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.

晶格规则是研究多元积分最接近的拟蒙特卡罗方法之一。秩为1的格规则可近似$s$积分完全由其生成向量指定 $\mathit{ž}\in {\mathbb{ž}}^s$ 及其点数 $ñ$。尽管关于“良好” rank-1格规则的存在有很多结果，但是对于维的良好生成矢量没有明确的构造$s\ge 3$。这就是为什么通常采用计算机搜索算法的原因。受Korobov于1963年和1982年的早期工作的启发，我们提出了两种搜索算法，以求出良好的晶格规则，并证明了所得规则在加权函数空间中的收敛速度可以任意接近最佳速度。而且，与大多数其他算法相反，我们不需要事先知道被积物的光滑度，生成矢量仍将恢复与特定被积物的光滑度相关的收敛速度，并且在权重的适当条件下，误差范围可以不依赖于陈述$s$。本文提出的搜索算法是众所周知的逐个组件（CBC）结构的两种变体，其中一种与逐个数字（DBD）结构相结合。在产品权重的情况下，我们使用快速构造算法给出两种算法的数值结果。他们证实了我们的理论发现。