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A Calderón problem with frequency-differential data in dispersive media


Authors: Sungwhan Kim and Alexandru Tamasan
Journal: Proc. Amer. Math. Soc. 144 (2016), 1265-1276
MSC (2010): Primary 35R30, 35J65, 65N21
DOI: https://doi.org/10.1090/proc12788
Published electronically: July 8, 2015
MathSciNet review: 3447677
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of identifying a complex valued coefficient $ \gamma (x,\omega )$ in the conductivity equation $ \nabla \cdot \gamma (\cdot ,\omega )\nabla u(\cdot ,\omega )=0$ from knowledge of the frequency differentials of the Dirichlet-to-Neumann map. For a frequency analytic $ \gamma (\cdot ,\omega )=\sum _{k=0}^\infty (\sigma _k+i\epsilon _k)\omega ^k$, in three dimensions and higher, we show that $ \left .\frac {d^j}{d\omega ^j}\Lambda _{\gamma (\cdot ,\omega )}\right \vert _{\omega =0}$ for $ j=0,1,\dots ,N$ recovers $ \sigma _0,\dots , \sigma _N$ and $ \epsilon _1,\dots ,\epsilon _N$. This problem arises in frequency differential electrical impedance tomography of dispersive media.


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  • [1] Kari Astala and Lassi Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), no. 1, 265-299. MR 2195135 (2007b:30019), https://doi.org/10.4007/annals.2006.163.265
  • [2] R. H. Bayford, Bioimpedance Tomography (Electrical impedance tomography), Annu. Rev. Biomed. Eng., 8 (2006), 63-91.
  • [3] Russell M. Brown and Gunther A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations 22 (1997), no. 5-6, 1009-1027. MR 1452176 (98f:35155), https://doi.org/10.1080/03605309708821292
  • [4] Liliana Borcea, Electrical impedance tomography, Inverse Problems 18 (2002), no. 6, R99-R136. MR 1955896, https://doi.org/10.1088/0266-5611/18/6/201
  • [5] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), no. 1, 19-33. MR 2387648 (2008m:30049), https://doi.org/10.1515/jiip.2008.002
  • [6] Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65-73. MR 590275 (81k:35160)
  • [7] Margaret Cheney, David Isaacson, and Jonathan C. Newell, Electrical impedance tomography, SIAM Rev. 41 (1999), no. 1, 85-101 (electronic). MR 1669729 (99k:78017), https://doi.org/10.1137/S0036144598333613
  • [8] K. R. Foster and H. P. Schwan, Dielectric properties of tissues and biological materials, a critical review, Crit. Rev. Biomed. Eng. 17 (1989), no. 1, 25-104.
  • [9] Elisa Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems 16 (2000), no. 1, 107-119. MR 1741230 (2001b:35292), https://doi.org/10.1088/0266-5611/16/1/309
  • [10] D. Holder, Electrical Impedance Tomography: Methods, History and Applications (Bristol, UK: IOP Publishing), 2005.
  • [11] Sungwhan Kim, Jeehyun Lee, Jin Keun Seo, Eung Je Woo, and Habib Zribi, Multifrequency trans-admittance scanner: mathematical framework and feasibility, SIAM J. Appl. Math. 69 (2008), no. 1, 22-36. MR 2447937 (2009i:35340), https://doi.org/10.1137/070683593
  • [12] Sungwhan Kim, Jin Keun Seo, and Taeyoung Ha, A nondestructive evaluation method for concrete voids: frequency differential electrical impedance scanning, SIAM J. Appl. Math. 69 (2009), no. 6, 1759-1771. MR 2496716 (2010e:35304), https://doi.org/10.1137/08071925X
  • [13] Sungwhan Kim, Eun Jung Lee, Eung Je Woo, and Jin Keun Seo, Asymptotic analysis of the membrane structure to sensitivity of frequency-difference electrical impedance tomography, Inverse Problems 28 (2012), no. 7, 075004, 17. MR 2944955, https://doi.org/10.1088/0266-5611/28/7/075004
  • [14] Kim Knudsen and Alexandru Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations 29 (2004), no. 3-4, 361-381. MR 2041600 (2005e:35250), https://doi.org/10.1081/PDE-120030401
  • [15] Robert Kohn and Michael Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), no. 3, 289-298. MR 739921 (85f:80008), https://doi.org/10.1002/cpa.3160370302
  • [16] Adrian I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), no. 3, 531-576. MR 970610 (90i:35283), https://doi.org/10.2307/1971435
  • [17] Adrian I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71-96. MR 1370758 (96k:35189), https://doi.org/10.2307/2118653
  • [18] J. K. Seo, J. Lee, S. W. Kim, H. Zribi, and E. J. Woo, Frequency-difference electrical impedance tomography (fdEIT): algorithm development and feasibility study, Inverse Problems, 29 (2008), 929-944.
  • [19] Jin Keun Seo, Bastian Harrach, and Eung Je Woo, Recent progress on frequency difference electrical impedance tomography, Mathematical methods for imaging and inverse problems, ESAIM Proc., vol. 26, EDP Sci., Les Ulis, 2009, pp. 150-161. MR 2498145 (2011e:78023), https://doi.org/10.1051/proc/2009011
  • [20] Erkki Somersalo, Margaret Cheney, and David Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math. 52 (1992), no. 4, 1023-1040. MR 1174044 (93f:35052), https://doi.org/10.1137/0152060
  • [21] Sungwhan Kim and Alexandru Tamasan, On a Calderón problem in frequency differential electrical impedance tomography, SIAM J. Math. Anal. 45 (2013), no. 5, 2700-2709. MR 3101088, https://doi.org/10.1137/130904739
  • [22] John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153-169. MR 873380 (88b:35205), https://doi.org/10.2307/1971291
  • [23] G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 1-39.

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Additional Information

Sungwhan Kim
Affiliation: Division of Liberal Arts, Hanbat National University, Korea
Email: sungwhan@hanbat.ac.kr

Alexandru Tamasan
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: tamasan@math.ucf.edu

DOI: https://doi.org/10.1090/proc12788
Keywords: Calder\'on problem, frequency differential electrical impedance tomography, complex geometrical optics
Received by editor(s): October 6, 2014
Received by editor(s) in revised form: March 22, 2015
Published electronically: July 8, 2015
Additional Notes: The second author was supported in part by the NSF Grant DMS 1312883.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2015 American Mathematical Society

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