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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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  • [DS03] I. Degani and J. Schiff, RCMS: Right correction Magnus series approach for integration of linear ordinary differential equations with highly oscillatory terms, Tech. report, Weizmann Institute of Science, 2003.
  • [IN05a] A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Royal Soc. A 461 (2005), 1383-1399. MR 2147752
  • [IN05b] -, On quadrature methods for highly oscillatory integrals and their implementation, BIT 44 (2005), 755-772.
  • [Ise96] A. Iserles, A first course in the numerical analysis of differential equations, Cambridge University Press, Cambridge, 1996.MR 1384977 (97m:65003)
  • [Ise02] -, Think globally, act locally: Solving highly-oscillatory ordinary differential equations, Appld Num. Anal. 43 (2002), 145-160.MR 1936107 (2003j:65066)
  • [Ise04a] -, On the method of Neumann series for highly oscillatory equations, BIT 44 (2004), 473-488. MR 2106011 (2005g:65101)
  • [Ise04b] -, On the numerical quadrature of highly-oscillating integrals I: Fourier transforms, IMA J. Num. Anal. 24 (2004), 365-391.MR 2068828 (2005d:65033)
  • [Ise05] -, On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators, IMA J. Num. Anal. 25 (2005), 25-44.MR 2110233 (2005i:65030)
  • [Lev96] D. Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Maths 67 (1996), 95-101. MR 1388139 (97a:65029)
  • [Mun91] J. R. Munkres, Analysis on Manifolds, Addison-Wesley, Reading, MA, 1991. MR 1079066 (92d:58001)
  • [Olv74] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [Olv05] S. Olver, Moment-free numerical integration of highly oscillatory functions, Tech. Report NA2005/04, DAMTP, University of Cambridge, 2005.
  • [Ste93] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • [STW90] A. H. Schatz, V. Thomee, and W. L. Wendland, Mathematical Theory of Finite and Boundary Elements Methods, Birkhauser, Boston, 1990.MR 1116555 (92f:65004)

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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.

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