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Rapid perturbational calculations for the Helmholtz equation in two dimensions

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  • 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.

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