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A remedy for the failure of the numerical steepest descent method on a class of oscillatory integrals

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Abstract

In this paper we demonstrate that the numerical method of steepest descent fails when applied in a straight forward fashion to the most commonly occurring highly oscillatory integrals in scattering theory. Through a polar change of variables, however, the integral can be reformulated so that it can be solved efficiently using a combination of oscillatory integration techniques and classical quadrature. The approach is described in detail and demonstrated numerically with some oscillatory integral examples. The numerical examples demonstrate that our approach avoids the failure in some special cases, such as in an acoustic scattering model oscillatory integral with observation point located in the illuminated region. This paves the way for using the framework of numerical steepest descent methods on a wider class of problems, like the 3D high frequency scattering from convex obstacles, up to now only handled in a satisfactory way by methods due to Ganesh and Hawkins (J Comp Phys 230:104–125, 2011).

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Notes

  1. For example, an Abel type regularisation will do: \(\int f(x)\mathrm{d}x:=\mathrm{lim}_{s\rightarrow 0}\int f(x)\mathrm{exp}(-s|x|^2)\mathrm{d} x\) [26].

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge , CB3 0WA, UK

    Andreas Asheim

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  1. Andreas Asheim
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Correspondence to Andreas Asheim.

Additional information

Communicated by Lothar Reichel.

Supported by the Norwegian Research Council’s Magne S. Espedahl Fellowship.

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Asheim, A. A remedy for the failure of the numerical steepest descent method on a class of oscillatory integrals. Bit Numer Math 54, 587–605 (2014). https://doi.org/10.1007/s10543-013-0463-z

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