Many questions in quantitative finance, uncertainty quantification, and other disciplines are answered by computing the population mean, $\mu := \mathbb{E}(Y)$, where instances of $Y:=f(\boldsymbol{X})$ may be generated by numerical simulation and $\boldsymbol{X}$ has a simple probability distribution. The population mean can be approximated by the sample mean, $\hat{\mu}_n := n^{-1} \sum_{i=0}^{n-1} f(\boldsymbol{x}_i)$ for a well chosen sequence of nodes, $\{\boldsymbol{x}_0, \boldsymbol{x}_1, \ldots\}$ and a sufficiently large sample size, $n$. Computing $\mu$ is equivalent to computing a $d$-dimensional integral, $\int f(\boldsymbol{x}) \varrho(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}$, where $\varrho$ is the probability density for $\boldsymbol{X}$. Quasi-Monte Carlo methods replace independent and identically distributed sequences of random vector nodes, $\{\boldsymbol{x}_i \}_{i = 0}^{\infty}$, by low discrepancy sequences. This accelerates the convergence of $\hat{\mu}_n$ to $\mu$ as $n \to \infty$. This tutorial describes low discrepancy sequences and their quality measures. We demonstrate the performance gains possible with quasi-Monte Carlo methods. Moreover, we describe how to formulate problems to realize the greatest performance gains using quasi-Monte Carlo. We also briefly describe the use of quasi-Monte Carlo methods for problems beyond computing the mean, $\mu$.

翻译：在量化金融、不确定性量化及其他领域中，许多问题可通过计算总体均值 $\mu := \mathbb{E}(Y)$ 来解答，其中 $Y:=f(\boldsymbol{X})$ 的实例可通过数值模拟生成，且 $\boldsymbol{X}$ 服从简单概率分布。总体均值可通过样本均值 $\hat{\mu}_n := n^{-1} \sum_{i=0}^{n-1} f(\boldsymbol{x}_i)$ 进行近似，其中 $\{\boldsymbol{x}_0, \boldsymbol{x}_1, \ldots\}$ 为精心选取的节点序列，$n$ 为足够大的样本量。计算 $\mu$ 等价于计算 $d$ 维积分 $\int f(\boldsymbol{x}) \varrho(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}$，其中 $\varrho$ 为 $\boldsymbol{X}$ 的概率密度函数。拟蒙特卡洛方法采用低差异序列替代独立同分布的随机向量节点序列 $\{\boldsymbol{x}_i \}_{i = 0}^{\infty}$，从而加速了 $\hat{\mu}_n$ 在 $n \to \infty$ 时向 $\mu$ 的收敛。本教程系统阐述低差异序列及其质量度量指标，展示拟蒙特卡洛方法可能带来的性能提升，并详细说明如何通过问题重构以最大化拟蒙特卡洛方法的效能。此外，本文还简要探讨了拟蒙特卡洛方法在均值计算 $\mu$ 之外的其他应用场景。