Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy

Joe, Stephen

doi:10.1007/3-540-31186-6_12

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Joe, S.(2006). Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy. In H. Niederreiter & D. Talay (Eds), Monte Carlo and Quasi-Monte Carlo Methods 2004 (pp. 181-196). Proceedings of the Sixth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing and of the Second International Conference on Monte Carlo and Probabilistic Methods for Partial Differential Equations. Berlin, Germany: Springer.

Permanent Research Commons link: https://hdl.handle.net/10289/2000

Abstract

The ‘goodness’ of a set of quadrature points in [0, 1]d may be measured by the weighted star discrepancy. If the weights for the weighted star discrepancy are summable, then we show that for n prime there exist n-point rank-1 lattice rules whose weighted star discrepancy is O(n−1+δ) for any δ>0, where the implied constant depends on δ and the weights, but is independent of d and n. Further, we show that the generating vector z for such lattice rules may be obtained using a component-by-component construction. The results given here for the weighted star discrepancy are used to derive corresponding results for a weighted Lp discrepancy.

Date

2006

Type

Conference Contribution

Publisher

Springer, Berlin

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Computing and Mathematical Sciences Papers [1455]