Comment on A.P. Calderón's paper: ``On an inverse boundary value problem'' [in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 6573, Soc. Brasil. Mat., Rio de Janeiro, 1980; MR0590275 (81k:35160)]
Authors:
David Isaacson and Eli L. Isaacson
Journal:
Math. Comp. 52 (1989), 553559
MSC:
Primary 35R30; Secondary 35K60
MathSciNet review:
962208
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Abstract 
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Abstract: Calderón determined a method to approximate the conductivity of a conducting body in (for ) based on measurements of boundary data. The approximation is good in the norm provided that the conductivity is a small perturbation from a constant. We calculate the approximation exactly for the case of homogeneous concentric conducting disks in with different conductivities. Here, the difference in the conductivities is the perturbation. We show that the approximation yields precise information about the spatial variation of , even when the perturbation is large. This ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications.
 [1]
B. H. Brown, D. C. Barber & A. D. Seagar, "Applied potential tomography: possible clinical applications," Clin. Phys. Physiol. Meas., v. 6, 1985, pp. 109121.
 [2]
AlbertoP.
Calderón, On an inverse boundary value problem, (Rio
de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980,
pp. 65–73. MR 590275
(81k:35160)
 [3]
D.
G. Gisser, D.
Isaacson, and J.
C. Newell, Electric current computed tomography and
eigenvalues, SIAM J. Appl. Math. 50 (1990),
no. 6, 1623–1634. MR 1080512
(91j:35280), http://dx.doi.org/10.1137/0150096
 [4]
Eugene
Isaacson and Herbert
Bishop Keller, Analysis of numerical methods, John Wiley &
Sons Inc., New York, 1966. MR 0201039
(34 #924)
 [5]
R. V. Kohn & A. McKenney, "A computational method for electrical impedance tomography," Preprint, CIMS, 1988.
 [6]
R.
V. Kohn and M.
Vogelius, Determining conductivity by boundary measurements. II.
Interior results, Comm. Pure Appl. Math. 38 (1985),
no. 5, 643–667. MR 803253
(86k:35155), http://dx.doi.org/10.1002/cpa.3160380513
 [7]
Robert
V. Kohn and Michael
Vogelius, Relaxation of a variational method for impedance computed
tomography, Comm. Pure Appl. Math. 40 (1987),
no. 6, 745–777. MR 910952
(89h:35339), http://dx.doi.org/10.1002/cpa.3160400605
 [8]
Adrian
I. Nachman, Reconstructions from boundary measurements, Ann.
of Math. (2) 128 (1988), no. 3, 531–576. MR 970610
(90i:35283), http://dx.doi.org/10.2307/1971435
 [9]
J. C. Newell, D. G. Gisser & D. Isaacson, "An electric current tomograph," IEEE Trans. Biomed. Engrg., v. BME35, 1988, pp. 828832.
 [10]
A.
G. Ramm, Characterization of the scattering data in
multidimensional inverse scattering problem, Inverse problems: an
interdisciplinary study (Montpellier, 1986) Adv. Electron. Electron
Phys., Suppl. 19, Academic Press, London, 1987, pp. 153–167. MR 1005569
(90h:35245)
 [11]
F. Santosa & M. Vogelius, "A back projection algorithm for electrical impedance imaging," Preprint, Univ. of Maryland, 1988.
 [12]
John
Sylvester and Gunther
Uhlmann, A uniqueness theorem for an inverse boundary value problem
in electrical prospection, Comm. Pure Appl. Math. 39
(1986), no. 1, 91–112. MR 820341
(87j:35377), http://dx.doi.org/10.1002/cpa.3160390106
 [13]
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), http://dx.doi.org/10.2307/1971291
 [14]
T. J. Yorkey, J. G. Webster & W. J. Tompkins, "Comparing reconstruction algorithms for electrical impedance tomography," IEEE Trans. Biomed. Engrg., v. BME34, 1987, pp. 843851.
 [1]
 B. H. Brown, D. C. Barber & A. D. Seagar, "Applied potential tomography: possible clinical applications," Clin. Phys. Physiol. Meas., v. 6, 1985, pp. 109121.
 [2]
 A. P. Calderón, "On an inverse boundary value problem," in Seminar on Numerical Analysis and its Applications to Continuum Physics (W. H. Meyer and M. A. Raupp, eds.), Brazilian Math. Society, Rio de Janeiro, 1980, pp. 6573. MR 590275 (81k:35160)
 [3]
 D. G. Gisser, D. Isaacson & J. C. Newell, "Electric current computed tomography and eigenvalues," SIAM J. Appl. Math., 1989. (To appear.) MR 1080512 (91j:35280)
 [4]
 E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 0201039 (34:924)
 [5]
 R. V. Kohn & A. McKenney, "A computational method for electrical impedance tomography," Preprint, CIMS, 1988.
 [6]
 R. V. Kohn & M. Vogelius, "Determining conductivity by boundary measurements II. Interior results," Comm. Pure Appl. Math., v. 38, 1985, pp. 643667. MR 803253 (86k:35155)
 [7]
 R. V. Kohn & M. Vogelius, "Relaxation of a variational method for impedance computed tomography," Comm. Pure Appl. Math., v. 40, 1987, pp. 745777. MR 910952 (89h:35339)
 [8]
 A. I. Nachman, "Reconstructions from boundary measurements," Ann. of Math., v. 128, 1988, pp. 531576. MR 970610 (90i:35283)
 [9]
 J. C. Newell, D. G. Gisser & D. Isaacson, "An electric current tomograph," IEEE Trans. Biomed. Engrg., v. BME35, 1988, pp. 828832.
 [10]
 A. G. Ramm, "Multidimensional inverse problems," Preprint, Kansas State Univ., 1987. MR 1005569 (90h:35245)
 [11]
 F. Santosa & M. Vogelius, "A back projection algorithm for electrical impedance imaging," Preprint, Univ. of Maryland, 1988.
 [12]
 J. Sylvester & G. Uhlmann, "A uniqueness theorem for an inverse boundary value problem in electrical prospection," Comm. Pure Appl. Math., v. 39, 1986, pp. 91112. MR 820341 (87j:35377)
 [13]
 J. Sylvester &. G. Uhlmann, "A global uniqueness theorem for an inverse boundary value problem," Ann. of Math., v. 125, 1987, pp. 153169. MR 873380 (88b:35205)
 [14]
 T. J. Yorkey, J. G. Webster & W. J. Tompkins, "Comparing reconstruction algorithms for electrical impedance tomography," IEEE Trans. Biomed. Engrg., v. BME34, 1987, pp. 843851.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819890962208X
PII:
S 00255718(1989)0962208X
Article copyright:
© Copyright 1989 American Mathematical Society
