Abstract
Quasi-Monte Carlo methods for integration over the s-dimensional unit cube in which only one vertex is a quadrature point are here modified by distributing the corresponding weight between all the vertices. One particular rule of this kind, namely the one that integrates all multilinear functions exactly, is shown to be optimal, in the sense that among all vertex-modified rules that integrate constants exactly it yields the least value of an L 2 version of the discrepancy. The sum of the squares of the vertex weights in that rule, the “vertex variance”, is shown to be related to the quality of the underlying quasi-Monte Carlo method. Numerical experiments confirm in a dramatic way the important role played by the vertex variance, even for the unmodified quasi-Monte Carlo method.
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References
Förster K.-J. (1988), On the minimal variance of quadrature formulas. In Numerical Integration HI, eds, II. Braß and G. Hämmerlin. Internat. Series of Numer. Math., Vol. 85, Birkhäuser Verlag, Basel, pp. 37–48.
Genz A, (1984), Testing multidimensional integration routines. In Tools, Methods and Languages for Scientific and Engineering Computation, eds. B. Ford, J.C. Rault and F. Thomasset. North-Holland, Amsterdam, pp. 81–94.
Genz A. (1987), A package for testing multiple integration subroutines. In Numerical Integration: Recent Developments, Software and Applications, eds. P. Keast and G. Fairweather. D. Reidel Publishing, Dordrecht, pp. 337–340.
Niederreiter II. (1992), Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia.
Niederreiter II. and Sloan I.H. (1993), Integration of non-periodic functions of two variables by Fibonacci lattice rules. J. Comp. Appl. Math., to appear.
Price J.F, and Sloan I.H. (1992), Pointwise convergence of multiple Fourier series: sufficient conditions and an application to numerical integration. J. Math. Anal. Appl. 169, 140–156.
Ritter K. (1970), Two-dimensional spline functions and best approximations of linear functionals. J. Approximation Th. 3, 352–368.
Sard A. (1963), Linear Approximation. American Math. Society, Providence.
Sloan I.H. (1992), Numerical integration in high dimensions — the lattice rule approach. In Numerical Integration: Recent Developments, Software and Applications, eds. T.O. Espelid and A. Genz. Kluwer Academic Publishers, Dordrecht, pp. 55–69.
Sloan LH. and Kachoyan P.J. (1987), Lattice methods for multiple integration: theory, error analysis and examples. SI AM J. Numer. Anal. 24, 116–128.
Zaremba S.K. (1968), Some applications of multidimensional integration by parts. Ann. Polon. Math. 21, 85–96.
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© 1993 Springer Basel AG
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Niederreiter, H., Sloan, I.H. (1993). Quasi-Monte Carlo Methods with Modified Vertex Weights. In: Brass, H., Hämmerlin, G. (eds) Numerical Integration IV. ISNM International Series of Numerical Mathematics, vol 112. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6338-4_20
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DOI: https://doi.org/10.1007/978-3-0348-6338-4_20
Publisher Name: Birkhäuser, Basel
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