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On the corner avoidance properties of various low-discrepancy sequences. (English) Zbl 1087.11050

For \(f:\overline {U}^s=[0,1]^s\to \mathbb{R}\) such that \(I=\int_{\overline {U}^s}f(\mathbf{x}) \,d\mathbf{x}\) exists and \((\mathbf{x}_n)_{n>0}\) with \(\mathbf{x}_n\!=\! (x_n^{(1)},...,x_n^{(s)})\!\in [0,1)^s\), the QMC estimator of \(I\) is \(\widehat{I}_N=\frac 1N \sum_{n=1}^N f(\mathbf{x}_n)\). For a given corner \(\mathbf{h}=(h_1,\dots,h_s)\in \{0,1\}^s\) of the unit cube, the minimal hyperbolic distance of the points \(\mathbf{x}_n\) to the corner \(\mathbf{h}\) is \(M_N(\mathbf{h})=\min_{1\leq n\leq N}\prod_{i=1}^s| h_i-x_n^{(i)}| \). The minimal hyperbolic distance is an important quality in the error analysis of QMC integration for functions with singularities. A corner avoidance property for the QMC sequence is \(M_N(\mathbf{h})\geq cN^{-r}\). Sobol’ (in 1973) and Owen (to appear) show that the Sobol’ and Halton sequences avoid a hyperbolically shaped region around the corners of the unit cube. The authors show \(r=1\) for generalized Niederreiter sequences in the origin case. Second, for the Halton sequence they prove that for any corner \(\mathbf{h}\) different from the origin there exist infinitely many \(N\) such that \(M_N(\mathbf{h})=O((N\log N)^{-1})\). They also show that \(M_N(\min):=\min_\mathbf{h} M_N(\mathbf{h})>cN^{-1-\varepsilon}\) for the Halton sequence. Finally, they prove that there exists a subsequence \(\mathbf{y}_n=\mathbf{x}_{N(n)}\) of the Faure sequence such that \(\prod_{i=1}^s (1-y_n^{(i)})\leq p^3N(n)^{-3/2}\) where \(p\) is the least prime larger or equal to the dimension \(s\).

MSC:

11K38 Irregularities of distribution, discrepancy
11K45 Pseudo-random numbers; Monte Carlo methods
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