**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30123

##### Topological Sensitivity Analysis for Reconstruction of the Inverse Source Problem from Boundary Measurement

**Authors:**
Maatoug Hassine,
Mourad Hrizi

**Abstract:**

**Keywords:**
Geometric inverse source problem,
heat equation,
topological sensitivity,
topological optimization,
Kohn-Vogelius
formulation.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1339982

**References:**

[1] 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.

[2] 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.

[3] Y. Alber and I. Ryazantseva, Nonlinear ill-posed problems of monotone type. Springer, 2006.

[4] G. Alessandrini and V. Isakov, Analicity and uniqueness for the inverse conductivity problem. 1996.

[5] 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.

[6] O. Andreikiv, O. Serhienko, et al, Acoustic-emission criteria for rapid analysis of internal defects in composite materials, Materials Science. 37(2001), 106-117.

[7] S. R. Arridge, Optical tomography in medical imaging, Inverse problems. 15(1999), R41.

[8] 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.

[9] J. Benoit, C. Chauvi`ere, and P. Bonnet, Source identification in time domain electromagnetics, Journal of Computational Physics. 231(2012), 3446-3456.

[10] 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.

[11] 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.

[12] 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.

[13] M. Choulli, Local stability estimate for an inverse conductivity problem, Inverse problems. 19(2003), 895.

[14] M. Choulli, On the determination of an inhomogeneity in an elliptic equation, Applicable Analysis. 85(2006), 693-699.

[15] M. Choulli, Une introduction aux probl‘emes inverses elliptiques et paraboliques, Berlin, Springer, 2009.

[16] M. Choulli and M. Yamamoto, Conditional stability in determining a heat source, Journal of inverse and ill-posed problems. 12(2004), 233244.

[17] A. Doicu, T. Trautmann, and F. Schreier, Numerical regularization for atmospheric inverse problems, Springer Science & Business Media, 2010.

[18] 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.

[19] 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.

[20] 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.

[21] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, springer, 2015.

[22] 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.

[23] W. Han, W. Cong, and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems. 22(2006), 1659.

[24] M. Hassine, S. Jan, and M. Masmoudi, From differential calculus to 01 topological optimization, SIAM Journal on Control and Optimization. 45(2007), 1965-1987.

[25] M. Hassine, M. Hrizi, One-iteration reconstruction algorithm for geometric inverse source problem, Submitted.

[26] 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.

[27] F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems. 14(1998), 67.

[28] F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation, Inverse Problems. 17(2001), 1465.

[29] V. Isakov. Inverse problems for partial differential equations, volume 127. Springer Science & Business Media, 2006.

[30] S. Larnier and M. Masmoudi, The extended adjoint method, ESAIM: Mathematical Modelling and Numerical Analysis. 47(2013), 83-108.

[31] 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.

[32] P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems. 25(2009), 075011.

[33] 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.

[34] G. Wang, Y. Li, and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical physics. 31(2004), 2289-2299.