques, Berlin, Springer, 2009.
 M. Choulli and M. Yamamoto, Conditional stability in determining a heat source, Journal of inverse and ill-posed problems. 12(2004), 233244.
 A. Doicu, T. Trautmann, and F. Schreier, Numerical regularization for atmospheric inverse problems, Springer Science & Business Media, 2010.
 A. El Badia and T. Ha-Duong, On an inverse source problem for the heat equation. application to a pollution detection problem, Journal of inverse and ill-posed problems. 10(2002), 585-599.
 J. Ferchichi, M. Hassine, and H. Khenous, Detection of point-forces location using topological algorithm in stokes flows, Applied Mathematics and Computation. 219(2013), 7056-7074.
 S. Garreau, P. Guillaume, and M. Masmoudi, The topological asymptotic for pde systems: the elasticity case, SIAM journal on control and optimization. 39(2001), 1756-1778.
 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, springer, 2015.
 W. Guo, K. Jia, D. Han, Q. Zhang, X. Liu, J. Feng, C. Qin, X. Ma, and J. Tian, Efficient sparse reconstruction algorithm for bioluminescence tomography based on duality and variable splitting, Applied optics. 51(2012), 5676-5685.
 W. Han, W. Cong, and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems. 22(2006), 1659.
 M. Hassine, S. Jan, and M. Masmoudi, From differential calculus to 01 topological optimization, SIAM Journal on Control and Optimization. 45(2007), 1965-1987.
 M. Hassine, M. Hrizi, One-iteration reconstruction algorithm for geometric inverse source problem, Submitted.
 A. T. Hayes, A. Martinoli, and R. M. Goodman, Swarm robotic odor localization, Off-line optimization and validation with real robots. Robotica. 21(2003), 427-441.
 F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems. 14(1998), 67.
 F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation, Inverse Problems. 17(2001), 1465.
 V. Isakov. Inverse problems for partial differential equations, volume 127. Springer Science & Business Media, 2006.
 S. Larnier and M. Masmoudi, The extended adjoint method, ESAIM: Mathematical Modelling and Numerical Analysis. 47(2013), 83-108.
 N. F. Martins, An iterative shape reconstruction of source functions in a potential problem using the mfs, Inverse Problems in Science and Engineering. 20(2012), 1175-1193.
 P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems. 25(2009), 075011.
 R. Unnthorsson, T. P. Runarsson, and M. T. Jonsson. Acoustic emission based fatigue failure criterion for cfrp, International Journal of Fatigue. 30(2008), 11-20.
 G. Wang, Y. Li, and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical physics. 31(2004), 2289-2299.