## Weighted compound integration rules with higher order convergence for all $$N$$.(English)Zbl 1240.65002

The subject of this paper is an improvement of the quasi-Monte-Carlo (QMC) integration rule $$Q_{qmc}(f)=\frac{1}{N}\sum_{k=0}^{N-1}f(x_k)$$ used for approximating a multivariate integral $$\int_{[0,1]^s}f(x)dx$$. It is known that its convergence is close to order $$1/N$$ (better than the convergence $$1/\sqrt{N}$$ assured by simple Monte-Carlo algorithms). The authors prove two important results:
(1) If the $$QMC$$ rule uses $$N$$ sampling points from an infinite sequence $$x_0,x_1,\dots$$, then the best convergence possible is $$1/N$$.
(2) If the points are selected with different weights, then it is possible to improve the performance of the $$QMC$$ rule.
The next result is proved and verified by examples: If the $$N$$ sampling points are partitioned into $$M$$ sets with $$N=\sum_{i=1}^MN_i$$, and for each set of size $$N_i$$ an integration rule $$Q_i$$ is used with a $${\mathcal O}(N_i^{-\alpha})$$ convergence rate $$(\alpha >1)$$, then the integration rule $$Q(f)=\sum_{i=1}^Mw_iQ_i(f)$$, where the weights $$w_i=N_i^a/(N_1^a+\dots+N_M^a)$$ are defined for $$a\geq \alpha$$ assures a $${\mathcal O}(N^{-\alpha})$$ convergence rate for all the values of $$N$$.
As a remark, this paper improves old results presented by H. Niederreiter [Diophantine Approx. Appl., Proc. Conf. Washington 1972, 129–199 (1973; Zbl 0268.65014)] and is a completion of similar current research [see for example J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens and F. Pillichshammer, “Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules”, arXiv:1105.2599]).

### MSC:

 65C05 Monte Carlo methods 11H06 Lattices and convex bodies (number-theoretic aspects) 11K45 Pseudo-random numbers; Monte Carlo methods 26B15 Integration of real functions of several variables: length, area, volume

Zbl 0268.65014

QSIMVN
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### References:

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