光滑度未知的周期函数格规则的逐位和逐分量构造
Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness
摘要：
格点规则是研究最多的拟蒙特卡罗方法之一。近似$s$维积分的秩1格点规则完全由其生成向量$\mathbf{z} \in \mathbb{Z}^s$及其点数$N$指定.虽然有很多结果的存在“好”的秩-1格规则，有没有明确的建设良好的生成向量的尺寸$s \ge 3$。这就是为什么人们通常诉诸计算机搜索算法。Korobov从1963年和1982年的早期工作的动机，我们提出了两个变种的搜索算法，良好的格规则，并表明所得到的规则表现出收敛速度加权函数空间，可以任意接近的最佳速率。此外，与大多数其他算法相反，我们不需要事先知道被积函数的光滑性，生成向量仍然会恢复与特定被积函数的光滑性相关的收敛速度，并且在适当的权重条件下，可以在不依赖于$s$的情况下声明误差界。本文提出的搜索算法是著名的组件的组件（CBC）建设，其中之一是结合数字的数字（DBD）建设的两个变种。我们目前的数值结果，这两种算法使用快速构造算法的情况下，产品的重量。它们证实了我们的理论发现。
英文摘要：
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 generating vector $\mathbf{z} \in \mathbb{Z}^s$ 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 for good generating vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate. Moreover, contrary to most other algorithms, we do not need to know the smoothness of our integrands in advance, the generating vector will still recover the convergence rate associated with the smoothness of the particular integrand, and, under appropriate conditions on the weights, the error bounds can be stated without dependence on $s$. The search algorithms presented in this paper are two variants of the well-known component-by-component (CBC) construction, one of which is combined with a digit-by-digit (DBD) construction. We present numerical results for both algorithms using fast construction algorithms in the case of product weights. They confirm our theoretical findings.
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