Abstract
In this paper, we present a Levin-type method for approximating multivariate highly oscillatory integrals, subject to a non-resonance condition. Unlike existing methods, we do not require the knowledge of moments, which enables us to derive an approximation when the oscillator is complicated, and when the domain is neither a simplex nor a polytope. The accuracy of this method improves as the frequency of oscillations increases. A special case of this method has the property that the asymptotic order increases with each additional sample point.
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References
Cheney W., Light W. (2000) A course in approximation theory. Brooks/Cole Publishing Company, Pacific Grove ,CA
Iserles A., Nørsett S.P. (2004) On quadrature methods for highly oscillatory integrals and their implementation. BIT 44, 755–772
Iserles A., Nørsett S.P. (2005) Efficient quadrature of highly oscillatory integrals using derivatives. Proc R Soc A 461, 1383–1399
Iserles, A., Nørsett, S.P. On the computation of highly oscillatory multivariate integrals with critical points. Technical Report NA2005/08, DAMTP, University of Cambridge 2005
Iserles, A., Nørsett, S.P. Quadrature methods for multivariate highly oscillatory integrals using derivatives. Technical Report NA2005/02, DAMTP, University of Cambridge, Math Comput (2005) (in press)
Levin D. (1997) Analysis of a collocation method for integrating rapidly oscillatory functions. J Comput Appl Maths 78, 131–138
Olver F.W.J. (1974) Asymptotics and special functions. Academic, New York, NY
Olver, S. Moment-free numerical integration of highly oscillatory functions. Technical Report NA2005/04, DAMTP, University of Cambridge, IMA J Num Anal 26, 213–227 (2006)
Powell M.J.D. (1981) Approximation theory and methods. Cambridge University Press, Cambridge
Press W.H., Flannery B.P., Teukolsky S., Vetterling W.T. (1988) Numerical recipes in C: the art of scientific computing. Cambridge Unversity Press, Cambridge
Stein E. (1993) Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ
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Olver, S. On the Quadrature of Multivariate Highly Oscillatory Integrals Over Non-polytope Domains. Numer. Math. 103, 643–665 (2006). https://doi.org/10.1007/s00211-006-0009-2
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DOI: https://doi.org/10.1007/s00211-006-0009-2