Application of nonlinear techniques to the inverse problem

  • V. H. Weston
Inverse Scattering Theory and Related Topics
Part of the Lecture Notes in Physics book series (LNP, volume 130)


The inverse problem for the reduced wave equation Δu + k2n2(x)u = 0, where n is real and continuous and n2−1 has compact support in ℝ3, is examined for the case where the scattering data consists of a set of measurements of the near or far field produced by a prescribed incident wave. The inverse problem is formu lated in terms of a system of functional equations, a quadratic nonlinear integral equation, plus an additional inequality or constraint. The general nonlinear theory of the complete system is examined.


Inverse Problem Integral Operator Generalize Inverse Incident Field Nonlinear Integral Equation 
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Footnotes and references

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    R. Leis: Vorlesungen über partielle Differentialgleichungen zweiter Ordunung (Bibliographisches Institut, Mannheim, Germany, 1967)Google Scholar
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    As a functional of v, these are nonlinear functional equations, since u depends on v. Otherwise, they are a linear functional of the product vu.Google Scholar
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    J.W. Hilgers: “Non-iterative methods for solving operator equations of the first kind,” MRC. Report 1413 (1974)Google Scholar
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    M.Z. Nashed: Generalized Inverses and Applications (Academic, New York, 1976)Google Scholar
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    G. Strang, G. Fix: An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, New Jersey, 1973)Google Scholar
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    M.M. Vainberg, V.A. Trenogin: Theory of Branching of Solutions of Non-linear Equations (Noordhoff, Groninger, 1974)Google Scholar
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    L.B. Rall: SIAM Rev. 11, 386 (1969)Google Scholar
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    M.M. Vainberg: Variational Methods for the Study of Nonlinear Operators (Holden-Day, San Francisco, 1964)Google Scholar
  9. [9]
    Unless otherwise specified the norm is the uniform (max) norm. Better estimates can be obtained using the L 2(D) normGoogle Scholar
  10. [10]
    D.W. Decker: “Topics in Bifurcation Theory,” thesis, Cal Tech (1978)Google Scholar
  11. [11]
    M.A. Krasnosel'skii: Topological Methods in the Theory of Nonlinear Integral Equations (Pergamon, New York, 1964)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • V. H. Weston
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest Lafayette

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