Calderón cavities inverse problem as a shape-from-moments problem

Munnier, Alexandre; Ramdani, Karim

doi:10.1090/qam/1505

Authors: Alexandre Munnier and Karim Ramdani
Journal: Quart. Appl. Math. 76 (2018), 407-435
MSC (2010): Primary 31A25, 45Q05, 65N21, 30E05
DOI: https://doi.org/10.1090/qam/1505
Published electronically: April 11, 2018
MathSciNet review: 3805035
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Abstract: In this paper, we address a particular case of Calderón’s (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e., heterogeneities of infinitely high conductivities). We aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non-iterative and uses two main ingredients. First, we show how to compute the so-called Generalized Pólia-Szegö tensors (GPST) of the cavities from the DtN of the cavities. Secondly, we show that the obtained shape from the GPST inverse problem can be transformed into a shape-from-moments problem, for some particular configurations. However, numerical results suggest that the reconstruction method is efficient for arbitrary geometries.

References

Ibrahim Akduman and Rainer Kress, Electrostatic imaging via conformal mapping, Inverse Problems 18 (2002), no. 6, 1659–1672. MR 1955911, DOI https://doi.org/10.1088/0266-5611/18/6/315
Habib Ammari and Hyeonbae Kang, Generalized polarization tensors, inverse conductivity problems, and dilute composite materials: a review, Inverse problems, multi-scale analysis and effective medium theory, Contemp. Math., vol. 408, Amer. Math. Soc., Providence, RI, 2006, pp. 1–67. MR 2262349, DOI https://doi.org/10.1090/conm/408/07685

H. Ammari and H. Kang, Polarization and moment tensors, Applied Mathematical Sciences, vol. 162, Springer, New York, 2007.

Habib Ammari, Josselin Garnier, Hyeonbae Kang, Mikyoung Lim, and Sanghyeon Yu, Generalized polarization tensors for shape description, Numer. Math. 126 (2014), no. 2, 199–224. MR 3150221, DOI https://doi.org/10.1007/s00211-013-0561-5

Habib Ammari and Hyeonbae Kang, Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics, vol. 1846, Springer-Verlag, Berlin, 2004. MR 2168949

Liliana Borcea, Electrical impedance tomography, Inverse Problems 18 (2002), no. 6, R99–R136. MR 1955896, DOI https://doi.org/10.1088/0266-5611/18/6/201

Laurent Bourgeois and Jérémi Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging 4 (2010), no. 3, 351–377. MR 2671101, DOI https://doi.org/10.3934/ipi.2010.4.351

Laurent Bourgeois and Jérémi Dardé, The “exterior approach” to solve the inverse obstacle problem for the Stokes system, Inverse Probl. Imaging 8 (2014), no. 1, 23–51. MR 3180411, DOI https://doi.org/10.3934/ipi.2014.8.23

Martin Brühl and Martin Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems 16 (2000), no. 4, 1029–1042. MR 1776481, DOI https://doi.org/10.1088/0266-5611/16/4/310

Fioralba Cakoni and David Colton, A qualitative approach to inverse scattering theory, Applied Mathematical Sciences, vol. 188, Springer, New York, 2014. MR 3137429

Fioralba Cakoni and Rainer Kress, Integral equation methods for the inverse obstacle problem with generalized impedance boundary condition, Inverse Problems 29 (2013), no. 1, 015005, 19. MR 3003012, DOI https://doi.org/10.1088/0266-5611/29/1/015005

Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR 590275

Y. Capdeboscq, J. Fehrenbach, F. de Gournay, and O. Kavian, Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements, SIAM J. Imaging Sci. 2 (2009), no. 4, 1003–1030. MR 2559157, DOI https://doi.org/10.1137/080723521

David Colton and Andreas Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems 12 (1996), no. 4, 383–393. MR 1402098, DOI https://doi.org/10.1088/0266-5611/12/4/003

David Colton, Michele Piana, and Roland Potthast, A simple method using Morozov’s discrepancy principle for solving inverse scattering problems, Inverse Problems 13 (1997), no. 6, 1477–1493. MR 1483999, DOI https://doi.org/10.1088/0266-5611/13/6/005

Raúl E. Curto and Lawrence A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), no. 4, 603–635. MR 1147276

Philip J. Davis, Plane regions determined by complex moments, J. Approximation Theory 19 (1977), no. 2, 148–153. MR 427637, DOI https://doi.org/10.1016/0021-9045%2877%2990037-5

Peter Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004. MR 2048384

Peter Ebenfelt, Björn Gustafsson, Dmitry Khavinson, and Mihai Putinar (eds.), Quadrature domains and their applications, Operator Theory: Advances and Applications, vol. 156, Birkhäuser Verlag, Basel, 2005.

Klaus Erhard and Roland Potthast, A numerical study of the probe method, SIAM J. Sci. Comput. 28 (2006), no. 5, 1597–1612. MR 2272180, DOI https://doi.org/10.1137/040607149

Avner Friedman and Michael Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rational Mech. Anal. 105 (1989), no. 4, 299–326. MR 973245, DOI https://doi.org/10.1007/BF00281494

Gene H. Golub, Peyman Milanfar, and James Varah, A stable numerical method for inverting shape from moments, SIAM J. Sci. Comput. 21 (1999/00), no. 4, 1222–1243. MR 1740393, DOI https://doi.org/10.1137/S1064827597328315

Björn Gustafsson, Chiyu He, Peyman Milanfar, and Mihai Putinar, Reconstructing planar domains from their moments, Inverse Problems 16 (2000), no. 4, 1053–1070. MR 1776483, DOI https://doi.org/10.1088/0266-5611/16/4/312

Björn Gustafsson, Mihai Putinar, Edward B. Saff, and Nikos Stylianopoulos, Bergman polynomials on an archipelago: estimates, zeros and shape reconstruction, Adv. Math. 222 (2009), no. 4, 1405–1460. MR 2554940, DOI https://doi.org/10.1016/j.aim.2009.06.010

Wolfgang Hackbusch, Integral equations, International Series of Numerical Mathematics, vol. 120, Birkhäuser Verlag, Basel, 1995. Theory and numerical treatment; Translated and revised by the author from the 1989 German original. MR 1350296

Houssem Haddar and Rainer Kress, Conformal mappings and inverse boundary value problems, Inverse Problems 21 (2005), no. 3, 935–953. MR 2146814, DOI https://doi.org/10.1088/0266-5611/21/3/009

H. Haddar and R. Kress, Conformal mapping and an inverse impedance boundary value problem, J. Inverse Ill-Posed Probl. 14 (2006), no. 8, 785–804. MR 2270700, DOI https://doi.org/10.1163/156939406779768319

H. Haddar and R. Kress, Conformal mapping and impedance tomography, Inverse Problems 26 (2010), no. 7, 074002, 18.

Houssem Haddar and Rainer Kress, A conformal mapping method in inverse obstacle scattering, Complex Var. Elliptic Equ. 59 (2014), no. 6, 863–882. MR 3195916, DOI https://doi.org/10.1080/17476933.2013.791687

Martin Hanke and Martin Brühl, Recent progress in electrical impedance tomography, Inverse Problems 19 (2003), no. 6, S65–S90. Special section on imaging. MR 2036522, DOI https://doi.org/10.1088/0266-5611/19/6/055

George C. Hsiao and Wolfgang L. Wendland, Boundary integral equations, Applied Mathematical Sciences, vol. 164, Springer-Verlag, Berlin, 2008. MR 2441884

Masaru Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations 23 (1998), no. 7-8, 1459–1474. MR 1642619, DOI https://doi.org/10.1080/03605309808821390

Masaru Ikehata, On reconstruction in the inverse conductivity problem with one measurement, Inverse Problems 16 (2000), no. 3, 785–793. MR 1766222, DOI https://doi.org/10.1088/0266-5611/16/3/314

M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inverse Ill-Posed Probl. 8 (2000), no. 4, 367–378. MR 1816720, DOI https://doi.org/10.1515/jiip.2000.8.4.367

M. Ikehata and S. Siltanen, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems 16 (2000), no. 4, 1043–1052. MR 1776482, DOI https://doi.org/10.1088/0266-5611/16/4/311

Olha Ivanyshyn and Rainer Kress, Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks, J. Integral Equations Appl. 18 (2006), no. 1, 13–38. MR 2264267, DOI https://doi.org/10.1216/jiea/1181075363

Hyeonbae Kang, Hyundae Lee, and Mikyoung Lim, Construction of conformal mappings by generalized polarization tensors, Math. Methods Appl. Sci. 38 (2015), no. 9, 1847–1854. MR 3353445, DOI https://doi.org/10.1002/mma.3195

Andreas Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr. 278 (2005), no. 3, 258–277. MR 2110531, DOI https://doi.org/10.1002/mana.200310239

Rainer Kress, Linear integral equations, 2nd ed., Applied Mathematical Sciences, vol. 82, Springer-Verlag, New York, 1999. MR 1723850

R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simulation 66 (2004), no. 4-5, 255–265. MR 2079577, DOI https://doi.org/10.1016/j.matcom.2004.02.006

Rainer Kress, Inverse problems and conformal mapping, Complex Var. Elliptic Equ. 57 (2012), no. 2-4, 301–316. MR 2886743, DOI https://doi.org/10.1080/17476933.2011.605446

Rainer Kress and William Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems 21 (2005), no. 4, 1207–1223. MR 2158105, DOI https://doi.org/10.1088/0266-5611/21/4/002

William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312

P. Milanfar, George C. Verghese, W. Clem Karl, and A.S. Willsky, Reconstructing polygons from moments with connections to array processing, Signal Processing, IEEE Transactions on 43 (1995), no. 2, 432–443.

Peyman Milanfar, Mihai Putinar, James Varah, Bjoern Gustafsson, and Gene H. Golub, Shape reconstruction from moments: theory, algorithms, and applications, Proc. SPIE, Advanced Signal Processing Algorithms, Architectures, and Implementations X, vol. 4116, 2000, pp. 406–416.

Alexandre Munnier and Karim Ramdani, Conformal mapping for cavity inverse problem: an explicit reconstruction formula, Appl. Anal. 96 (2017), no. 1, 108–129. MR 3581691, DOI https://doi.org/10.1080/00036811.2016.1208816

Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706

Roland Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems 22 (2006), no. 2, R1–R47. MR 2216404, DOI https://doi.org/10.1088/0266-5611/22/2/R01

Mihai Putinar, A two-dimensional moment problem, J. Funct. Anal. 80 (1988), no. 1, 1–8. MR 960218, DOI https://doi.org/10.1016/0022-1236%2888%2990060-2

Olaf Steinbach, Numerical approximation methods for elliptic boundary value problems, Springer, New York, 2008. Finite and boundary elements; Translated from the 2003 German original. MR 2361676

Guo Chun Wen, Conformal mappings and boundary value problems, Translations of Mathematical Monographs, vol. 106, American Mathematical Society, Providence, RI, 1992. Translated from the Chinese by Kuniko Weltin. MR 1187758