随机拟蒙特卡罗积分
Randomized quasi-Monte Carlo integration
浏览论文内容
中文总结 AI 辅助
本文针对统计领域读者，梳理随机拟蒙特卡罗（RQMC）的历史与当前方向，阐述其作为数值积分方法的特性及误差表现，为相关研究提供讲解。
中文摘要 AI 辅助
拟蒙特卡罗采样是一种数值积分方法，使用[0,1]^s中具有空间填充特性的点，相比普通蒙特卡罗方法能给出更优的估计。对于Hardy-Krause意义下有界变差的被积函数，n个采样点可得到任意ε>0下O(n^{-1+ε})的误差。随机拟蒙特卡罗（RQMC）点各自均匀分布但整体呈空间填充特性，独立重复可提供方差估计；对于足够光滑的被积函数，该随机化可得到O(n^{-3/2+ε})的均方根误差。本文为统计领域读者讲解RQMC，梳理部分历史并介绍当前研究方向。
英文摘要
Quasi-Monte Carlo sampling is a numerical integration method that uses points with a space-filling property in $[0,1]^s$ designed to give better estimates than plain Monte Carlo methods do. For integrands of bounded variation in the sense of Hardy and Krause, errors of $O(n^{-1+ε})$ for any $ε>0$ are obtained from $n$ sample points. Randomized quasi-Monte Carlo (RQMC) points are individually uniformly distributed but collectively space-filling and then independent replications provide variance estimates. For smooth enough integrands the randomization can give a root mean squared error of $O(n^{-3/2+ε})$. This article explains RQMC for a statistical readership recounting some history and presenting some current directions.
