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

Roland Griesmaier

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 6, page 1193-1218
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Griesmaier, Roland. "A general perturbation formula for electromagnetic fields in presence of low volume scatterers." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1193-1218. <http://eudml.org/doc/276349>.

@article{Griesmaier2011,
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. },
author = {Griesmaier, Roland},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Perturbation formulas; electromagnetic scattering; low volume scatterers; asymptotic expansions; perturbation formulas, electromagnetic scattering, low volume scatterers},
language = {eng},
month = {7},
number = {6},
pages = {1193-1218},
publisher = {EDP Sciences},
title = {A general perturbation formula for electromagnetic fields in presence of low volume scatterers},
url = {http://eudml.org/doc/276349},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Griesmaier, Roland
TI - A general perturbation formula for electromagnetic fields in presence of low volume scatterers
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/7//
PB - EDP Sciences
VL - 45
IS - 6
SP - 1193
EP - 1218
AB - 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.
LA - eng
KW - Perturbation formulas; electromagnetic scattering; low volume scatterers; asymptotic expansions; perturbation formulas, electromagnetic scattering, low volume scatterers
UR - http://eudml.org/doc/276349
ER -

References

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  1. R.A. Adams, Sobolev Spaces, Pure Appl. Math.65. Academic Press, New York (1975).  
  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.  Zbl1132.78308
  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.  Zbl1036.35050
  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.  Zbl1149.35337
  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).  Zbl1220.35001
  6. H. Ammari and A. Khelifi, Electromagnetic scattering by small dielectric inhomogeneities. J. Math. Pures Appl.82 (2003) 749–842.  Zbl1033.78006
  7. H. Ammari, S. Moskow and M.S. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM: COCV9 (2003) 49–66.  Zbl1075.78010
  8. H. Ammari and J.K. Seo, An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math.30 (2003) 679–705.  Zbl1040.78008
  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.  Zbl1042.78002
  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.  Zbl1026.78005
  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.  Zbl0974.65113
  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 Problems25 (2009) 065004.  Zbl1173.35721
  14. E. Beretta and E. Francini, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneitiesContemp. Math.333, edited by G. Uhlmann and G. Alessandrini, Amer. Math. Soc., Providence (2003).  Zbl1148.35354
  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.  Zbl1089.78003
  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.  Zbl0974.78006
  17. M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math.93 (2003) 635–654.  Zbl1016.65079
  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.  Zbl1137.35346
  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.  Zbl1137.35347
  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).  Zbl1072.35198
  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.  Zbl1130.35011
  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 Problems14 (1998) 553–595.  Zbl0916.35132
  23. M. Cheney, The linear sampling method and the MUSIC algorithm. Inverse Problems17 (2001) 591–595.  Zbl0991.35105
  24. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. John Wiley & Sons, New York (1983).  Zbl0522.35001
  25. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Spectral Theory and Applications3. Springer-Verlag, Berlin (1990).  Zbl0683.35001
  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.  Zbl0684.35087
  27. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften224. 2nd edition, Springer-Verlag, Berlin (1998).  
  28. R. Griesmaier, An asymptotic factorization method for inverse electromagnetic scattering in layered media. SIAM J. Appl. Math.68 (2008) 1378–1403.  Zbl1156.35339
  29. R. Griesmaier, Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Probl. Imaging3 (2009) 389–403.  Zbl1194.78017
  30. R. Griesmaier, Reconstruction of thin tubular inclusions in three-dimensional domains using electrical impedance tomography. SIAM J. Imaging Sci.3 (2010) 340–362.  Zbl1193.78012
  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.  Zbl1179.78063
  32. R. Griesmaier and M. Hanke, MUSIC-characterization of small scatterers for normal measurement data. Inverse Problems25 (2009) 075012.  Zbl1167.35542
  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 Monographs102, translated by V. Minachin, American Mathematical Society, Providence, RI (1992).  
  35. A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Ser. Math. Appl.36. Oxford University Press, New York (2008).  Zbl1222.35001
  36. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000).  Zbl0948.35001
  37. P. Monk, Finite Element Methods for Maxwell's Equations. Numer. Math. Sci. Comput. Oxford University Press, New York (2003).  Zbl1024.78009
  38. F. Murat and L. Tartar, H-convergence, Progress in Nonlinear Differential Equations and Their Applications31, edited by A. Cherkaev and R. Kohn. Birkhäuser, Boston (1997).  Zbl0913.35020
  39. W.-K. Park and D. Lesselier, MUSIC-type imaging of a thin penetrable inclusion from its multi-static response matrix. Inverse Problems25 (2009) 075002.  Zbl1180.35571
  40. W. Rudin, Real and complex analysis. McGraw-Hill Book Co., New York (1966).  Zbl0142.01701
  41. W. Rudin, Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973).  
  42. D. Volkov, Numerical methods for locating small dielectric inhomogeneities. Wave Motion38 (2003) 189–206.  Zbl1163.74456
  43. C. Weber, Regularity theorems for Maxwell's equations. Math. Methods Appl. Sci.3 (1981) 523–536.  Zbl0477.35020

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