An Initial Value Approach to the Inverse Helmholtz Problem at Fixed Frequency

  • Frank Natterer


The inverse Helmholtz problem calls for the determination of the function f in \(\Omega \subseteq \bold{R}^{n}\) from the boundary values \(g(\theta, x) = u(x), x \in \partial \Omega\) of the solution u of the Helmholtz equation
$$\bigtriangleup u(x)+k^{2}(1+f(x))u(x) = 0$$
$$u(x) = e^{ikx.\boldsymbol{\theta}}+w(x)$$
in R n, where w satisfies the Sommerfeld radiation condition. The number k > 0 is the frequency of the incident plane wave with direction \(\theta \in S^{n-1}\). We assume f = 0 outside Ω. g is measured for a single fixed frequency k and all \(\theta \in S^{n-1}\).


Inverse Problem Helmholtz Equation Hankel Function Incident Plane Wave Ultrasonic Image 
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  1. 1.
    Borup, D.T. — Johnson, S.A. — Kim, W.W. — Berggren, M.J.: Nonperturbative diffraction tomography via Gauss-Newton iteration applied to the scattering integral equation, Ultrasonic Imaging 14, 69–85 (1992).CrossRefGoogle Scholar
  2. 2.
    Colton, D. — Monk, P.: A modified dual space method for solving the electromagnetic inverse scattering problem for an infinite cylinder, Inverse Problems 10, 87–108 (1994).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Devaney, A.J.: A filtered backpropagation algorithm for diffraction tomography, Ultrasonic Imaging 4, 336–350 (1982).CrossRefGoogle Scholar
  4. 4.
    Gutman, S. — Klibanov, M.: Regularized Quasi-Newton method for inverse scattering problems, Math. Comput. Modelling 18, No. 1, pp. 5–31, Pergamon Press Ltd. (1993).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hahn, W.: Stability of motion. Springer 1967.Google Scholar
  6. 6.
    Hartman, P.: Ordinary differential equations, Wiley 1964.Google Scholar
  7. 7.
    Kak, A.C. — Slaney, M.: Principles Computerized Tomographic Imaging, IEEE Press, New York 1987.Google Scholar
  8. 8.
    Kleinman, R.E. — van den Berg, P.M.: A modified gradient method for twodimensional problems in tomography, J. Comp. Appl. Math. 42, 17–35 (1992).MATHCrossRefGoogle Scholar
  9. 9.
    Morse, P.M. — Feshbach, H.: Methods of theoretical physics, McGraw-Hill 1953.Google Scholar
  10. 10.
    Nachman, A.I.: Global uniqueness for a two-dimensional inverse boundary value problem. Department of Mathematics, Preprint Series, Number 19, University of Rochester (1993).Google Scholar
  11. 11.
    Natterer, F.: Finite Difference Methods for Inverse Problems, in Ang, D.D. et al. (eds.): Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology. Publications of the HoChiMinh City Mathematical Society, Vol. 2, 1995.Google Scholar
  12. 12.
    Natterer, F. — Wübbeling, F.: A propagation — backpropagation method for ultrasound tomography. Inverse Problems 11, 1225–1232 (1995).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Stenger, F. — O’Reilly, M.: Sinc inversion of the Helmholtz equation without computing the forward solution, Preprint, Department of Computer Science, University of Utah, Salt Lake City, Utah 84112.Google Scholar

Copyright information

© Springer-Verlag Wien 1997

Authors and Affiliations

  • Frank Natterer
    • 1
  1. 1.Institut für Numerische und instrumentelle MathematikUniversität MünsterMünsterGermany

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