A general perturbation formula for electromagnetic fields in presence of low volume scatterers
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 6, p. 1193-1218

In several practically interesting applications of electromagnetic scattering theory like, e.g., scattering from small point-like objects such as buried artifacts or small inclusions in non-destructive testing, scattering from thin curve-like objects such as wires or tubes, or scattering from thin sheet-like objects such as cracks, the volume of the scatterers is small relative to the volume of the surrounding medium and with respect to the wave length of the applied electromagnetic fields. This smallness typically causes problems when solving direct scattering problems due to the need to discretize the objects and also when solving inverse scattering problems because small objects have very little effect on electromagnetic fields. In this paper we consider an asymptotic representation formula for scattered electromagnetic waves caused by low volume objects contained in some otherwise homogeneous three-dimensional bounded domain, assuming only that the scatterers are measurable and well-separated from the boundary of the domain. The formula yields a very general asymptotic model for electromagnetic scattering due to low volume objects that can either be used to simulate the corresponding electromagnetic fields or as the foundation of efficient reconstruction methods for inverse scattering problems with low volume scatterers. Our analysis extends results originally obtained in [Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal. 37 (2003) 159-173] for steady state voltage potentials to time-harmonic Maxwell's equations.

DOI : https://doi.org/10.1051/m2an/2011015
Classification:  35C20,  35Q60,  35J20
Keywords: perturbation formulas, electromagnetic scattering, low volume scatterers, asymptotic expansions
@article{M2AN_2011__45_6_1193_0,
author = {Griesmaier, Roland},
title = {A general perturbation formula for electromagnetic fields in presence of low volume scatterers},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {45},
number = {6},
year = {2011},
pages = {1193-1218},
doi = {10.1051/m2an/2011015},
zbl = {1277.78021},
mrnumber = {2833178},
language = {en},
url = {http://www.numdam.org/item/M2AN_2011__45_6_1193_0}
}

Griesmaier, Roland. A general perturbation formula for electromagnetic fields in presence of low volume scatterers. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 6, pp. 1193-1218. doi : 10.1051/m2an/2011015. http://www.numdam.org/item/M2AN_2011__45_6_1193_0/

[1] R.A. Adams, Sobolev Spaces, Pure Appl. Math. 65. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] H. Ammari, E. Iakovleva, D. Lesselier and G. Perrusson, MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions. SIAM J. Sci. Comput. 29 (2007) 674-709. | MR 2306264 | Zbl 1132.78308

[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 | Zbl 1036.35050

[4] H. Ammari and H. Kang, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities. J. Math. Anal. Appl. 296 (2004) 190-208. | MR 2070502 | Zbl 1149.35337

[5] H. Ammari and H. Kang, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Appl. Math. Sci. 162. Springer-Verlag, Berlin (2007). | MR 2327884 | Zbl 1220.35001

[6] H. Ammari and A. Khelifi, Electromagnetic scattering by small dielectric inhomogeneities. J. Math. Pures Appl. 82 (2003) 749-842. | MR 2005296 | Zbl 1033.78006

[7] H. Ammari, S. Moskow and M.S. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM: COCV 9 (2003) 49-66. | Numdam | MR 1957090 | Zbl 1075.78010

[8] H. Ammari and J.K. Seo, An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30 (2003) 679-705. | MR 1977849 | Zbl 1040.78008

[9] H. Ammari, M.S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. the full Maxwell equations. J. Math. Pures Appl. 80 (2001) 769-814. | MR 1860816 | Zbl 1042.78002

[10] H. Ammari and D. Volkov, Asymptotic formulas for perturbations in the eigenfrequencies of the full maxwell equations due to the presence of imperfections of small diameter. Asympt. Anal. 30 (2002) 331-350. | MR 1932037 | Zbl 1026.78005

[11] H. Ammari and D. Volkov, The leading order term in the asymptotic expansion of the scattering amplitude of a collection of finite number of dielectric inhomogeneities of small diameter. Int. J. Multiscale Comput. Engrg. 3 (2005) 149-160.

[12] D.N. Arnold, R.S. Falk and R. Winther, Multigrid in H(div) and H(curl). Numer. Math. 85 (2000) 197-217. | MR 1754719 | Zbl 0974.65113

[13] E. Beretta, Y. Capdeboscq, F. De Gournay and E. Francini, Thin cylindrical conductivity inclusions in a 3-dimensional domain: a polarization tensor and unique determination from boundary data. Inverse Problems 25 (2009) 065004. | MR 2506849 | Zbl 1173.35721

[14] E. Beretta and E. Francini, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities Contemp. Math. 333, edited by G. Uhlmann and G. Alessandrini, Amer. Math. Soc., Providence (2003). | MR 2032006 | Zbl 1148.35354

[15] E. Beretta, E. Francini and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. a rigorous error analysis. J. Math. Pures Appl. 82 (2003) 1277-1301. | MR 2020923 | Zbl 1089.78003

[16] E. Beretta, A. Mukherjee and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys. 52 (2001) 543-572. | MR 1856987 | Zbl 0974.78006

[17] 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 | Zbl 1016.65079

[18] Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal. 37 (2003) 159-173. | Numdam | MR 1972656 | Zbl 1137.35346

[19] Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. Math. Model. Numer. Anal. 37 (2003) 227-240. | Numdam | MR 1991198 | Zbl 1137.35347

[20] Y. Capdeboscq and M.S. Vogelius, A review of some recent work on impedance imaging for inhomogeneities of low volume fraction. Contemp. Math. 362, edited by C. Conca, R. Manasevich, G. Uhlmann and M.S. Vogelius, Amer. Math. Soc., Providence (2004). | MR 2091492 | Zbl 1072.35198

[21] Y. Capdeboscq and M.S. Vogelius, Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities. Asymptot. Anal. 50 (2006) 175-204. | MR 2294598 | Zbl 1130.35011

[22] D. Cedio-Fengya, S. Moskow and M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553-595. | MR 1629995 | Zbl 0916.35132

[23] M. Cheney, The linear sampling method and the MUSIC algorithm. Inverse Problems 17 (2001) 591-595. | MR 1861470 | Zbl 0991.35105

[24] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. John Wiley & Sons, New York (1983). | MR 700400 | Zbl 0522.35001

[25] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Spectral Theory and Applications 3. Springer-Verlag, Berlin (1990). | MR 1064315 | Zbl 0766.47001

[26] A. Friedman and M.S. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Rational Mech. Anal. 105 (1989) 299-326. | MR 973245 | Zbl 0684.35087

[27] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224. 2nd edition, Springer-Verlag, Berlin (1998). | Zbl 0562.35001

[28] R. Griesmaier, An asymptotic factorization method for inverse electromagnetic scattering in layered media. SIAM J. Appl. Math. 68 (2008) 1378-1403. | MR 2407129 | Zbl 1156.35339

[29] R. Griesmaier, Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Probl. Imaging 3 (2009) 389-403. | MR 2557912 | Zbl 1194.78017

[30] R. Griesmaier, Reconstruction of thin tubular inclusions in three-dimensional domains using electrical impedance tomography. SIAM J. Imaging Sci. 3 (2010) 340-362. | MR 2679431 | Zbl 1193.78012

[31] R. Griesmaier and M. Hanke, An asymptotic factorization method for inverse electromagnetic scattering in layered media II: A numerical study. Contemp. Math. 494 (2008) 61-79. | MR 2581766 | Zbl 1179.78063

[32] R. Griesmaier and M. Hanke, MUSIC-characterization of small scatterers for normal measurement data. Inverse Problems 25 (2009) 075012. | MR 2519864 | Zbl 1167.35542

[33] E. Iakovleva, S. Gdoura, D. Lesselier and G. Perrusson, Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging. IEEE Trans. Antennas Propag. 55 (2007) 2598-2609

[34] A.M. Il'In, Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs 102, translated by V. Minachin, American Mathematical Society, Providence, RI (1992). | MR 1182791 | Zbl 0754.34002

[35] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Ser. Math. Appl. 36. Oxford University Press, New York (2008). | MR 2378253 | Zbl 1222.35001

[36] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). | MR 1742312 | Zbl 0948.35001

[37] P. Monk, Finite Element Methods for Maxwell's Equations. Numer. Math. Sci. Comput. Oxford University Press, New York (2003). | MR 2059447 | Zbl 1024.78009

[38] F. Murat and L. Tartar, H-convergence, Progress in Nonlinear Differential Equations and Their Applications 31, edited by A. Cherkaev and R. Kohn. Birkhäuser, Boston (1997). | MR 1493039 | Zbl 0920.35019

[39] W.-K. Park and D. Lesselier, MUSIC-type imaging of a thin penetrable inclusion from its multi-static response matrix. Inverse Problems 25 (2009) 075002. | MR 2519854 | Zbl 1180.35571

[40] W. Rudin, Real and complex analysis. McGraw-Hill Book Co., New York (1966). | MR 210528 | Zbl 0278.26001

[41] W. Rudin, Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973). | MR 365062 | Zbl 0253.46001

[42] D. Volkov, Numerical methods for locating small dielectric inhomogeneities. Wave Motion 38 (2003) 189-206. | MR 1994816 | Zbl 1163.74456

[43] C. Weber, Regularity theorems for Maxwell's equations. Math. Methods Appl. Sci. 3 (1981) 523-536. | MR 657071 | Zbl 0477.35020