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 $\mathit{z}\in {\mathbb{Z}}^s$ 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.

格规则是研究最多的用于逼近多元积分的准蒙特卡罗方法之一。一个等级 1 的格子规则来近似一个$秒$-维积分完全由其生成向量指定 $\mathit{z}\in {\mathbb{Z}}^{秒}$ 和它的点数 $N$. 虽然有很多关于“好”秩 1 格规则的存在的结果，但对于维度的良好生成向量没有明确的构造$秒\ge 3$. 这就是人们通常求助于计算机搜索算法的原因。受 Korobov 1963 年和 1982 年早期工作的启发，我们提出了两种用于良好格规则的搜索算法的变体，并表明所得规则在加权函数空间中表现出可以任意接近最佳速率的收敛速度。此外，与大多数其他算法相反，我们不需要提前知道我们的被积函数的平滑度，生成向量仍然会恢复与特定被积函数的平滑度相关的收敛速度，并且在权重的适当条件下，可以在不依赖于$秒$. 本文提出的搜索算法是众所周知的逐个组件 (CBC) 构造的两种变体，其中一种与逐位 (DBD) 构造相结合。我们在产品权重的情况下使用快速构造算法为两种算法提供数值结果。他们证实了我们的理论发现。