Biblioteca de la Universidad Complutense de Madrid

Solving inhomogeneous inverse problems by topological derivative methods


Carpio, Ana y Rapún, M.L. (2008) Solving inhomogeneous inverse problems by topological derivative methods. Inverse problems, 24 (4). ISSN 0266-5611

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We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.

Tipo de documento:Artículo
Palabras clave:Level set methods; Transmission problems; Obstacle scattering; Helmotz-equation; Shape optimization; Integral-equations; Tomography; Uniqueness; Reconstruction; Approximation
Materias:Ciencias > Matemáticas > Ecuaciones diferenciales
Código ID:14934

[1] Bonnet M and Constantinescu A 2005 Inverse problems in elasticity Inverse Problems 21 R1–50

[2] Bonnet M and Guzina B B 2004 Sounding of finite bodies by way of topological derivative Int. J. Numer. Methods Eng. 61 2344–73

[3] Burger M, Hackl B and Ring W 2004 Incorporating Topological derivatives into level set methods J. Comput. Phys. 194 344–62

[4] Cheney M, Isaacson D and Newell J C 1999 Electrical impedance tomography SIAM Rev. 41 85–101

[5] Carpio A and Rap´unML 2008 Domain reconstruction using photothermal techniques J. Comput. Phys. at press

[6] Carpio A and Rap´un M L 2007 Topological derivative based methods for non-destructive testing Proc. ENUMATH 2007 (Berlin: Springer) at press

[7] Colton D, Gieberman K andMonk P 2000 A regularized sampling method for solving three dimensional inverse scattering problems SIAM J. Sci. Comput. 21 2316–30

[8] Colton D and Kirsch A 1996 A simple method for solving inverse scattering problems in the resonance region Inverse Problems 12 383–93

[9] Colton D and Kress R 1992 Inverse Acoustic and Electromagnetic Scattering Theory (Berlin: Springer)

[10] CostabelMand Stephan E 1985 A direct boundary integral equation method for transmission problems J. Math. Anal. Appl. 106 367–413

[11] Devaney A J 1984 Geophysical diffraction tomography IEEE Trans. Geosci. Remote Sens. 22 3–13

[12] Dorn O and Lesselier D 2006 Level set methods for inverse scattering Inverse Problems 22 R67–131

[13] Delattre B, Ivaldi D and Stolz C 2002 Application du contrˆole optimal `a l’identification d’un chargement thermique Rev. Eur. Elem. Finish. 11 393–404

[14] Feijoo G R 2004 A new method in inverse scattering based on the topological derivative Inverse Problems 20 1819–40

[15] Feijoo G R, Oberai A A and Pinsky P M 2004 An application of shape optimization in the solution of inverse

acoustic scattering problems Inverse Problems 20 199–228

[16] Garreau S, Guillaume P and Masmoudi M 2001 The topological asymptotic for PDE systems: the elasticity case SIAM J. Control Optim. 39 1756–78

[17] Gerlach T and Kress R 1996 Uniqueness in inverse obstacle scattering with conductive boundary condition Inverse Problems 12 619–25

[18] Guzina B B and Bonnet M 2006 Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics Inverse Problems 22 1761–85

[19] Guzina B B and Chikichev I 2007 From imaging to material identification: a generalized concept of topological sensitivity J. Mech. Phys. Solids 55 245–79

[20] H¨ahner P 2000 On the uniqueness of the shape of a penetrable, anisotropic obstacle J. Comput. Appl. Math. 116 167–80

[21] Hettlich F 1995 Fr´echet derivatives in inverse obstacle scattering Inverse Problems 11 371–82

[22] Keller J B and Givoli D 1989 Exact non-reflecting boundary conditions J. Comput. Phys. 82 172–92

[23] Kirsch A 1993 The domain derivative and two applications in inverse scattering theory Inverse Problems 9 81–93

[24] Kirsch A and Kress R 1993 Uniqueness in inverse obstacle scattering Inverse Problems 9 285–9

[25] Kleinman R E and Martin P 1988 On single integral equations for the transmission problem of acoustics SIAM J. Appl. Math. 48 307–25

[26] Kleinman R E and van der Berg P M 1992 A modified gradient method for two dimensional problems in tomography J. Comput. Appl. Math. 42 17–35

[27] Kress R and Roach G F 1978 Transmission problems for the Helmholtz equation J. Math. Phys. 19 1433–7

[28] Liseno A and Pierri R 2004 Imaging of voids bymeans of a physical optics based shape reconstruction algorithm J. Opt. Soc. Am. A 21 968–74

[29] 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

[30] Masmoudi M 1987 Outils pour la conception optimale des formes Th`ese d’Etat en Sciences Math´ematiques Universit´e de Nice

[31] Natterer F and Wubbeling F 1995 A propagation backpropagation method for ultrasound tomography Inverse Problems 11 1225–32

[32] Peters A, Berger H U, Chase J and Van Houten E 2006 Digital-image based elasto-tomography: nonlinear mechanical property reconstruction of homogeneous gelatine phantoms Int J. Inf. Syst. Sci. 2 512–21

[33] 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

[34] Rap´un M L and Sayas F J 2006 Boundary integral approximation of a heat diffusion problem in time-harmonic regime Numer. Algorithms 41 127–60

[35] Rap´un M L and Sayas F J 2006 Indirect methods with Brakhage–Werner potentials for Helmholtz transmission problems Numerical Mathematics and Advanced Applications (ENUMATH 2005 Berlin: Springer) pp 1146–54

[36] Rap´un M L and Sayas F J 2008 Exterior Dirichlet and Neumann problems for the Helmholtz equation as limits of transmission problems Integral Methods in Science and Engineering (Boston, MA: Birkhauser) pp 207–16

[37] Rap´un M L and Sayas F J 2006 A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media ESAIM Math. Model. Numer. Anal. 40 871–96

[38] Samet B, Amstutz S and Masmoudi M 2003 The topological asymptotic for the Helmholtz equation SIAM J. Control Optim. 42 1523–44

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Última Modificación:28 Oct 2016 08:30

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