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A general perturbation formula for electromagnetic fields in presence of low volume scatterers

Published online by Cambridge University Press:  22 July 2011

Roland Griesmaier
Roland Griesmaier*
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA. [email protected]
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Abstract

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.

Keywords

Perturbation formulaselectromagnetic scatteringlow volume scatterersasymptotic expansions
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 6 , November 2011 , pp. 1193 - 1218
Copyright
© EDP Sciences, SMAI, 2011

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References

R.A. Adams, Sobolev Spaces, Pure Appl. Math. 65. Academic Press, New York (1975).
Ammari, H., Iakovleva, E., Lesselier, D. and Perrusson, G., MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions. SIAM J. Sci. Comput. 29 (2007) 674–709. CrossRef
Ammari, H. and 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 (2003) 1152–1166. CrossRef
Ammari, H. and Kang, H., Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities. J. Math. Anal. Appl. 296 (2004) 190–208. CrossRef
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).
Ammari, H. and Khelifi, A., Electromagnetic scattering by small dielectric inhomogeneities. J. Math. Pures Appl. 82 (2003) 749–842. CrossRef
Ammari, H., Moskow, S. and Vogelius, M.S., Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM: COCV 9 (2003) 49–66. CrossRef
Ammari, H. and Seo, J.K., An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30 (2003) 679–705. CrossRef
Ammari, H., Vogelius, M.S. and Volkov, D., 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. CrossRef
Ammari, H. and Volkov, D., 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.
Ammari, H. and Volkov, D., 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. CrossRef
Arnold, D.N., Falk, R.S. and Winther, R., Multigrid in H(div) and H(curl). Numer. Math. 85 (2000) 197–217. CrossRef
Beretta, E., Capdeboscq, Y., de Gournay, F. and Francini, E., Thin cylindrical conductivity inclusions in a 3-dimensional domain: a polarization tensor and unique determination from boundary data. Inverse Problems 25 (2009) 065004. CrossRef
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).
Beretta, E., Francini, E. and Vogelius, M.S., 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. CrossRef
Beretta, E., Mukherjee, A. and Vogelius, M.S., Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys. 52 (2001) 543–572. CrossRef
Brühl, M., Hanke, M. and Vogelius, M.S., A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93 (2003) 635–654. CrossRef
Capdeboscq, Y. and Vogelius, M.S., 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. CrossRef
Capdeboscq, Y. and Vogelius, M.S., Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. Math. Model. Numer. Anal. 37 (2003) 227–240. CrossRef
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).
Capdeboscq, Y. and Vogelius, M.S., Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities. Asymptot. Anal. 50 (2006) 175–204.
Cedio-Fengya, D., Moskow, S. and Vogelius, M.S., Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553–595. CrossRef
Cheney, M., The linear sampling method and the MUSIC algorithm. Inverse Problems 17 (2001) 591–595. CrossRef
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. John Wiley & Sons, New York (1983).
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Spectral Theory and Applications 3. Springer-Verlag, Berlin (1990).
Friedman, A. and Vogelius, M.S., Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Rational Mech. Anal. 105 (1989) 299–326. CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224. 2nd edition, Springer-Verlag, Berlin (1998).
Griesmaier, R., An asymptotic factorization method for inverse electromagnetic scattering in layered media. SIAM J. Appl. Math. 68 (2008) 1378–1403. CrossRef
Griesmaier, R., Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Probl. Imaging 3 (2009) 389–403. CrossRef
Griesmaier, R., Reconstruction of thin tubular inclusions in three-dimensional domains using electrical impedance tomography. SIAM J. Imaging Sci. 3 (2010) 340–362. CrossRef
Griesmaier, R. and Hanke, M., An asymptotic factorization method for inverse electromagnetic scattering in layered media II: A numerical study. Contemp. Math. 494 (2008) 61–79. CrossRef
Griesmaier, R. and Hanke, M., MUSIC-characterization of small scatterers for normal measurement data. Inverse Problems 25 (2009) 075012. CrossRef
Iakovleva, E., Gdoura, S., Lesselier, D. and Perrusson, G., Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging. IEEE Trans. Antennas Propag. 55 (2007) 2598–2609 CrossRef
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).
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Ser. Math. Appl. 36. Oxford University Press, New York (2008).
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000).
P. Monk, Finite Element Methods for Maxwell's Equations. Numer. Math. Sci. Comput. Oxford University Press, New York (2003).
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).
Park, W.-K. and Lesselier, D., MUSIC-type imaging of a thin penetrable inclusion from its multi-static response matrix. Inverse Problems 25 (2009) 075002. CrossRef
W. Rudin, Real and complex analysis. McGraw-Hill Book Co., New York (1966).
W. Rudin, Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973).
Volkov, D., Numerical methods for locating small dielectric inhomogeneities. Wave Motion 38 (2003) 189–206. CrossRef
Weber, C., Regularity theorems for Maxwell's equations. Math. Methods Appl. Sci. 3 (1981) 523–536. CrossRef