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Highly oscillatory quadrature: the story so far. (English) Zbl 1119.65317
Bermúdez de Castro, Alfredo (ed.) et al., Numerical mathematics and advanced applications. Proceedings of ENUMATH 2005, the 6th European conference on numerical mathematics and advanced applications, Santiago de Compostela, Spain, July 18–22, 2005. Berlin: Springer (ISBN 3-540-34287-7/hbk). 97-118 (2006).
Summary: The last few years have witnessed substantive developments in the computation of highly oscillatory integrals in one or more dimensions. The availability of new asymptotic expansions and a Stokes-type theorem allow for a comprehensive analysis of a number of old (although enhanced) and new quadrature techniques: the asymptotic, Filon-type and Levin-type methods. All these methods share the surprising property that their accuracy increases with growing oscillation. These developments are described in a unified fashion, taking the multivariate integral Ω f(x)e iωg(x) dV as our point of departure.
MSC:
65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures
65T40Trigonometric approximation and interpolation (numerical methods)