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Application of nonlinear techniques to the inverse problem

  • Inverse Scattering Theory and Related Topics
  • Conference paper
  • First Online:
Book cover Mathematical Methods and Applications of Scattering Theory

Part of the book series: Lecture Notes in Physics ((LNP,volume 130))

Abstract

The inverse problem for the reduced wave equation Δu + k 2 n 2(x)u = 0, where n is real and continuous and n 2−1 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.

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Footnotes and references

  1. R. Leis: Vorlesungen über partielle Differentialgleichungen zweiter Ordunung (Bibliographisches Institut, Mannheim, Germany, 1967)

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  2. 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.

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  3. J.W. Hilgers: “Non-iterative methods for solving operator equations of the first kind,” MRC. Report 1413 (1974)

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  4. M.Z. Nashed: Generalized Inverses and Applications (Academic, New York, 1976)

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  5. G. Strang, G. Fix: An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, New Jersey, 1973)

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  6. M.M. Vainberg, V.A. Trenogin: Theory of Branching of Solutions of Non-linear Equations (Noordhoff, Groninger, 1974)

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  7. L.B. Rall: SIAM Rev. 11, 386 (1969)

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  8. M.M. Vainberg: Variational Methods for the Study of Nonlinear Operators (Holden-Day, San Francisco, 1964)

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  9. Unless otherwise specified the norm is the uniform (max) norm. Better estimates can be obtained using the L 2(D) norm

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  10. D.W. Decker: “Topics in Bifurcation Theory,” thesis, Cal Tech (1978)

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  11. M.A. Krasnosel'skii: Topological Methods in the Theory of Nonlinear Integral Equations (Pergamon, New York, 1964)

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Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, 47907, West Lafayette, Indiana

    V. H. Weston

Authors
  1. V. H. Weston
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Editor information

John A. DeSanto Albert W. Sáenz Woodford W. Zachary

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© 1980 Springer-Verlag

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Weston, V.H. (1980). Application of nonlinear techniques to the inverse problem. In: DeSanto, J.A., Sáenz, A.W., Zachary, W.W. (eds) Mathematical Methods and Applications of Scattering Theory. Lecture Notes in Physics, vol 130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-10023-7_125

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  • DOI: https://doi.org/10.1007/3-540-10023-7_125

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10023-2

  • Online ISBN: 978-3-540-38184-6

  • eBook Packages: Springer Book Archive

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