Carpio Rodríguez, Ana María and Rapún , M.L. (2008) Topological Derivatives for Shape Reconstruction. Inverse problems and imaging, 1943 . pp. 85-133. ISSN 0075-8434
Restricted to Repository staff only until 2020.
Topological derivative methods are used to solve constrained optimization reformulations of inverse scattering problems. The constraints take the form of Helmholtz or elasticity problems with different boundary conditions at the interface between the surrounding medium and the scatterers. Formulae for the topological derivatives are found by first computing shape derivatives and then performing suitable asymptotic expansions in domains with vanishing holes. We discuss integral methods for the numerical approximation of the scatterers using topological derivatives and implement a fast iterative procedure to improve the description of their number, size, location and shape.
|Uncontrolled Keywords:||Inverse obstacle scattering; Boundary inegral-equations; Level set methods; Transmission problems; Helmholtz-equation; Anisotropic elasticity; Sampling method; Tomography; Waves; Optimization|
|Subjects:||Sciences > Mathematics > Differential equations|
1. Anagnostopoulos KA and Charalambopoulos A 2006 The linear sampling method for the transmission problem in two-dimensional anisotropic elasticity Inverse problems
2. Bonnet M and Constantinescu A 2005 Inverse problems in elasticity Inverse problems 21 R1-R50
3. Burger M, Hackl B and Ring W 2004 Incorporating topological derivatives into level set methods J. Comp. Phys. 194 344–362
4. Carpio A and Rapun ML, Solving inhomogeneous inverse problems by topological derivative methods, submitted, 2007
5. Carpio A and Rapun ML, Shape reconstruction in anisotropic elastic media. In preparation
6. Chandler GA and Sloan IH 1990. Spline qualocation methods for boundary integral equations. Numer. Math. 58, 537–567
7. Charalambopoulos A 2002 On the interior transmission problem in nondissipative,homogeneous, anisotropic elasticity J. Elast. 67 149–170
8. Colton D 1984 The inverse scattering problem for time-harmonic acoustic waves,SIAM Review 26 323–350.
9. Colton D, Gieberman K and Monk P 2000 A regularized sampling method for solving three dimensional inverse scattering problems SIAM J. Sci. Comput. 21 2316–2330
10. Colton D and Kress R 1983 Integral equation methods in scattering theory John Wiley & Sons. New York.
11. Colton D and Kress R 1992 Inverse acoustic and electromagnetic scattering theory Springer Berlin.
12. Colton D and Kirsch A 1996 A simple method for solving inverse scattering problems in the resonance region Inverse problems 12 383–393
13. Costabel M and Stephan E 1985 A direct boundary integral equation method for transmission problems J. Math. Anal. Appl. 106 367–413
14. Devaney AJ 1984 Geophysical diffraction tomography, IEEE Trans. Geosci. Remote Sens. 22 3–13
15. Dom´ınguez V, Rap´un ML and Sayas FJ 2007 Dirac delta methods for Helmholtz transmission problems. To appear in Adv. Comput. Math.
16. Domınguez V and Sayas FJ 2003. An asymptotic series approach to qualocation methods. J. Integral Equations Appl. 15, 113–151
17. Dorn O and Lesselier D 2006 Level set methods for inverse scattering Inverse Problems 22 R67–R131
18. Feijoo GR 2004 A new method in inverse scattering based on the topological derivative Inverse Problems 20 1819–1840
19. Feijoo GR, Oberai AA and Pinsky PM 2004 An application of shape optimization in the solution of inverse acoustic scattering problems Inverse problems 20 199–228
20. Gachechiladze A and Natroshvili D 2001 Boundary variational inequality approach in the anisotropic elasticity for the Signorini problem Georgian Math. J. 8 469-492
21. Garreau S, Guillaume P and Masmoudi M 2001 The topological asymptotic for PDE systems: the elasticity case SIAM J. Control Optim. 39 1756–1778
22. Gegelia T and Jentsch L 1994 Potential methods in continuum mechanics Georgian Math. J. 599-640
23. Gerlach T and Kress R 1996 Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Problems 12 619–625
24. Guzina BB and Bonnet M 2006 Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics Inverse Problems 22 1761–1785
25. Guzina BB, Bonnet M 2004 Topological derivative for the inverse scattering of elastic waves Q. Jl. Mech. Appl. Math. 57 161–179
26. Guzina BB, Chikichev I 2007 From imaging to material identification: A generalized concept of topological sensitivity J. Mech. Phys. Sol. 55 245–279
27. Hettlich F 1995 Fr´echet derivatives in inverse obstacle scattering Inverse problems 11 371–382
28. Keller JB and Givoli D 1989 Exact non-reflecting boundary conditions J. Comput. Phys. 82 172–192
29. Kirsch A, Kress R, Monk P and Zinn A 1988 Two methods for solving the inverse acoustic scattering problem Inverse problems 4 749–770
30. Kirsch A and Kress R 1993 Uniqueness in inverse obstacle scattering Inverse Problems 9 285–299
31. Kirsch A 1993 The domain derivative and two applications in inverse scattering theory Inverse Problems 9 81–93
32. Kleinman RE and Martin P 1988 On single integral equations for the transmission problem of acoustics SIAM J. Appl. Math 48 307–325
33. Kleinman RE and van der Berg PM 1992 A modified gradient method for two dimensional problems in tomography J. Comput. Appl. Math. 42 17–35
34. Kress R and Roach GF 1978 Transmission problems for the Helmholtz equation J. Math. Phys. 19 1433–1437
35. Kupradze VD, Gegelia TG, Basheleuishvili MO and Burchauladze TV, Three dimensional problems of the mathematical theory of elasticity and thermoelasticity,
North-Holland Ser. Appl. Math. Mech. 25, North-Holland, Amsterdam, 1979.
36. Liseno A and Pierri R 2004 Imaging of voids by means of a physical optics based shape reconstruction algorithm J. Opt. Soc. Am. A 21 968–974
37. Litman A, Lesselier D and Santosa F 1998 Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level set Inverse problems 14 685–706
38. Masmoudi M 1987 Outils pour la conception optimale des formes Th`ese d’Etat en Sciences Math´ematiques, Universit´e de Nice
39. Masmoudi M, Pommier J and Samet B 2005 The topological asymptotic expansion for the Maxwell equations and some applications Inverse Problems 21 547–564
40. Meddahi S and Sayas FJ 2005 Analysis of a new BEM–FEM coupling for two dimensional fluid–solid iteraction Num. Methods Part. Diff. Eq. 21 1017–1154
41. Natterer F and Wubbeling F 1995 A propagation backpropagation method for ultrasound tomography Inverse problems 11 1225-1232
42. Natroshvili D 1995 Mixed interface problems for anisotropic elastic bodies Georgian Math. J. 2 631-652
43. Natroshvili D 1996 Two-dimensional steady state oscillation problems o nisotropic elasticity Georgian Math. J. 3 239-262
44. Potthast R 1996 Fr´echet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain J. Inverse Ill-Posed Problems 4
45. Rap´un ML and Sayas FJ 2006 Boundary integral approximation of a heat diffusion problem in time–harmonic regime Numer. Algorithms 41 127–160
46. Rap´un ML and Sayas FJ 2006 Indirect methods with Brakhage–Werner potentials for Helmholtz transmission problems. In Numerical Mathematics and advanced
applications. ENUMATH 2005. Springer 1146–1154
47. Rap´un ML and Sayas FJ 2007 Exterior Dirichlet and Neumann problems for the Hemholtz equation as limits of transmission problems. Submitted.
48. Rap´un ML and Sayas FJ 2007 Boundary element simulation of thermal waves. Arch. Comput. Methods. Engrg 14 3–46
49. Samet B, Amstutz S and Masmoudi M 2003 The topological asymptotic for the Helmholtz equation SIAM J. Control Optimim 42 1523–1544 50. Santosa F 1996 A level set approach for inverse problems involving obstacles
ESAIM Control, Optim. Calculus Variations 1 17–33
51. Sloan IH 2000. Qualocation. J. Comput. Appl. Math. 125, 461–478
52. Sokolowski J and Zol´esio JP 1992 Introduction to shape optimization. Shape sensitivity analysis (Heidelberg: Springer)
53. Sokowloski J and Zochowski A 1999 On the topological derivative in shape optimization SIAM J. Control Optim. 37 1251–1272
54. Torres RH and Welland GV 1993 The Helmholtz equation and transmission problems with Lipschitz interfaces Indiana Univ. Math. J. 42 1457–1485
55. von Petersdorff T 1989 Boundary integral equations for mixed Dirichlet, Neumann and transmission problems. Math. Methods Appl. Sci. 11 185–213
|Deposited On:||24 Apr 2012 12:51|
|Last Modified:||24 Apr 2012 12:51|
Repository Staff Only: item control page