An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands
An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands
复制标题
平滑被积函数最优阶拟蒙特卡罗规则的显式构造
DOI：
10.1137/16m1060807
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发表时间：
2016
影响因子：
2.9
通讯作者：
Takehito Yoshiki
中科院分区：
文献类型：
--
作者：
Takashi Goda;Kosuke Suzuki;Takehito Yoshiki
作者在最近的一篇论文中指出，在由光滑函数组成的再生核Hilbert空间中，存在一个拟Monte Carlo（QMC）规则，它能使数值积分达到最佳的收敛速度。在本文中，我们提供了这样的最优阶QMC规则的显式构造。我们的方法是同时利用再生核的沃尔什系数的衰减和稀疏性。这可以通过将Dick的数字交织合成应用于具有Chen和Skriganov的大的最小Hamming和Niederreiter-Rosenbloom-Tsfasman度量的数字网络来完成。据我们所知，我们的构造给出了第一个QMC规则，该规则在该函数空间中实现了最佳收敛。
In a recent paper by the authors, it is shown that there exists a quasi--Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter--Rosenbloom--Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.