Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 877-897.

This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.

DOI : 10.1016/j.anihpc.2013.07.008
Classification : 35R30, 35C20
Mots clés : Generalized polarization tensors, Inhomogeneous conductivity, Neumann-to-Dirichlet map, Asymptotic expansion, Inverse conductivity problem
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     title = {Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {877--897},
     publisher = {Elsevier},
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Ammari, Habib; Deng, Youjun; Kang, Hyeonbae; Lee, Hyundae. Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 877-897. doi : 10.1016/j.anihpc.2013.07.008. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.008/

[1] H. Ammari, T. Boulier, J. Garnier, W. Jing, H. Kang, H. Wang, Target identification using dictionary matching of generalized polarization tensors, Found. Comput. Math. (2013), arXiv:1204.3035 | MR | Zbl

[2] H. Ammari, T. Boulier, J. Garnier, H. Kang, H. Wang, Tracking of a mobile target using generalized polarization tensors, SIAM J. Imaging Sci. 6 (2013), 1477 -1498 | MR | Zbl

[3] H. Ammari, J. Garnier, H. Kang, M. Lim, K. Sølna, Multistatic imaging of extended targets, SIAM J. Imaging Sci. 5 (2012), 564 -600 | MR | Zbl

[4] H. Ammari, J. Garnier, H. Kang, M. Lim, S. Yu, Generalized polarization tensors for shape description, Numer. Math. (2013), http://dx.doi.org/10.1007/s00211-013-0561-5 | Zbl

[5] H. Ammari, J. Garnier, K. Sølna, Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging, Proc. Am. Math. Soc. 141 (2013), 3431 -3446 | MR | Zbl

[6] H. Ammari, H. Kang, Properties of generalized polarization tensors, Multiscale Model. Simul. 1 (2003), 335 -348 | MR | Zbl

[7] H. Ammari, 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 no. 5 (2003), 1152 -1166 | MR | Zbl

[8] H. Ammari, H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lect. Notes Math. vol. 1846 , Springer-Verlag, Berlin (2004) | MR | Zbl

[9] H. Ammari, H. Kang, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Appl. Math. Sci. vol. 162 , Springer-Verlag, New York (2007) | MR | Zbl

[10] H. Ammari, H. Kang, Expansion methods, Handbook of Mathematical Mehtods of Imaging, Springer (2011), 447 -499 | Zbl

[11] H. Ammari, H. Kang, E. Kim, M. Lim, Reconstruction of closely spaced small inclusions, SIAM J. Numer. Anal. 42 (2005), 2408 -2428 | MR | Zbl

[12] H. Ammari, H. Kang, H. Lee, M. Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem, Commun. Math. Phys. 317 (2013), 485 -502 | MR | Zbl

[13] H. Ammari, H. Kang, K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal. 41 (2005), 119 -140 | MR | Zbl

[14] H. Ammari, H. Kang, M. Lim, H. Zribi, The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion, Math. Comput. 81 (2012), 367 -386 | MR | Zbl

[15] K. Astala, L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. Math. 163 (2006), 265 -299 | MR | Zbl

[16] G. Bao, S. Hou, P. Li, Recent studies on inverse medium scattering problems, Lect. Notes Comput. Sci. Eng. 59 (2007), 165 -186 | MR | Zbl

[17] J. Bikowski, K. Knudsen, J.L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Probl. 27 (2011) | MR | Zbl

[18] L. Borcea, Electrical impedance tomography, Inverse Probl. 18 (2002), R99 -R136 | MR | Zbl

[19] L. Borcea, G. Papanicolaou, F.G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sci. 1 (2008), 75 -114 | MR | Zbl

[20] M. Brühl, M. Hanke, M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities, Numer. Math. 93 no. 4 (2003), 635 -654 | MR | Zbl

[21] Y. Capdeboscq, A.B. Karrman, J.-C. Nédélec, Numerical computation of approximate generalized polarization tensors, Appl. Anal. 91 (2012), 1189 -1203 | MR | Zbl

[22] Y. Capdeboscq, M.S. Vogelius, A general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Math. Model. Numer. Anal. 37 (2003), 159 -173 | EuDML | Numdam | MR | Zbl

[23] M. Cheney, D. Isaacson, J.C. Newell, Electrical impedance tomography, SIAM Rev. 41 (1999), 85 -101 | MR | Zbl

[24] G. Dassios, R.E. Kleinman, On Kelvin inversion and low-frequency scattering, SIAM Rev. 31 (1989), 565 -585 | MR | Zbl

[25] H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht (1996) | MR | Zbl

[26] G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ (1976) | MR | Zbl

[27] A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989), 299 -326 | MR | Zbl

[28] H. Kang, J.K. Seo, Recent progress in the inverse conductivity problem with single measurement, Inverse Problems and Related Fields, CRC Press, Boca Raton, FL (2000), 69 -80 | MR | Zbl

[29] R.V. Kohn, H. Shen, M.S. Vogelius, M.I. Weinstein, Cloaking via change of variables in electric impedance tomography, Inverse Probl. 24 (2008) | MR | Zbl

[30] S.M. Kozlov, On the domain of variations of added masses, polarization and effective characteristics of composites, J. Appl. Math. Mech. 56 (1992), 102 -107 | MR | Zbl

[31] R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites, J. Mech. Phys. Solids 41 (1993), 809 -833 | MR | Zbl

[32] G.W. Milton, The Theory of Composites, Cambridge Monogr. Appl. Comput. Math. , Cambridge University Press (2001) | MR | Zbl

[33] L.J. Mueller, S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comput. 24 (2003), 1232 -1266 | MR | Zbl

[34] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Stud. vol. 27 , Princeton University Press, Princeton, NJ (1951) | MR | Zbl

[35] S. Nagayasu, G. Uhlmann, J.-N. Wang, Depth dependent stability estimates in electrical impedance tomography, Inverse Probl. 25 (2009) | MR | Zbl

[36] J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Appl. Math. Sci. vol. 144 , Springer-Verlag, New York (2001) | MR | Zbl

[37] S. Siltanen, J.L. Mueller, D. Isaacson, Reconstruction of high contrast 2-D conductivities by the algorithm of A. Nachman, Contemp. Math. vol. 278 , Amer. Math. Soc., Providence, RI (2001), 241 -254 | MR | Zbl

[38] J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125 (1987), 153 -169 | MR | Zbl

[39] G.C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572 -611 | MR | Zbl

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