We study the approximation of integrals $\int_D f(\boldsymbol{x}^\top A) \mathrm{d} \mu(\boldsymbol{x})$, where $A$ is a matrix, by quasi-Monte Carlo (QMC) rules $N^{-1} \sum_{k=0}^{N-1} f(\boldsymbol{x}_k^\top A)$. We are interested in cases where the main cost arises from calculating the products $\boldsymbol{x}_k^\top A$. We design QMC rules for which the computation of $\boldsymbol{x}_k^\top A$, $k = 0, 1, \ldots, N-1$, can be done fast, and for which the error of the QMC rule is similar to the standard QMC error. We do not require that $A$ has any particular structure. For instance, this approach can be used when approximating the expected value of a function with a multivariate normal random variable with a given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients. The speed-up of the computation time is sometimes better and sometimes worse than the fast QMC matrix-vector product from [Dick, Kuo, Le Gia, and Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015)]. As in that paper, our approach applies to (polynomial) lattice point sets, but also to digital nets (we are currently not aware of any approach which allows one to apply the fast method from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets). Our method does not use FFT, instead we use repeated values in the quadrature points to derive a reduction in the computation time. This arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can also be used to also reduce the computation time of the QMC rule.

翻译：我们研究积分$\int_D f(\boldsymbol{x}^\top A) \mathrm{d} \mu(\boldsymbol{x})$（其中$A$为矩阵）的准蒙特卡洛（QMC）逼近方法，即采用规则$N^{-1} \sum_{k=0}^{N-1} f(\boldsymbol{x}_k^\top A)$。我们关注的主要问题在于计算乘积$\boldsymbol{x}_k^\top A$时占主导的计算成本。为此，我们设计了一种QMC规则，使得对$k = 0, 1, \ldots, N-1$计算$\boldsymbol{x}_k^\top A$可实现快速运算，同时保证该QMC规则的误差与标准QMC误差相当。该方法无需对$A$施加任何特定结构。例如，在逼近具有给定协方差矩阵的多元正态随机变量函数的期望值，或逼近随机系数偏微分方程解的期望值时，均可采用此方法。与[Dick, Kuo, Le Gia, and Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015)]中的快速QMC矩阵-向量积相比，本方法的计算加速效果有时更优，有时略逊。与前述论文类似，我们的方法不仅适用于（多项式）格点集，也适用于数字网（据我们所知，目前尚无方法能将上述Dick、Kuo、Le Gia和Schwab论文中的快速方法应用于数字网）。本方法无需使用快速傅里叶变换（FFT），而是通过利用求积节点中的重复值来降低计算时间。这一特性源于格点规则和多项式格点规则的缩减CBC构造。已有的研究表明，缩减CBC构造能缩短CBC构造的计算时间。本文进一步证明，该构造同样可用于减少QMC规则整体的计算时间。