LAT：适用于任意样本量的拉丁化非周期铺砌

中文摘要 AI 辅助

拟蒙特卡洛方法允许计算机实验以远少于原始蒙特卡洛的模拟次数运行。然而，基数为二的数字网格（如Sobol'序列）仅在样本量为2的幂时达到平衡，而拉丁超立方采样（LHS）虽接受任意样本量，但仅控制一维边际。本工作提出一种适用于任意样本量的空间填充设计，称为LAT（拉丁化非周期铺砌）。单位超立方体沿其最长边按黄金分割递归切割，形成N个体积相等的单元。随后在每个单元中放置一个点，并使边际呈拉丁分布，同时每个点保持在其所属单元内。该构造仅使用整数分割，计算复杂度为O(dNlogN)。研究表明该递归遵循斐波那契词，并分析了其非周期性。LAT通过四种L2差异度以及解析函数和工程仿真器上的积分误差进行评估，并与蒙特卡洛、LHS、Halton、Sobol'和秩1格进行比较。当被积函数存在交互作用时，LAT优于蒙特卡洛和LHS。Sobol'在2的幂样本量下仍更精确，但在其他样本量下其误差大一到两个数量级。LAT的精度不依赖于样本量。最后，对于任意N，这些单元构成超立方体的一个划分。这使得可以局部细化设计、通过分割单元搜索最优解，并对非矩形区域进行采样。

英文摘要

Quasi-Monte Carlo methods allow computer experiments to be run with far fewer simulations than crude Monte Carlo. A digital net in base two, such as Sobol', is however only balanced when the number of samples is a power of two, and Latin Hypercube Sampling (LHS) accepts any sample size but only controls the one-dimensional margins. This work proposes a space- filling design defined for any sample size, referred to as LAT for Latinized aperiodic tiling. The unit hypercube is cut recursively across its longest edge following the golden section into N cells of equal volume. One point is then placed in each cell, and the margins are made Latin while every point stays inside its own cell. The construction only uses integer splits and costs O(dNlog N). The recursion is shown to follow the Fibonacci word and its aperiodicity is analysed. LAT is assessed with four L2-discrepancies and with the integration error on analytical functions and engineering emulators, and compared to Monte Carlo, LHS, Halton, Sobol' and a rank-1 lattice. LAT is better than Monte Carlo and LHS as soon as the integrand has interactions. Sobol' remains more accurate at the powers of two, but its error is one to two orders of magnitude larger at other sample sizes. The accuracy of LAT does not depend on the sample size. Finally, the cells form a partition of the hypercube for any N. This allows one to refine the design locally, to search for an optimum by splitting cells and to sample non-rectangular regions.

发表机构

• Consulting Manao GmbH（Manao咨询有限公司）

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