The selection of points for numerical integration of the Monte Carlo type, largely used in analysis of engineering problems, is developed. It is achieved by modification of the metric in the minimax optimality criterion. The standard minimax criterion ensures the design exhibits good space-filling property and therefore reduces the variance of the estimator of the integral. We, however, show that the points are not selected with the same probability over the space of sampling probabilities: some regions are over- or under-sampled when designs are generated repetitively. This violation of statistical uniformity may lead to systematically biased integral estimators.

We propose that periodic metric be considered for calculation of the minimax distance. Such periodic minimax criterion provides statistically uniform designs leading to unbiased integration results and also low estimator variance due to retained space-filling property. These conclusions are demonstrated by examples integrating analytical functions.

The designs are constructed by two different algorithms: (i) a new time-stepping algorithm resembling a damped system of attracted particles developed here, and (ii) the heuristic swapping of coordinates. The designs constructed by the time-stepping algorithm are attached to the paper as a supplementary material. The computer code for construction of the designs is attached too.

开发了主要用于工程问题分析的蒙特卡洛数值积分点的选择。这是通过修改最小最大最优标准中的度量来实现的。标准的minimax准则可确保设计表现出良好的空间填充特性，从而减小积分估计量的方差。但是，我们表明，在抽样概率的空间中，选择这些点的概率不相同：当重复生成设计时，某些区域会被过度抽样或欠抽样。违反统计统一性可能会导致系统地估计积分估计量。

我们建议考虑周期性度量来计算最小最大距离。这种周期性的极小值准则提供了统计上统一的设计，从而导致无偏积分结果，并且由于保留了空间填充特性而导致估计量方差低。这些结论通过集成分析功能的示例得到证明。

设计是通过两种不同的算法构建的：（i）一种新的时间步长算法，类似于此处开发的吸引颗粒的阻尼系统，以及（ii）启发式坐标交换。通过时间步长算法构建的设计作为补充材料附在纸上。随附用于构建设计的计算机代码。