A Neural Network Approach for Solving Inverse Problems in NDE

  • I. Elshafiey
  • L. Udpa
  • S. S. Udpa
Part of the Advances in Cryogenic Engineering book series (volume 28)


Solution to inverse problems is of interest in many fields of science and engineering. In nondestructive evaluation [1], for example, inverse techniques are used to obtain quantitative estimates of the size, shape and nature of defects in materials. Inv.:rse scattering problems in electromagnetics deal with estimation of scatterer information from knowledge of incident and scattered fields. Inverse problems are frequently described by Fredholm integral equations in the form
$$ \smallint _a^bk(x,y)z(y)dy = u(x) (c \leqslant x \leqslant d)$$
where u(x) represents the measured data, z(y) represents the source function or the system states or parameters, and k(x,y) represents the kernel of the transformation. The objective of inverse problem is then to solve for the source or state function from known measurements. This problem is sensitive to the system parameters z, to the shape of the kernel k, and to the accuracy of the measurements u.


Inverse Problem Fredholm Integral Equation Neural Network Approach Hopfield Neural Network Hopfield Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    L. Udpa and W. Lord, “A Discussion of the Inverse problem in Electromagnetic Nondestructive Testing,” Review of Progress in Quantitative NDE, edited by D. O. Thompson and D. E. Chimenti (Plenum Press, New York, 1986), Vol. 5A, pp.375–382.Google Scholar
  2. 2.
    I. Elshafiey, L. Udpa and S. S. Udpa, “A Neural Network Approach for Solving Integral Equations,” Proceedings of IEEE Int. Symposium on Circuits and Systems, Singapore, pp. 1416–1419 (1991).Google Scholar
  3. 3.
    B. R. Hunt, “The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computers.” IEEE Trans. Comput. Vol. C-22, No. 9, pp. 805–812 (1973).CrossRefGoogle Scholar
  4. 4.
    A. N. Tihonov, “Solution of Incorrectly Formulated Problems and the Regularization Method.” Soviet Mathematics, Vol. 4, p. 1035–1038 (1963).MathSciNetGoogle Scholar
  5. 5.
    J. J. Hopfield, “Neurons with Graded Response Have Collective Computational Properties like Those of Two-State Neurons,” Proc. Natl. Acad. Sei. USA, Vol. 81, p. 3088–3092 (1984).CrossRefGoogle Scholar
  6. 6.
    T. M. Habashy, W. C. Chew and E. Y. Chow, “Simultaneous Reconstruction of Permittivity and Conductivity Profiles in a Radially Inhomogeneous Slab.” Radio Science, Vol. 21, No. 4, p. 635–645 (1986).CrossRefGoogle Scholar
  7. 7.
    W. C. Chew, “Response of a Current Loop Antenna in an Invaded Borehole.” Geophysics, Vol. 49, No. 1, p. 81–91 (1984).CrossRefGoogle Scholar
  8. 8.
    S. Coen, K. M. Kenneth, and D. J. Angelakos, “Inverse Scattering Technique Applied to Remote Sensing of Layered Media.” IEEE Trans. Antennas Propagat., Vol. AP-29, No. 2, p. 298–306 (1981).CrossRefGoogle Scholar
  9. 9.
    D. W. Marquardt, “Solution of Nonlinear Chemical Engineering Models.” Chemical Engineering Progress, Vol. 55, No. 6, p. 65–70 (1959).Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • I. Elshafiey
    • 1
  • L. Udpa
    • 1
  • S. S. Udpa
    • 1
  1. 1.Department of Electrical Engineering and Computer Engineering and Center for NDEIowa State UniversityAmesUSA

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