Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Calderón determined a method to approximate the conductivity $ \sigma $ of a conducting body in $ {R^n}$ (for $ n \geq 2$) based on measurements of boundary data. The approximation is good in the $ {L_\infty }$ 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 $ {R^2}$ 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 $ \sigma $, even when the perturbation is large. This ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications.


References [Enhancements On Off] (What's this?)

  • [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.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 35R30, 35K60

Retrieve articles in all journals with MSC: 35R30, 35K60


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1989-0962208-X
Article copyright: © Copyright 1989 American Mathematical Society