On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation

By Daan Huybrechs, Stefan Vandewalle. SIAM Journal on Numerical Analysis, 2006

314 citations recorded by the source corpus.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3ef1f667611ef09ee7b6d1e58efd4c43e9f7d617

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In Applied Mathematics

In Atomic and Molecular Physics, and Optics

In Numerical Analysis

Citations in the corpus

Highly Oscillatory Quadrature: The Story soFar (2006, methodology): “Another option is quadrature methods based on altogether new principles, e.g. the recent technique of Huybrechs & Vandewalle (2005), who approximate (1) in a single dimension using a complex-valued path along which eiωg(x) does not oscillate.”
Moment-free numerical approximation of highly oscillatory integrals with stationary points (2007, background): “It is possible to compute the integrals by moving to the complex plane and integrating along the path of steepest descent [ 3 ], however, this suers from the requirement that both f and g are analytic, in order to deform the integration path, and requires the knowledge or computation of the path of ”
Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals (2011, background): “More recently, the question of computing highly oscillatory integrals has enjoyed a substantial renaissance and a number of authors have been concerned in particular with identifying quadrature rules for (1) with error which decays with high negative powers of k.”
From high oscillation to rapid approximation I: Modified Fourier expansions (2008, background): “The quadrature of highly oscillatory integrals of the formI[f ; g] = ∫ b a f(x)eiωg(x)dx, where g is a real function andω ≫ 1, has received much attention in the last few years (Huybrechs & Vandewalle 2006, Iserles & Nørsett 2005, Olver 2006).”
Numerical approximation of vector-valued highly oscillatory integrals (2007, background): “[ 2 ] by going to the complex plane, and integrating along a path that approximated the path of steepest descent.”
An improved Levin quadrature method for highly oscillatory integrals (2010, methodology): “Among the oscillatory quadrature methods, the numerical steepest descent method [10], the Filon method [8], and the Levin method [16] are three classical ones.”
Fast, numerically stable computation of oscillatory integrals with stationary points (2010, background): “Numerical steepest descent [10, 7] simultaneously achieves high asymptotic order and numerical stability.”