article

Superinterpolation in Highly Oscillatory Quadrature

Authors:

Daan Huybrechs,

Sheehan OlverAuthors Info & Claims

Foundations of Computational Mathematics, Volume 12, Issue 2

Pages 203 - 228

Published: 01 April 2012 Publication History

Abstract

Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small number of critical points. We show that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order, but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary points as well, through a simple change of variables.

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Cited By

Majidian H(2019)Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformationsBIT10.1007/s10543-018-0730-059:1(155-181)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s10543-018-0730-0
Majidian H(2018)Automatic computing of oscillatory integralsNumerical Algorithms10.1007/s11075-017-0343-877:3(867-884)Online publication date: 1-Mar-2018
https://dl.acm.org/doi/10.1007/s11075-017-0343-8
Majidian H(2017)FilonClenshawCurtis formulas for highly oscillatory integrals in the presence of stationary pointsApplied Numerical Mathematics10.1016/j.apnum.2017.02.003117:C(87-102)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1016/j.apnum.2017.02.003
Show More Cited By

Superinterpolation in Highly Oscillatory Quadrature
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
    2. Numerical analysis
2. Theory of computation

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Published In

cover image Foundations of Computational Mathematics

Foundations of Computational Mathematics Volume 12, Issue 2

April 2012

123 pages

ISSN:1615-3375

EISSN:1615-3383

Issue’s Table of Contents

Copyright © Copyright © 2012 SFoCM.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 April 2012

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Cited By

Majidian H(2019)Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformationsBIT10.1007/s10543-018-0730-059:1(155-181)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s10543-018-0730-0
Majidian H(2018)Automatic computing of oscillatory integralsNumerical Algorithms10.1007/s11075-017-0343-877:3(867-884)Online publication date: 1-Mar-2018
https://dl.acm.org/doi/10.1007/s11075-017-0343-8
Majidian H(2017)FilonClenshawCurtis formulas for highly oscillatory integrals in the presence of stationary pointsApplied Numerical Mathematics10.1016/j.apnum.2017.02.003117:C(87-102)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1016/j.apnum.2017.02.003
He GZhang C(2017)On the numerical approximation for Fourier-type highly oscillatory integrals with Gauss-type quadrature rulesApplied Mathematics and Computation10.1016/j.amc.2017.03.021308:C(96-104)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1016/j.amc.2017.03.021
Zhao LHuang C(2017)An adaptive Filon-type method for oscillatory integrals without stationary pointsNumerical Algorithms10.1007/s11075-016-0219-375:3(753-775)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1007/s11075-016-0219-3
Siraj-ul-Islam Zaman S(2015)New quadrature rules for highly oscillatory integrals with stationary pointsJournal of Computational and Applied Mathematics10.1016/j.cam.2014.09.019278:C(75-89)Online publication date: 15-Apr-2015
https://dl.acm.org/doi/10.1016/j.cam.2014.09.019

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