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Quadrature methods for multivariate highly oscillatory integrals using derivatives

Authors: Arieh Iserles and Syvert P. Nørsett
Journal: Math. Comp. 75 (2006), 1233-1258
MSC (2000): Primary 65D32; Secondary 41A60, 41A63
Published electronically: March 8, 2006
MathSciNet review: 2219027
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Abstract: While there exist effective methods for univariate highly oscillatory quadrature, this is not the case in a multivariate setting. In this paper we embark on a project, extending univariate theory to more variables. Inter alia, we demonstrate that, in the absence of critical points and subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices, a direct counterpart of the univariate case. This provides a convenient avenue towards the generalization of asymptotic and Filon-type methods, as formerly introduced by the authors in a single dimension, to simplices and, more generally, to polytopes. The nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth: in effect, its violation is equivalent to the presence of stationary points in a single dimension. We further explore this issue and propose a technique that often can be used in this situation. Yet, much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals.

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Additional Information

Arieh Iserles
Affiliation: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Syvert P. Nørsett
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

Received by editor(s): February 17, 2005
Received by editor(s) in revised form: July 28, 2005
Published electronically: March 8, 2006
Dedicated: We dedicate this paper to the memory of Germund Dahlquist
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.