论文标题
有关CBC-DBD构建晶格规则的注释
A note on the CBC-DBD construction of lattice rules with general positive weights
论文作者
论文摘要
晶格规则是最突出的准蒙特卡洛方法，可近似多元积分。 排名$ 1 $的晶格规则近似于$ s $二维的积分由其\ emph {genating vector} $ \ boldsymbol {z} \ in \ mathbb {z}^s $及其点数〜$ n $。 尽管存在``良好''等级的存在很多结果 - $ 1 $晶格规则，但没有明确的构造良好的生成矢量$ s \ ge 3 $。 这就是为什么人们通常求助于计算机搜索算法的原因。在Ebert等人的最新论文中。在《复杂性杂志》中，我们显示了一个划分数字的组成部分数字（CBC-DBD） 良好生成媒介的构造级别-1晶格规则，用于在加权Korobov类中集成功能。但是，该论文的结果是 限于产品重量。在本文中，我们将将这一结果推广到任意的积极权重，从而回答了一个开放的问题 在Ebert等人的论文中。我们还提供了一个简短的部分，介绍了如何在POD权重的情况下实施该算法，通过该算法，我们可以看到CBC-DBD结构与经典CBC结构具有竞争力。
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its \emph{generating vector} $\boldsymbol{z} \in \mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of ``good'' rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in the paper of Ebert et al. We also include a short section on how the algorithm can be implemented in the case of POD weights, by which we see that the CBC-DBD construction is competitive with the classical CBC construction.
