Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume
ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003), p. 49-66

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.

DOI : https://doi.org/10.1051/cocv:2002071
Classification:  35J25,  35R30,  65R99
Keywords: electromagnetic imaging, small inhomogeneities, numerical reconstruction algorithms
@article{COCV_2003__9__49_0,
     author = {Ammari, Habib and Moskow, Shari and Vogelius, Michael S.},
     title = {Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     pages = {49-66},
     doi = {10.1051/cocv:2002071},
     zbl = {1075.78010},
     mrnumber = {1957090},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2003__9__49_0}
}
Ammari, Habib; Moskow, Shari; Vogelius, Michael S. Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003) pp. 49-66. doi : 10.1051/cocv:2002071. http://www.numdam.org/item/COCV_2003__9__49_0/

[1] 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

[2] S. Andrieux and A. Ben Abda, Identification of planar cracks by complete overdetermined data: Inversion formulae. Inverse Problems 12 (1996) 553-563. | MR 1413418 | Zbl 0858.35131

[3] S. Andrieux, A. Ben Abda and M. Jaoua, On the inverse emerging plane crack problem. Math. Meth. Appl. Sci. 21 (1998) 895-907. | MR 1634847 | Zbl 0916.35129

[4] H.D. Bui, A. Constantinescu and H. Maigre, Diffraction acoustique inverse de fissure plane : solution explicite pour un solide borné. C. R. Acad. Sci. Paris Sér. II 327 (1999) 971-976. | Zbl 0966.74038

[5] E. Beretta, A. Mukherjee and M. Vogelius, Asymptotic formuli for steady state voltage potentials in the presence of conductivity imperfection of small area. ZAMP 52 (2001) 543-572. | MR 1856987 | Zbl 0974.78006

[6] M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Preprint (2001). | MR 1961882 | Zbl 1016.65079

[7] 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. | MR 590275

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

[9] I. Daubechies, Ten Lectures on Wavelets. SIAM, Philadelphia (1992). | MR 1162107 | Zbl 0776.42018

[10] G.B. Folland, Introduction to Partial Differential Equations. Princeton University Press, Princeton (1976). | MR 599578 | Zbl 0325.35001

[11] A. Friedman and M. 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

[12] S. He and V.G. Romanov, Identification of small flaws in conductors using magnetostatic measurements. Math. Comput. Simul. 50 (1999) 457-471. | MR 1768346

[13] M.S. Joshi and S.R. Mcdowall, Total determination of material parameters from electromagnetic boundary information. Pacific J. Math. (to appear). | MR 1748184 | Zbl 1012.78012

[14] K. Miller, Stabilized numerical analytic prolongation with poles. SIAM J. Appl. Math. 18 (1970) 346-363. | MR 260143 | Zbl 0211.19303

[15] P. Ola, L. Païvärinta and E. Somersalo, An inverse boundary value problem in electrodynamics. Duke Math. J. 70 (1993) 617-653. | MR 1224101 | Zbl 0804.35152

[16] M.S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter 34 (2000) 723-748. | Numdam | MR 1784483 | Zbl 0971.78004

[17] D. Volkov, An inverse problem for the time harmonic Maxwell Equations, Ph.D. Thesis. Rutgers University (2001).