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

PDF
Restringido a Repository staff only hasta 2020. 5MB |

Official URL: http://www.springerlink.com/content/q47k22718j526414/fulltext.pdf

## Abstract

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.

Item Type: | Article |
---|---|

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 |

ID Code: | 14944 |

References: | 1. Anagnostopoulos KA and Charalambopoulos A 2006 The linear sampling method for the transmission problem in two-dimensional anisotropic elasticity Inverse problems 22 553–577 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 67–84 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 10:51 |

Last Modified: | 06 Feb 2014 10:12 |

Repository Staff Only: item control page