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Boundary integral formulae for the reconstruction of electricand electromagnetic inhomogeneities of small volume

Published online by Cambridge University Press:  15 September 2003

Habib Ammari ,
Shari Moskow  and
Michael S. Vogelius
Habib Ammari
Affiliation:
CMAP, École Polytechnique, 91128 Palaiseau, France.
Shari Moskow
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA.
Michael S. Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA; [email protected].
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Abstract

In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.

Keywords

Electromagnetic imagingsmall inhomogeneitiesnumerical reconstruction algorithms.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 49 - 66
Copyright
© EDP Sciences, SMAI, 2003

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References

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
Andrieux, S. and Ben Abda, A., Identification of planar cracks by complete overdetermined data: Inversion formulae. Inverse Problems 12 (1996) 553-563. CrossRef
Andrieux, S., Ben Abda, A. and Jaoua, M., On the inverse emerging plane crack problem. Math. Meth. Appl. Sci. 21 (1998) 895-907. 3.0.CO;2-1>CrossRef
Bui, H.D., Constantinescu, A. and Maigre, H., Diffraction acoustique inverse de fissure plane : solution explicite pour un solide borné. C. R. Acad. Sci. Paris Sér. II 327 (1999) 971-976.
Beretta, E., Mukherjee, A. and Vogelius, M., Asymptotic formuli for steady state voltage potentials in the presence of conductivity imperfection of small area. ZAMP 52 (2001) 543-572. CrossRef
M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Preprint (2001).
A.P. Calderon, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics. Soc. Brasileira de Matemática, Rio de Janeiro (1980) 65-73.
Cedio-Fengya, D.J., Moskow, S. and Vogelius, M.S., Identification of conductivity inperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553-595. CrossRef
I. Daubechies, Ten Lectures on Wavelets. SIAM, Philadelphia (1992).
G.B. Folland, Introduction to Partial Differential Equations. Princeton University Press, Princeton (1976).
Friedman, A. and Vogelius, M., Identification of Small Inhomogeneities of Extreme Conductivity by Boundary Measurements: A Theorem on Continuous Dependence. Arch. Rational Mech. Anal. 105 (1989) 299-326. CrossRef
He, S. and Romanov, V.G., Identification of small flaws in conductors using magnetostatic measurements. Math. Comput. Simul. 50 (1999) 457-471. CrossRef
M.S. Joshi and S.R. McDowall, Total determination of material parameters from electromagnetic boundary information. Pacific J. Math. (to appear).
Miller, K., Stabilized numerical analytic prolongation with poles. SIAM J. Appl. Math. 18 (1970) 346-363. CrossRef
Ola, P., Païvärinta, L. and Somersalo, E., An inverse boundary value problem in electrodynamics. Duke Math. J. 70 (1993) 617-653. CrossRef
M.S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. ESAIM: M 2 AN 34 (2000) 723-748.
D. Volkov, An inverse problem for the time harmonic Maxwell Equations, Ph.D. Thesis. Rutgers University (2001).