Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T20:44:05.967Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter

Published online by Cambridge University Press:  15 April 2002

Michael S. Vogelius  and
Darko Volkov
Michael S. Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. ([email protected])
Darko Volkov
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. ([email protected])
Get access

Abstract

We consider solutions to the time-harmonic Maxwell's Equationsof a TE (transverse electric) nature. For such solutions we providea rigorous derivation of the leading order boundary perturbationsresulting from the presence of a finite number of interior inhomogeneitiesof small diameter. We expect that these formulas will form the basis for very effective computational identification algorithms, aimed at determininginformation about the inhomogeneities from electromagnetic boundary measurements.

Keywords

Maxwell equationsinverse problems.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 723 - 748
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Ammari, S. Moskow and M. Vogelius, Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. Preprint, Rutgers University (1999); Inverse Problems (submitted).
P.M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations. Prentice-Hall, Englewood Cliffs, New Jersey (1971).
L. Baratchart, J. Leblond, F. Mandréa and E.B. Saff, How can meromorhic approximation help to solve some 2D inverse problems for the Laplacian? Inverse Problems 15 (1999) 79-90.
J. Blitz, Electrical and Magnetic Methods of Nondestructive Testing. IOP Publishing, Adam Hilger, New York (1991).
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
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. Krieger Publishing Co., Malabar, Florida (1992).
Dobson, D. and Santosa, F., Nondestructive evaluation of plates using eddy current methods. Internat. J. Engrg. Sci. 36 (1998) 395-409. CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Ed., Springer-Verlag, New York (1983).
D. Griffiths, Introduction to Electrodynamics, 2nd Ed., Prentice Hall, Upper Saddle River, New Jersey (1989).
Gylys-Colwell, F., An inverse problem for the Helmholtz equation. Inverse Problems 12 (1996) 139-156. CrossRef
J.D. Jackson, Classical Electrodynamics, 2nd Ed., Wiley, New York (1975).
Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements. Comm. Pure Appl. Math. 37 (1984) 289-298. II. Interior results. Comm. Pure Appl. Math. 38 (1985) 643-667. CrossRef
Lassas, M., The impedance imaging problem as a low-frequency limit. Inverse Problems 13 (1997) 1503-1518. CrossRef
N.N. Lebedev, Special Functions & Their Applications. Dover Publications, New York (1972).
Nachman, A., Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. 143 (1996) 71-96. CrossRef
Ola, P., Päivärinta, L. and Somersalo, E., An inverse boundary value problem in electrodynamics. Duke Math. J. 70 (1993) 617-653. CrossRef
A. Sahin and E.L. Miller, Electromagnetic scattering-based array processing methods for near-field object characterization. Preprint, Northeastern University (1998).
Somersalo, E., Isaacson, D. and Cheney, M., A linearized inverse boundary value problem for Maxwell's equations. J. Comput. Appl. Math. 42 (1992) 123-136. CrossRef
J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection. Comm. Pure Appl. Math. 39 (1986) 91-112.
Sylvester, J. and Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125 (1987) 153-169. CrossRef
G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed., Cambridge University Press, London (1962).