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Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

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Abstract

We propose a variant of the numerical method of steepest descent for oscillatory integrals by using a low-cost explicit polynomial approximation of the paths of steepest descent. A loss of asymptotic order is observed, but in the most relevant cases the overall asymptotic order remains higher than a truncated asymptotic expansion at similar computational effort. Theoretical results based on number theory underpinning the mechanisms behind this effect are presented.

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Authors and Affiliations

  1. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Andreas Asheim

  2. Department of Computer Science, KU Leuven, 3001, Leuven, Belgium

    Daan Huybrechs

Authors
  1. Andreas Asheim
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  2. Daan Huybrechs
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Correspondence to Andreas Asheim.

Additional information

Communicated by Arieh Iserles.

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Asheim, A., Huybrechs, D. Asymptotic Analysis of Numerical Steepest Descent with Path Approximations. Found Comput Math 10, 647–671 (2010). https://doi.org/10.1007/s10208-010-9068-y

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  • DOI: https://doi.org/10.1007/s10208-010-9068-y

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