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), 65-73, Soc. Brasil. Mat., Rio de Janeiro, 1980; MR0590275 (81k:35160)]

Authors:
David Isaacson and Eli L. Isaacson

Journal:
Math. Comp. **52** (1989), 553-559

MSC:
Primary 35R30; Secondary 35K60

MathSciNet review:
962208

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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. 109-121.**[2]**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****[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**, 10.1137/0150096**[4]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[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**, 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**, 10.1002/cpa.3160400605**[8]**Adrian I. Nachman,*Reconstructions from boundary measurements*, Ann. of Math. (2)**128**(1988), no. 3, 531–576. MR**970610**, 10.2307/1971435**[9]**J. C. Newell, D. G. Gisser & D. Isaacson, "An electric current tomograph," IEEE*Trans. Biomed. Engrg.*, v. BME-35, 1988, pp. 828-832.**[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**, 10.2307/3146577**[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**, 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**, 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. BME-34, 1987, pp. 843-851.

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0962208-X

Article copyright:
© Copyright 1989
American Mathematical Society