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The Construction of cubature rules for multivariate highly oscillatory integrals

Authors: Daan Huybrechs and Stefan Vandewalle
Journal: Math. Comp. 76 (2007), 1955-1980
MSC (2000): Primary 65D32; Secondary 41A60, 41A63
Published electronically: April 27, 2007
MathSciNet review: 2336276
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Abstract | References | Similar Articles | Additional Information

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.

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  • 1. N. Bleistein and R. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, 1975.
  • 2. P. J. Davis and P. Rabinowitz. Methods of Numerical Integration. Computer Science and Applied Mathematics. Academic Press Inc., 1984. MR 760629 (86d:65004)
  • 3. L. N. G. Filon. On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh, 49:38-47, 1928.
  • 4. W. Gautschi. Orthogonal Polynomials: Computation and Approximation. Clarendon Press, Oxford, 2004. MR 2061539 (2005e:42001)
  • 5. P. Henrici. Applied and Computational Complex Analysis Volume I. Wiley & Sons, 1974. MR 0372162 (51:8378)
  • 6. D. Huybrechs and S. Vandewalle. On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (2006), no. 3, 1026-1048. MR 2231854
  • 7. A. Iserles and S. P. Nørsett. Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461:1383-1399, 2005. MR 2147752 (2006b:65030)
  • 8. A. Iserles and S. P. Nørsett. On quadrature methods for highly oscillatory integrals and their implementation., BIT Numerical Mathematics 44, 4:755-772, 2005. MR 2211043 (2006k:65060)
  • 9. A. Iserles and S. P. Nørsett. Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comp. 75 (2006), no. 255, 1233-1258. MR 2219027 (2007e:65024)
  • 10. A. Iserles and S. P. Nørsett. On the computation of highly oscillatory multivariate integrals with critical points. Technical Report NA08, University of Cambridge, 2005.
  • 11. D. Levin. Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math., 67(1):95-101, 1996. MR 1388139 (97a:65029)
  • 12. S. Olver. Moment-free numerical integration of highly oscillatory functions, IMA J. Num. Anal., 26(2):213-227, 2006. MR 2218631 (2006k:65064)
  • 13. S. Olver. On the quadrature of multivariate highly oscillatory integrals over non-polytope domains, Numerische Mathematik 103 (2006), no. 4, 643-665. MR 2221066
  • 14. E. M. Stein. Harmonic analysis: Real-variable methods, orthogonality and oscillatory integrals. Princeton University Press, Princeton, New York, 1993. MR 1232192 (95c:42002)
  • 15. R. Wong. Asymptotic approximation of integrals. SIAM, 2001. MR 1851050 (2002f:41023)

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Additional Information

Daan Huybrechs
Affiliation: Scientific Computing, Department of Computer Science, K.U.Leuven, Belgium

Stefan Vandewalle
Affiliation: Scientific Computing, Department of Computer Science, K.U.Leuven, Belgium

Keywords: Cubature formulas, oscillatory functions, steepest descent
Received by editor(s): November 21, 2005
Received by editor(s) in revised form: April 12, 2006
Published electronically: April 27, 2007
Additional Notes: The first author was supported by The Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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