On the numerical quadrature of highly‐oscillating integrals I: Fourier transforms

By Arieh Iserles. 2004

181 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/fb8e27204670eb5f7624b203974aa8fa9a9eb06b

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

The Theory of Scale Functions for Spectrally Negative Lévy Processes (2011, methodology): “We will just mention here an excellent textbook by Cohen [33], very helpful reviews and research articles by Abate, Choudhury, Whitt and Valko [1,3,4] and works of Filon [41], Bailey and Swarztrauber [12] and Iserles [50].”
A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line (2008, methodology): “There exist powerful methods for the numerical evaluation of Fourier-type integrals when they cannot be computed analytically (see for example Iserles 2004).”
Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition (2005, methodology): “In a similar situation, Iserles [9] proposes to use Filon quadrature.”
Numerical approximation of vector-valued highly oscillatory integrals (2007, methodology): “This compares with an error of order O(1) of the traditional quadrature methods [3].”
On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation (2006, methodology): “Methods have been devised that compute an accurate approximation to the value of the integral with low computational complexity and with a number of operations that actually decreases as ω increases to infinity [12, 13, 14, 15, 16, 17, 18].”
The Magnus expansion and some of its applications (2008, methodology): “In those cases, the norm of B(t) is considerably smaller than ‖A(t)‖, but B(t) is still highly oscillatory, so that especially adapted quadrature rules have to be used in conjunction with the Magnus expansion [117,121].”
Fourth-Order Time-Stepping for Stiff PDEs (2005, background): “They have been independently derived several times [5, 12, 14, 21, 46, 48]—indeed Iserles has pointed out to us that in the ODE context, related ideas go back as far as Filon in 1928 [18, 30]—but the most comprehensive treatment, and in particular the exponential time-differencing fourthorder Runge–”
From high oscillation to rapid approximation I: Modified Fourier expansions (2008, methodology): “This is perhaps the place to emphasize that a Filon-type method with s is not simply an approximation to the asymptotic method with s + 1: as can be confirmed from Tables 1 and 2, it is typically much better! The reason is that Filon-type methods are far superior for small values of n, before the on”