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Advances in Computational Mathematics

, Volume 36, Issue 2, pp 235–265 | Cite as

Second-order topological expansion for electrical impedance tomography

Article

Abstract

Second-order topological expansions in electrical impedance tomography problems with piecewise constant conductivities are considered. First-order expansions usually consist of local terms typically involving the state and the adjoint solutions and their gradients estimated at the point where the topological perturbation is performed. In the case of second-order topological expansions, non-local terms which have a higher computational cost appear. Interactions between several simultaneous perturbations are also considered. The study is aimed at determining the relevance of these non-local and interaction terms from a numerical point of view. A level set based shape algorithm is proposed and initialized by using topological sensitivity analysis.

Keywords

Electrical impedance tomography Inverse problem Shape and topological derivative Level sets 

AMS 2000 Subject Classifications

49Q10 49K24 49N45 

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References

  1. 1.
    Ammari, H., Kang, H.: High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. SIAM J. Math. Anal. 34(5), 1152–1166 (electronic) (2003)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ammari, H., Moskow, S., Vogelius, M.S.: Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM Control Optim. Calc. Var. 9, 49–66 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bonnaillie-Noël, V., Dambrine, M., Tordeux, S., Vial, G.: On moderately close inclusions for the Laplace equation. C. R. Math. Acad. Sci. Paris 345(11), 609–614 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Bonnet, M.: Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions. Int. J. Solids Struct. 46(11–12), 2275–2292 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Borcea, L.: Electrical impedance tomography. Inverse Probl. 18(6), R99–R136 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brühl, M., Hanke, M., Vogelius, M.S.: A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93(4), 635–654 (2003)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    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 Probl. 14(3), 553–595 (1998)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cheney, M., Isaacson, D., Newell, J.C.: Electrical impedance tomography. SIAM Rev. 41(1), 85–101 (electronic) (1999)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chung, E.T., Chan, T.F., Tai, X.-C.: Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205(1), 357–372 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    de Faria, J.R., Novotny, A.A., Feijóo, R.A., Taroco, E., Padra, C.: Second order topological sensitivity analysis. Int. J. Solids Struct. 44(14–15), 4958–4977 (2007)MATHCrossRefGoogle Scholar
  11. 11.
    Delfour, M.C., Zolésio, J.-P.: Shapes and geometries. In: Advances in Design and Control, vol. 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)Google Scholar
  12. 12.
    Eschenauer, H., Kobelev, V., Schumacher, A.: Bubble method for topology and shape optimization of structures. J. Struct. Optim. 8, 42–51 (1994)CrossRefGoogle Scholar
  13. 13.
    Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39(6), 1756–1778 (electronic) (2001)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Henrot, A.: Extremum problems for eigenvalues of elliptic operators. In: Frontiers in Mathematics. Birkhäuser, Basel (2006)Google Scholar
  15. 15.
    Hintermüller, M., Laurain, A.: Electrical impedance tomography: from topology to shape. Control Cybern. 37(4), 913–933 (2008)MATHGoogle Scholar
  16. 16.
    Il′in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems. In: Transl. Math. Monog., vol. 102. American Mathematical Society, Providence (1992)Google Scholar
  17. 17.
    Mazja, W.G., Nasarow, S.A., Plamenewski, B.A.: Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. I. In: Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien, vol. 82. Akademie, Berlin (1991)Google Scholar
  18. 18.
    Nazarov, S.A., Sokolowski, J.: Spectral problems in shape optimization. Singular boundary perturbations. Asymptot. Anal. 56(3–4), 159–196 (2008)MathSciNetMATHGoogle Scholar
  19. 19.
    Nazarov, S.A., Sokolowski, J.: Shape sensitivity analysis of eigenvalues revisited. Control Cybern. 37(4), 999–1012 (2008)MathSciNetMATHGoogle Scholar
  20. 20.
    Osher, S., Fedkiw, R.: Level set methods and dynamic implicit surfaces. In: Applied Mathematical Sciences, vol. 153. Springer, New York (2003)Google Scholar
  21. 21.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Sokołowski, J., Żochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (electronic) (1999)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sokołowski, , Zolésio, J.-P.: Introduction to shape optimization. In: Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992). Shape sensitivity analysisGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsHumboldt-University of BerlinBerlinGermany
  2. 2.Department of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  3. 3.Laboratório Nacional de Computação Científica LNCC/MCTPetropolisBrazil

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