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Mathematics of Computation

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The asymptotic efficiency
of randomized nets for quadrature

Authors: Fred J. Hickernell and Hee Sun Hong
Journal: Math. Comp. 68 (1999), 767-791
MSC (1991): Primary 65D30, 65D32
MathSciNet review: 1609662
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Abstract: An $ \mathcal{L}_{2}$-type discrepancy arises in the average- and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by $ K(x,y)$, which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized $(0,m,s)$-nets in base $b$. For moderately smooth $ K(x,y)$ the discrepancy is $ \operatorname{O}(N^{-1}[\log(N)]^{(s-1)/2})$, and for $ K(x,y)$ with greater smoothness the discrepancy is $ \operatorname{O}(N^{-3/2}[\operatorname{log}(N)]^{(s-1)/2})$, where $N=b^{m}$ is the number of points in the net. Numerical experiments indicate that the $(t,m,s)$-nets of Faure, Niederreiter and Sobol' do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions.

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Additional Information

Fred J. Hickernell
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Hee Sun Hong
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Keywords: $\mathcal{L}_{2}$-discrepancy, multidimensional integration, $(t, m, s)$-nets, number-theoretic nets and sequences
Received by editor(s): March 6, 1997
Received by editor(s) in revised form: September 11, 1997
Additional Notes: This research was supported by a HKBU FRG grant 95-96/II-01.
Article copyright: © Copyright 1999 American Mathematical Society