This paper investigates the construction of space-filling designs for computer experiments. The space-filling property is characterized by the covering and separation radii of a design, which are integrated through the unified criterion of quasi-uniformity. We focus on a special class of designs, known as quasi-Monte Carlo (QMC) lattice point sets, and propose two construction algorithms. The first algorithm generates rank-1 lattice point sets as an approximation of quasi-uniform Kronecker sequences, where the generating vector is determined explicitly. As a byproduct of our analysis, we prove that this explicit point set achieves an isotropic discrepancy of $O(N^{-1/d})$. The second algorithm utilizes Korobov lattice point sets, employing the Lenstra--Lenstra--Lovász (LLL) basis reduction algorithm to identify the generating vector that ensures quasi-uniformity. Numerical experiments are provided to validate our theoretical claims regarding quasi-uniformity. Furthermore, we conduct empirical comparisons between various QMC point sets in the context of Gaussian process regression, showcasing the efficacy of the proposed designs for computer experiments.

翻译：本文研究了计算机实验中空间填充设计的构造方法。空间填充特性通过设计的覆盖半径与分离半径来表征，二者通过拟均匀性的统一准则进行整合。我们聚焦于一类特殊设计——拟蒙特卡洛格点集，并提出两种构造算法。第一种算法生成秩-1格点集作为拟均匀Kronecker序列的近似，其中生成向量通过显式方式确定。作为分析的副产品，我们证明该显式点集能达到$O(N^{-1/d})$的各向同性差异度。第二种算法采用Korobov格点集，利用Lenstra--Lenstra--Lovász基约化算法寻找确保拟均匀性的生成向量。数值实验验证了关于拟均匀性的理论结论。此外，我们在高斯过程回归框架下对不同拟蒙特卡洛点集进行了实证比较，展示了所提设计在计算机实验中的有效性。