I-binomial scrambling of digital nets and sequences
I-binomial scrambling of digital nets and sequences
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DOI：
10.1016/s0885-064x(03)00035-9
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发表时间：
2003-12
期刊：
影响因子：
--
通讯作者：
S. Tezuka;H. Faure
中科院分区：
文献类型：
--
作者：
S. Tezuka;H. Faure
以期望误差表示的积分问题的计算复杂性是近年来信息复杂性研究的一个重要课题。在这种情况下，我们假设一些积分规则的样本空间，我们从中随机选择一个。最流行的样本空间是基于欧文的随机置乱方案，其理论优势是某些光滑函数的快速收敛速度。本文考虑利用i-二项式性质的概念来降低欧文随机置乱所需的随机性。首先建立了数字（0，s）-序列具有i-二项式性质的一组充分必要条件。在此基础上定义了左、右i-二项式随机数。我们证明了Owen的关键引理（引理4，SIAM J. Numer. Anal. 34（1997）1884）对于左i-二项置乱仍然有效，并且由此得出结论，迄今为止对于欧文置乱获得的关于积分问题的期望误差的所有结果对于左i-二项置乱也成立。
The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owen's random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions. This paper considers a reduction of randomness required for Owen's random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owen's key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i-binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owen's scrambling also hold with the left i-binomial scrambling.