Skip to main content
SpringerLink
Log in

Low Frequency Electromagnetic Fields in High Contrast Media

  • Chapter
Surveys on Solution Methods for Inverse Problems

  • 623 Accesses

Abstract

Using variational principles we construct discrete network approximations for the Dirichlet to Neumann or Neumann to Dirichlet maps of high contrast, low frequency electromagnetic media.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allers, A., Santosa, F.: Stability and resolution analysis of a linearized problem in electrical impedance tomography, Inverse Problems 7 (1990) 515–533.

    Article  MathSciNet  Google Scholar 

  2. Alessandrini, G.: Stable determination of conductivity by boundary measurements, App. Anal. 27 (1988) 153–172.

    Article  MATH  MathSciNet  Google Scholar 

  3. Alumbaugh, D.L., Morrison, H. F.: Electromagnetic conductivity imaging with an iterative Born inversion, IEEE Transactions on Geosciences and Remote Sensing, 31 (1993) 758–763.

    Article  Google Scholar 

  4. Alumbaugh, L., Morrison, H. F.: Monitoring subsurface changes over time with cross-well electromagnetic tomography, Geophysical Prospecting, 43 (1995) 873–902.

    Article  Google Scholar 

  5. Bender, C. M., Orszag, S. A.: Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, Inc., New York, 1978.

    MATH  Google Scholar 

  6. Berryman, J. G., Kohn, R. V.: Variational constraints for electrical impedance tomography, Phys. Rev. Lett. 65 (1990) 325–328.

    Article  Google Scholar 

  7. Berryman, J. G.: Convexity properties of inverse problems with variational constraints, J. Franklin Inst. 328 (1991) 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  8. Borcea, L.: Asymptotic Analysis of Quasistatic Transport in High Contrast Conductive Media, SIAM J. Appl. Math. 59 (1999) 597–639.

    Article  MATH  MathSciNet  Google Scholar 

  9. Borcea, L., Berryman, J. G., Papanicolaou, G. C: High Contrast Impedance Tomography, Inverse Problems 12 (1996) 935–958.

    Article  MathSciNet  Google Scholar 

  10. Borcea, L., Berryman, J. G., Papanicolaou, G. C: Matching pursuit for imaging high contrast conductive media, Inverse Problems 15 (1999) 811–849.

    Article  MATH  MathSciNet  Google Scholar 

  11. Borcea, L., Papanicolaou, G. C: Network approximation for transport properties of high contrast materials, SIAM J. Appl. Math 58 (1998) 501–539.

    Article  MATH  MathSciNet  Google Scholar 

  12. Borcea L., Papanicolaou, G.C.: A Hybrid Numerical Method for High Contrast Conductivity Problems, Journal of Computational and Applied Mathematics 87 (1997) 61–78.

    Article  MATH  MathSciNet  Google Scholar 

  13. Calderón, A. P.: On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica Rinftyo de Janeiro (1980) 65–73.

    Google Scholar 

  14. Cheney, M., Isaacson, D.: Distinguishability in impedance imaging, IEEE Trans. Biomed. Eng. 39 (1992) 852–860.

    Article  Google Scholar 

  15. Cherkaeva, E., Tripp, A.: Inverse conductivity problem for noisy measurements, Inverse Problems 12 (1996) 869–883.

    Article  MATH  MathSciNet  Google Scholar 

  16. Cherkaev, A. V., Gibiansky, L. V.: Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli, J. Math. Phys. 35 (1994) 127–145.

    Article  MATH  MathSciNet  Google Scholar 

  17. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I, Wiley, New York (1953) pp. 240–242 (Dirichlet’s principle) and pp. 267–268 (Thomson’s principle).

    Google Scholar 

  18. Curtis, E. B., Morrow, J. A.: Determining the resistors in a network, SIAM J. Appl. Math. 50 (1990) 918–930.

    Article  MATH  MathSciNet  Google Scholar 

  19. Curtis, E.B., Ingerman, D., Morrow, J. A.: Circular planar graphs and resistor networks, Linear Algebra Appl. 283 (1998) 115–150.

    Article  MATH  MathSciNet  Google Scholar 

  20. Dines, K. A., Lytle, R. J.: Analysis of electrical conductivity imaging, Geophysics 46 (1981) 1025–1036.

    Article  Google Scholar 

  21. Dobson, D. C., Santosa, F.: Resolution and stability analysis of an inverse problem in electrical impedance tomography: dependence on the input current patterns, SIAM J. Appl. Math. 54 (1994) 1542–1560.

    Article  MATH  MathSciNet  Google Scholar 

  22. Fannjiang, A., Papanicolaou, G. C: Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 (1994) 333–408.

    Article  MATH  MathSciNet  Google Scholar 

  23. Fucks, L. F., Cheney, M., Isaacson, D., Gisser, D. G., Newell, J. C: Detection and imaging of electric conductivity and permittivity at low frequency, IEEE Trans. Biomed. Eng. 38 (1991) 1106–1110.

    Article  Google Scholar 

  24. Grünbaum, F. A., Zubelli, J. P.: Diffuse tomography: computational aspects of the isotropic case, Inverse Problems 8 (1992) 421–433.

    Article  MATH  MathSciNet  Google Scholar 

  25. Habashy, T. M., Groom, R. W., Spies, B. R.: Beyond the Born and Rytov approximations: nonlinear approach to electromagnetic scattering, Journal of Geophysical Research 98 (1993) 1759–1775.

    Article  Google Scholar 

  26. Isaacson, D.: Distinguishability of conductivities by electric current computed tomography, IEEE Trans. Med. Imag. MI-5 (1986) 91–95.

    Article  Google Scholar 

  27. Isaacson, D., Cheney, D.: Current problems in impedance imaging, Inverse problems in partial differential equations, editors: Colton, D., Ewing, R., Rundell, W., SIAM (1990) 141–149.

    Google Scholar 

  28. Isakov, V.: Uniqueness and stability in multi-dimensional inverse problems, Inverse Problems 9 (1993) 579–621.

    Article  MATH  MathSciNet  Google Scholar 

  29. Isakov, V.: Inverse problems for partial differential equations, Springer-Verlag, New York (1998).

    MATH  Google Scholar 

  30. Ito, K., Kunisch, K.: The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM J. Control and Optimization 28 (1990) 113–136.

    Article  MATH  MathSciNet  Google Scholar 

  31. Jackson, J. D.: Classical Electrodynamics, second ed., Wiley, New York (1974).

    Google Scholar 

  32. Kallman, J. S., Berryman, J. G.: Weighted least-squares methods for electrical impedance tomography, IEEE Trans. Med. Imaging 11 (1992) 284–292.

    Article  Google Scholar 

  33. Keller, G. V.: Rock and Mineral Properties, Electromagnetic Methods in Applied Geophysics, Vol. 1, Theory, (ed. Nabighian, M. N.) (1988) 13–52.

    Google Scholar 

  34. Keller, J. B.: Conductivity of a Medium Containing a Dense Array of Perfectly Conducting Spheres or Cylinders or Nonconducting Cylinders, J. Appl. Phys. 34 (1963) 991–993.

    Article  MATH  Google Scholar 

  35. Keller, J. B.: Effective conductivity of periodic composites composed of two very unequal conductors, J. Math. Phys. 28 (1987) 2516–2520.

    Article  MATH  MathSciNet  Google Scholar 

  36. Kevorkian, J., Cole, J. D.: Multiple scale and singular perturbation methods, Springer Verlag, New York (1996).

    Google Scholar 

  37. Kohn, R. V., McKenney, A.: Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems 6 (1990) 389–414.

    Article  MATH  MathSciNet  Google Scholar 

  38. Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements, Comm. Pure App. Math. 38 (1985) 643–667.

    Article  MATH  MathSciNet  Google Scholar 

  39. Kohn, R., Vogelius, M.: Relaxation of a variational method for impedance computed tomography, Comm. Pure Appl. Math. 40 (1987) 745–777.

    Article  MATH  MathSciNet  Google Scholar 

  40. Kozlov, S. M.: Geometric aspects of averaging, Russian Math. Surveys 44 (1989) 91–144.

    Article  MATH  MathSciNet  Google Scholar 

  41. Mallat, S., Zhang, Z.: Matching pursuit in a time-frequency dictionary, IEEE Trans. Signal Processing 41 (1993) 3397–3415.

    Article  MATH  Google Scholar 

  42. Nachman, A. I.: Global uniqueness for a two-dimensional inverse boundary problem, Annals of Mathematics 142 (1995) 71–96.

    MathSciNet  Google Scholar 

  43. Santosa, F., Vogelius, M.: A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math. 50 (1990) 216–243.

    Article  MATH  MathSciNet  Google Scholar 

  44. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125 (1987) 153–169.

    Article  MATH  MathSciNet  Google Scholar 

  45. Sylvester, J., Uhlmann, G.: The Dirichlet to Neumann map and applications, Inverse problems in partial differential equations, editors: Colton, D., Ewing, R., Rundell, W., SIAM (1990) 101–139.

    Google Scholar 

  46. Wexler, A., Pry, B., Neuman, M.: Impedance-computed tomography algorithm and system, Appl. Opt. 24 (1985) 3985–3982.

    Article  Google Scholar 

  47. Yorkey, T. J., Webster, J. G., Tompkins, W. J.: Comparing reconstruction algorithms for electrical impedance tomography, IEEE Trans. Biomed. Eng. BME-34 (1987) 843–852.

    Google Scholar 

  48. Zhou, Q., Becker, A., Morrison, H. F.: Audio-frequency electromagnetic tomography in 2-D, Geophysics (1993) 482–495.

    Google Scholar 

  49. Ziman, J. M.: Principles of the Theory of Solids, Cambridge University Press (1971).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computational and Applied Mathematics, Rice University, MS 134, 6100 Main Street, Houston, TX, 77005-1892, USA

    Liliana Borcea

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    George C. Papanicolaou

Authors
  1. Liliana Borcea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George C. Papanicolaou
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    David Colton

  2. Institut für Mathematik, Johannes-Kepler-Universität, Linz, Austria

    Heinz W. Engl

  3. Fachbereich Mathematik, Universität des Saarlandes, Saarbrücken, Federal Republic of Germany

    Alfred K. Louis

  4. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, USA

    Joyce R. McLaughlin

  5. Department of Mathematics, Texas A & M University, College Station, Texas, USA

    William Rundell

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag/Wien

About this chapter

Cite this chapter

Borcea, L., Papanicolaou, G.C. (2000). Low Frequency Electromagnetic Fields in High Contrast Media. In: Colton, D., Engl, H.W., Louis, A.K., McLaughlin, J.R., Rundell, W. (eds) Surveys on Solution Methods for Inverse Problems. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6296-5_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-6296-5_11

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83470-1

  • Online ISBN: 978-3-7091-6296-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation