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The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion

Authors: Habib Ammari, Hyeonbae Kang, Mikyoung Lim and Habib Zribi
Journal: Math. Comp. 81 (2012), 367-386
MSC (2010): Primary 35R30, 49Q10, 49Q12
Published electronically: August 15, 2011
MathSciNet review: 2833499
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Abstract: With each $ \mathcal{C}^2$-domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain and its material parameter. They generalize the concept of Polarization Tensor (PT), which can be seen as the first-order GPT. It is known that given an arbitrary shape, one can find an equivalent ellipse or ellipsoid with the same PT. In this paper we consider the problem of recovering finer details of the shape of a given domain using higher-order polarization tensors. We design an optimization approach which solves the problem by minimizing a weighted discrepancy functional. In order to compute the shape derivative of this functional, we rigorously derive an asymptotic expansion of the perturbations of the GPTs that are due to a small deformation of the boundary of the domain. Our derivations are based on the theory of layer potentials. We perform some numerical experiments to demonstrate the validity and the limitations of the proposed method. The results clearly show that our approach is very promising in recovering fine shape details.

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  • 1. H. Ammari, E. Beretta, E. Francini, H. Kang, and M. Lim,
    Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements,
    Math. Comp., 79 (2010), 1757-1777. MR 2630011
  • 2. H. Ammari, E. Beretta, E. Francini, H. Kang, and M. Lim,
    Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case,
    J. Math. Pures Appl., 94 (2010), 322-339. MR 2679030
  • 3. H. Ammari and H. Kang,
    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 (2003), 1152-1166. MR 2001663 (2004e:35027)
  • 4. H. Ammari and H. Kang,
    Properties of generalized polarization tensors,
    SIAM Multiscale Model. Simul., 1 (2003), 335-348. MR 1990200 (2004g:78029)
  • 5. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin, 2004. MR 2168949 (2006k:35295)
  • 6. H. Ammari and H. Kang, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York, 2007. MR 2327884 (2009f:35339)
  • 7. H. Ammari, H. Kang, E. Kim, and J.Y. Lee, The generalized polarization tensors for resolved imaging. Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements, Math. Comp., to appear.
  • 8. H. Ammari, H. Kang, E. Kim, and M. Lim, Reconstruction of closely spaced small inclusions, SIAM J. Numer. Anal., 42 (2005), 2408-2428. MR 2139399 (2005m:78011)
  • 9. H. Ammari, H. Kang, M. Lim, and H. Zribi, Conductivity interface problems. Part I: Small perturbations of an interface, Trans. Amer. Math. Soc., 362 (2010), 2435-2449. MR 2584606
  • 10. H. Ammari, H. Kang, and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymp. Anal., 41 (2005), 119-140. MR 2129229 (2006b:35333)
  • 11. G. Bao, F. Ma, and Y. Chen, An error estimate for recursive linearization of the inverse scattering problems, J. Math. Anal. Appl., 247 (2000), 255-271.
  • 12. L. Borcea, G. Papanicolaou, and F.G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sci., 1 (2008), 75-114. MR 2475826 (2010a:15028)
  • 13. M. Brühl, M. Hanke, and M.S. Vogelius,
    A direct impedance tomography algorithm for locating small inhomogeneities,
    Numer. Math., 93 (2003), 635-654. MR 1961882 (2004b:65169)
  • 14. Y. Chen and V. Rokhlin, On the inverse scattering problem for the Helmholtz equation in one dimension, Inverse Problems, 8 (1992), 365-391. MR 1166487 (93c:34034)
  • 15. R.R. Coifman, M. Goldberg, T. Hrycak, M. Israel, and V. Rokhlin, An improved operator expansion algorithm for direct and inverse scattering computations, Waves Random Media, 9 (1999), 441-457. MR 1705850 (2000f:65151)
  • 16. D. Colton and R. Kress,
    Integral Equation Methods in Scattering Theory,
    John Wiley & Sons Inc, 1983. MR 700400 (85d:35001)
  • 17. R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $ L^2$ pour les courbes Lipschitziennes, Ann. Math., 116 (1982), 361-387. MR 672839 (84m:42027)
  • 18. E.B. Fabes, M. Jodeit, and N.M. Riviére,
    Potential techniques for boundary value problems on $ \mathcal{C}^{1}$ domains,
    Acta Math., 141 (1978), 165-186. MR 501367 (80b:31006)
  • 19. A. Friedman and M.S. Vogelius,
    Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence,
    Arch. Rat. Mech. Anal., 105 (1989), 299-326. MR 973245 (90c:35198)
  • 20. G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, New Jersey, 1976. MR 0599578 (58:29031)
  • 21. V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877. MR 951742 (90f:35205)
  • 22. G.W. Milton,
    The Theory of Composites,
    Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2001. MR 1899805 (2003d:74077)
  • 23. H. Kang and J.K. Seo, Recent progress in the inverse conductivity problem with single measurement, in Inverse Problems and Related Fields, CRC Press, Boca Raton, FL, 69-80, 2000. MR 1761339 (2001f:35427)
  • 24. M. Lim, K. Louati, and H. Zribi,
    Reconstructing small perturbations of scatterers from electric or acoustic far-field measurements.
    Math. Meth. Appl. Sci., 31 (2008), 1315-1332. MR 2431429 (2009h:35449)
  • 25. G. Pólya and G. Szegö,
    Isoperimetric Inequalities in Mathematical Physics,
    Annals of Mathematical Studies Number 27, Princeton University Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
  • 26. G.C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. MR 769382 (86e:35038)

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Additional Information

Habib Ammari
Affiliation: Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France

Hyeonbae Kang
Affiliation: Department of Mathematics, Inha University, Incheon 402-751, Korea

Mikyoung Lim
Affiliation: Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea

Habib Zribi
Affiliation: Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea

Keywords: Generalized polarization tensor, asymptotic expansions, shape recovery
Received by editor(s): August 18, 2010
Received by editor(s) in revised form: December 2, 2010
Published electronically: August 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society