The Construction of cubature rules for multivariate highly oscillatory integrals

Huybrechs, Daan; Vandewalle, Stefan

doi:10.1090/S0025-5718-07-01937-0

The Construction of cubature rules for multivariate highly oscillatory integrals
HTML articles powered by AMS MathViewer

by Daan Huybrechs and Stefan Vandewalle;

Math. Comp. 76 (2007), 1955-1980

DOI: https://doi.org/10.1090/S0025-5718-07-01937-0

Published electronically: April 27, 2007

PDF | Request permission

Abstract:

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule. The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

References

N. Bleistein and R. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, 1975.
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
L. N. G. Filon. On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh, 49:38–47, 1928.
Walter Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004. Oxford Science Publications. MR 2061539
Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 372162
Daan Huybrechs and Stefan Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (2006), no. 3, 1026–1048. MR 2231854, DOI 10.1137/050636814
Arieh Iserles and Syvert P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2057, 1383–1399. MR 2147752, DOI 10.1098/rspa.2004.1401
A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT 44 (2004), no. 4, 755–772. MR 2211043, DOI 10.1007/s10543-004-5243-3
Arieh Iserles and Syvert P. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comp. 75 (2006), no. 255, 1233–1258. MR 2219027, DOI 10.1090/S0025-5718-06-01854-0
A. Iserles and S. P. Nørsett. On the computation of highly oscillatory multivariate integrals with critical points. Technical Report NA08, University of Cambridge, 2005.
David Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math. 67 (1996), no. 1, 95–101. MR 1388139, DOI 10.1016/0377-0427(94)00118-9
Sheehan Olver, Moment-free numerical integration of highly oscillatory functions, IMA J. Numer. Anal. 26 (2006), no. 2, 213–227. MR 2218631, DOI 10.1093/imanum/dri040
Sheehan Olver, On the quadrature of multivariate highly oscillatory integrals over non-polytope domains, Numer. Math. 103 (2006), no. 4, 643–665. MR 2221066, DOI 10.1007/s00211-006-0009-2
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
R. Wong, Asymptotic approximations of integrals, Classics in Applied Mathematics, vol. 34, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Corrected reprint of the 1989 original. MR 1851050, DOI 10.1137/1.9780898719260