December  2017, 11(6): 1027-1046. doi: 10.3934/ipi.2017047

Some remarks on the small electromagnetic inhomogeneities reconstruction problem

Sorbonne University, Université de Technologie de Compiègne, Laboratoire de Mathématiuqes Appliquées de Compiègne LMAC, 60205 Compiègne Cedex, France

* Corresponding author: Abdellatif El Badia

Received  October 2016 Revised  July 2017 Published  September 2017

This work considers the problem of recovering small electromagnetic inhomogeneities in a bounded domain $Ω \subset \mathbb{R}^3$, from a single Cauchy data, at a fixed frequency. This problem has been considered by several authors, in particular in [4]. In this paper, we revisit this work with the objective of providing another identification method and establishing stability results from a single Cauchy data and at a fixed frequency. Our approach is based on the asymptotic expansion of the boundary condition derived in [4] and the extension of the direct algebraic algorithm proposed in [1].

Citation: Batoul Abdelaziz, Abdellatif El Badia, Ahmad El Hajj. Some remarks on the small electromagnetic inhomogeneities reconstruction problem. Inverse Problems & Imaging, 2017, 11 (6) : 1027-1046. doi: 10.3934/ipi.2017047
References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, Journal of Mathematical Analysis and Applications, 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.  Google Scholar

[2]

H. AmmariM. S Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter Ⅱ, Journal de Mathématiques Pures et Appliquées, 80 (2001), 769-814.  doi: 10.1016/S0021-7824(01)01217-X.  Google Scholar

[3]

H. Ammari and H. Kang, A new method for reconstructing electromagnetic inhomogeneities of small volume, Inverse problems, 19 (2003), 63-71.  doi: 10.1088/0266-5611/19/1/304.  Google Scholar

[4]

H. Ammari and H. Kang, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, Journal of Mathematical Analysis and Applications, 296 (2004), 190-208.  doi: 10.1016/j.jmaa.2004.04.003.  Google Scholar

[5]

H. AmmariH. KangE. KimM. Lim and K. Louati, A direct algorithm for ultrasound imaging of internal corrosion, SIAM Journal on Numerical Analysis, 49 (2011), 1177-1193.  doi: 10.1137/100784710.  Google Scholar

[6]

M. BrühlM. Hanke and M. S Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities, Numerische Mathematik, 93 (2003), 635-654.  doi: 10.1007/s002110200409.  Google Scholar

[7]

D. J. Cedio-FengyaS. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595.  doi: 10.1088/0266-5611/14/3/011.  Google Scholar

[8]

M. CheneyD. Isaacson and J. C Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.  Google Scholar

[9]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[10]

A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663.  doi: 10.1088/0266-5611/16/3/308.  Google Scholar

[11]

A. El Badia and T. Nara, Inverse dipole source problem for time-harmonic Maxwell equations: algebraic algorithm and Hölder stability, Inverse Problems, 29 (2013), 015007, 19pp. doi: 10.1088/0266-5611/29/1/015007.  Google Scholar

[12]

A. El Badia and A. El Hajj, Stability estimates for an inverse source problem of Helmholtz's equation from single Cauchy data at a fixed frequency, Inverse Problems, 29 (2013), 125008, 20pp. doi: 10.1088/0266-5611/29/12/125008.  Google Scholar

[13]

A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Archive for Rational Mechanics and Analysis, 105 (1989), 299-326.  doi: 10.1007/BF00281494.  Google Scholar

[14]

P. C. Hansen, Rank-deficient and Discrete Ill-Posed Problems, Philadelphia, PA, 1998. doi: 10.1137/1.9780898719697.  Google Scholar

[15]

H. Kang and H. Lee, Identification of simple poles via boundary measurements and an application of EIT, Inverse Problems, 20 (2004), 1853-1863.  doi: 10.1088/0266-5611/20/6/010.  Google Scholar

[16]

A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040.  doi: 10.1088/0266-5611/18/4/306.  Google Scholar

[17]

O. KwonJ. K. Seo and J. R. Yoon, A real time algorithm for the location search of discontinuous conductivities with one measurement, Communications on Pure and Applied Mathematics, 55 (2002), 1-29.  doi: 10.1002/cpa.3009.  Google Scholar

[18]

T. D. MastA. I. Nachman and R. C. Waag, Focusing and imaging using eigenfunctions of the scattering operator, The Journal of the Acoustical Society of America, 102 (1997), 715-725.  doi: 10.1121/1.419898.  Google Scholar

[19]

M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, ESAIM: Mathematical Modelling and Numerical Analysis, 34 (2000), 723-748.  doi: 10.1051/m2an:2000101.  Google Scholar

[20]

D. Volkov, An Inverse Problem for the Time Harmonic Maxwell's Equations, Ph. D thesis, Rutgers The State University of New Jersey -New Brunswick, 2001.  Google Scholar

show all references

References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, Journal of Mathematical Analysis and Applications, 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.  Google Scholar

[2]

H. AmmariM. S Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter Ⅱ, Journal de Mathématiques Pures et Appliquées, 80 (2001), 769-814.  doi: 10.1016/S0021-7824(01)01217-X.  Google Scholar

[3]

H. Ammari and H. Kang, A new method for reconstructing electromagnetic inhomogeneities of small volume, Inverse problems, 19 (2003), 63-71.  doi: 10.1088/0266-5611/19/1/304.  Google Scholar

[4]

H. Ammari and H. Kang, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, Journal of Mathematical Analysis and Applications, 296 (2004), 190-208.  doi: 10.1016/j.jmaa.2004.04.003.  Google Scholar

[5]

H. AmmariH. KangE. KimM. Lim and K. Louati, A direct algorithm for ultrasound imaging of internal corrosion, SIAM Journal on Numerical Analysis, 49 (2011), 1177-1193.  doi: 10.1137/100784710.  Google Scholar

[6]

M. BrühlM. Hanke and M. S Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities, Numerische Mathematik, 93 (2003), 635-654.  doi: 10.1007/s002110200409.  Google Scholar

[7]

D. J. Cedio-FengyaS. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595.  doi: 10.1088/0266-5611/14/3/011.  Google Scholar

[8]

M. CheneyD. Isaacson and J. C Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.  Google Scholar

[9]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[10]

A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663.  doi: 10.1088/0266-5611/16/3/308.  Google Scholar

[11]

A. El Badia and T. Nara, Inverse dipole source problem for time-harmonic Maxwell equations: algebraic algorithm and Hölder stability, Inverse Problems, 29 (2013), 015007, 19pp. doi: 10.1088/0266-5611/29/1/015007.  Google Scholar

[12]

A. El Badia and A. El Hajj, Stability estimates for an inverse source problem of Helmholtz's equation from single Cauchy data at a fixed frequency, Inverse Problems, 29 (2013), 125008, 20pp. doi: 10.1088/0266-5611/29/12/125008.  Google Scholar

[13]

A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Archive for Rational Mechanics and Analysis, 105 (1989), 299-326.  doi: 10.1007/BF00281494.  Google Scholar

[14]

P. C. Hansen, Rank-deficient and Discrete Ill-Posed Problems, Philadelphia, PA, 1998. doi: 10.1137/1.9780898719697.  Google Scholar

[15]

H. Kang and H. Lee, Identification of simple poles via boundary measurements and an application of EIT, Inverse Problems, 20 (2004), 1853-1863.  doi: 10.1088/0266-5611/20/6/010.  Google Scholar

[16]

A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040.  doi: 10.1088/0266-5611/18/4/306.  Google Scholar

[17]

O. KwonJ. K. Seo and J. R. Yoon, A real time algorithm for the location search of discontinuous conductivities with one measurement, Communications on Pure and Applied Mathematics, 55 (2002), 1-29.  doi: 10.1002/cpa.3009.  Google Scholar

[18]

T. D. MastA. I. Nachman and R. C. Waag, Focusing and imaging using eigenfunctions of the scattering operator, The Journal of the Acoustical Society of America, 102 (1997), 715-725.  doi: 10.1121/1.419898.  Google Scholar

[19]

M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, ESAIM: Mathematical Modelling and Numerical Analysis, 34 (2000), 723-748.  doi: 10.1051/m2an:2000101.  Google Scholar

[20]

D. Volkov, An Inverse Problem for the Time Harmonic Maxwell's Equations, Ph. D thesis, Rutgers The State University of New Jersey -New Brunswick, 2001.  Google Scholar

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