Topological and shape gradient strategy for solving geometrical inverse problems

Journal of Mathematical Analysis and Applications

Volumn 400, Issue 2, 2013, Pages 724-742

  Chaabane, S ^a   Masmoudi, M ^b   Meftahi, H ^c  

^a Faculté des Sciences de Sfax   (Tunisia)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^c UNIVERSITÉ DE HAUTE ALSACE   (France)

Author keywords

Inverse problem; Shape gradient; Topological asymptotic expansion; Topological gradient

Indexed keywords

EID: 84872017798     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2012.11.044     Document Type: Article

References (33)

1. Barcelo B., Fabes E.B., Seo J.K. The inverse conductivity problem with one measurement, uniqueness for convex polyhedra. Proc. Amer. Math. Soc. 1994, 122:183-189.
2. Bellout H., Friedman A., Isakov V. Inverse problem in potential theory. Trans. Amer. Math. Soc. 1992, 332:271-296.
3. Friedman A., Isakov V. On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 1989, 38:553-580.
4. Isakov V., Powell J. On the inverse conductivity problem with one measurement. Inverse Problems 1990, 6:311-318.
5. Isakov V. On uniqueness of recovery of a discontinuous conductivity coefficient. Comm. Pure Appl. Math. 1988, 41(7):865-877.
6. Alessandrini G., Di Cristo M. Stable determination of inclusion by boundary measurements. SIAM J. Math. Anal. 2005, 37(1):200-217.
7. Hettlich F., Rundell W. Iterative methods for the reconstruction of an inverse potential problems. Inverse Problems 1996, 12:251-266.
8. Hettlich F., Rundell W. Recovery of the support of source term in elliptic differential equation. Inverse Problems 1997, 13:959-976.
9. Hettlich F., Rundell W. The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 1998, 14:67-82.
10. S. Garreau, P. Guillaume, M. Masmoudi, The topological sensitivity for linear isotropic elasticity, in: European conference on computational Mechanics, ECCM99, 1999, rapport MIP no. 99.45.
11. Guillaume Ph., Sididris K. The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 2002, 41:1042-1072.
12. Guillaume Ph., Sididris K. Topological sensitivity and shape optimization for the Stokes equations. SIAM J. Control Optim. 2004, 43:1-31.
13. Masmoudi M. The topological asymptotic. GAKUTO Internat. Ser. Math. Sci. Appl. 2001, vol. 16:53-72. Gakkotosho, Tokyo, Japan. R. Glowinski, H. Kawarada, J. Periaux (Eds.).
14. Masmoudi M. The topological asymptotic. International Series GAKUTO 2002.
15. A. Schumacher, Topologieoptimisierung von Buteilstrukturen unter Verwundung von Lopchpositionierungkriterien, Thesis, Univerität Gesamthochschule Siegen, 1995.
16. Novotny A.A., Feijóo R.A., Taroco E., Padra C. Topological sensitivity analysis. Comput. Methods Appl. Mech. Engrg. 2003, 192:803-829.
17. Cedio-Fengya D.J., Moskow S., Vogelius M.S. Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 2008, 14:553-595.
18. Alves C., Ammari H. Boundary integral formulas for the reconstruction of imperfections of small diameter in an elastic medium. SIAM J. Appl. Math. 2001, 62:94-106.
19. Ammari H. An inverse initial boundary value problem for the wave equation in the presence of imperfections of small volume. SIAM J. Control Optim. 2003, 41:1194-1211.
20. Ammari H., Volkov D. Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter. Asymptot. Anal. 2002, 30:331-350.
21. Ammari H., Moskow S. Asymptotic expansions for eigenvalues in the presence of small inhomogeneities. Math. Methods Appl. Sci. 2003, 26:67-75.
22. Ammari H., Kang H. A new method for reconstructing electromagnetic inhomogeneities of small volume. Inverse Problems 2003, 19:63-71.
23. Vogelius M.S., Volkov D. Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. M2AN Math. Model. Numer. Anal. 2000, 34:723-748.
24. Ammari H., Garnier J., Jugnon V., Kang H. Stability and resolution analysis for a topological derivative based imaging functional. SIAM J. Control Optim. 2012, 50(1):48-76.
25. Ammari H., Garnier J., Kang H., Park W.K., Solna K. Imaging schemes for perfectly conducting cracks. SIAM J. Appl. Math. 2011, 71(1):68-91.
26. Ammari H., Garnier J., Kang H., Lim M., Solna K. Multistatic imaging of extended targets. SIAM J. Imaging Sci. 2012, 5:564-600.
27. Ammari H., Beretta E., Francini E., Kang H., Lim M. Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements. Math. Comp. 2010, 79(271):1757-1777.
28. Ammari H., Kang H., Lim M., Zribi H. Conductivity interface problems. I. Small perturbations of an interface. Trans. Amer. Math. Soc. 2010, 362(5):2435-2449.
29. Evans Lawrence C. Partial Differential Equations 1998, American Mathematical Society, Providence, RI.
30. Amstutz S., Horchani I., Masmoudi M. Crack detection by the topological gradient method. Control Cybernet. 2005, 34(1):81-101.
31. Amstutz S. Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 2006, 49:87-108. IOS Press.
32. Ammari H., Kang H. Reconstruction of Small Inhomogeneities From Boundary Measurements. Lecture Notes in Mathematics 2004, Springer-Verlag.
33. Z. Belhachmi, H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability (submitted for publication).