We present a preconditioned Monte Carlo method for computing high-dimensional multivariate normal and Student-t probabilities arising in spatial statistics. The approach combines a tile-low-rank representation of covariance matrices with a block-reordering scheme for efficient quasi-Monte Carlo simulation. The tile-low-rank representation decomposes the high-dimensional problem into many diagonal-block-size problems and low-rank connections. The block-reordering scheme reorders between and within the diagonal blocks to reduce the impact of integration variables from right to left, thus improving the Monte Carlo convergence rate. Simulations up to dimension 65,536 suggest that the new method can improve the run time by an order of magnitude compared with the hierarchical quasi-Monte Carlo method and two orders of magnitude compared with the dense quasi-Monte Carlo method. Our method also forms a strong substitute for the approximate conditioning methods as a more robust estimation with error guarantees. An application study to wind stochastic generators is provided to illustrate that the new computational method makes the maximum likelihood estimation feasible for high-dimensional skew-normal random fields.

我们提出了一种预处理的蒙特卡洛方法，用于计算高维多元正态和学生t空间统计中出现的概率。该方法将协方差矩阵的平铺低秩表示与有效的准蒙特卡洛模拟的块重排序方案结合在一起。平铺低秩表示将高维问题分解为许多对角块大小问题和低秩连接。块重排序方案在对角块之间和对角块内进行重排序，以减小积分变量从右到左的影响，从而提高了蒙特卡洛收敛速度。高达65,536维的仿真表明，与分层准蒙特卡洛方法相比，该新方法可以将运行时间缩短一个数量级，而与密集准蒙特卡洛方法相比，则可以将运行时间缩短两个数量级。我们的方法还可以作为近似条件方法的有力替代品，可以更可靠地估计误差，并且具有误差保证。通过对风力随机发电机的应用研究表明，该新的计算方法使得最大似然估计对于高维偏态正态随机场可行。