Weighted compound integration rules with higher order convergence for all \(N\) (Q663491)

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 \textit{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 \textit{J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens} and \textit{F. Pillichshammer}, ``Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules'', \url{arXiv:1105.2599}]).