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2007, Volume 18, Issue 4: 627-636. Doi: 10.3934/dcds.2007.18.627
This issue Previous Article An invariant set generated by the domain topology for parabolic semiflows with small diffusion Next Article Bifurcation of relaxation oscillations in dimension two

Rapid perturbational calculations for the Helmholtz equation in two dimensions

  • 1.

    Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1110

  • 2.

    Centro de Investigación en Matemáticas, CIMAT A. C., Jalisco s/n, Mineral de Valenciana, C.P. 36240, Guanajuato, Guanajuato

Received:  May 2006
Revised:  April 2007
Published:  October 2007
Abstract Related Papers Cited by
    Abstract
  • Existing approaches to the solution of the inverse scattering problems in two and three dimensions rely on linearization of the Helmholtz equation, which requires the knowledge of the Fr\'echet derivative of the far field with respect to the index of refraction. We present an efficient algorithm for this perturbational calculation in two dimensions. Our method is based on the merging and splitting procedures already established for the solution of the Lippmann-Schwinger equation [2], [3], [4]. For an $m$-by-$m$ wavelength problem, the algorithm obtains perturbations to scattered waves for $m$ distinct incident waves in $O(m^3)$ steps.
    Mathematics Subject Classification: Primary: 65R20, 49N45; Secondary: 45Q05.

    Citation:
    shu

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