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November  2007, 18(4): 627-636. doi: 10.3934/dcds.2007.18.627

Rapid perturbational calculations for the Helmholtz equation in two dimensions

Sang-Yeun Shim 1, , Marcos Capistran 2, and  Yu Chen 1,

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1110, United States, United States
2. 

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

Received  May 2006 Revised  April 2007 Published  May 2007

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.
Keywords: scattering matrix, merging formulae, fast direct algorithms..
Mathematics Subject Classification: Primary: 65R20, 49N45; Secondary: 45Q0.
Citation: Sang-Yeun Shim, Marcos Capistran, Yu Chen. Rapid perturbational calculations for the Helmholtz equation in two dimensions. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 627-636. doi: 10.3934/dcds.2007.18.627
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