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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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: 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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Additional Information

Daan Huybrechs
Affiliation: Scientific Computing, Department of Computer Science, K.U.Leuven, Belgium
Email: daan.huybrechs@cs.kuleuven.be

Stefan Vandewalle
Affiliation: Scientific Computing, Department of Computer Science, K.U.Leuven, Belgium
Email: stefan.vandewalle@cs.kuleuven.be

DOI: http://dx.doi.org/10.1090/S0025-5718-07-01937-0
PII: S 0025-5718(07)01937-0
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.