In this paper, we develop and test a fast numerical algorithm, called MDI-LR, for efficient implementation of quasi-Monte Carlo lattice rules for computing $d$-dimensional integrals of a given function. It is based on the idea of converting and improving the underlying lattice rule into a tensor product rule by an affine transformation and adopting the multilevel dimension iteration approach which computes the function evaluations (at the integration points) in the tensor product multi-summation in cluster and iterates along each (transformed) coordinate direction so that a lot of computations can be reused. The proposed algorithm also eliminates the need for storing integration points and computing function values independently at each point. Extensive numerical experiments are presented to gauge the performance of the algorithm MDI-LR and to compare it with standard implementation of quasi-Monte Carlo lattice rules. It is also showed numerically that the algorithm MDI-LR can achieve a computational complexity of order $O(N^2d^3)$ or better, where $N$ represents the number of points in each (transformed) coordinate direction and $d$ standard for the dimension. Thus, the algorithm MDI-LR effectively overcomes the curse of dimensionality and revitalizes QMC lattice rules for high-dimensional integration.

翻译：本文开发并测试了一种名为MDI-LR的快速数值算法，用于高效实现拟蒙特卡罗格点规则以计算给定函数的$d$维积分。该算法基于以下思想：通过仿射变换将底层格点规则转化并改进为张量积规则，并采用多层次维度迭代方法——该方法在张量积多重求和过程中对集群内的函数求值（在积分点处）进行计算，并沿每个（变换后的）坐标方向迭代，从而可复用大量计算结果。所提出的算法还消除了存储积分点及独立计算每个点处函数值的需求。通过大量数值实验评估了MDI-LR算法的性能，并将其与拟蒙特卡罗格点规则的标准实现进行了对比。数值结果还表明，MDI-LR算法可达到$O(N^2d^3)$或更优的计算复杂度，其中$N$表示每个（变换后的）坐标方向上的点数，$d$表示维度。因此，MDI-LR算法有效克服了维度诅咒，重振了QMC格点规则在高维积分中的应用。