In this paper, we present the fundamentals of a hierarchical algorithm for computing the N-dimensional integral \(\phi (\mathbf {a}, \mathbf {b}; A) = {\int \limits }_{\mathbf {a}}^{\mathbf {b}} H(\mathbf {x}) f(\mathbf {x} | A) \text {d} \mathbf {x}\) representing the expectation of a function H(X) where f(x|A) is the truncated multi-variate normal (TMVN) distribution with zero mean, x is the vector of integration variables for the N-dimensional random vector X, A is the inverse of the covariance matrix Σ, and a and b are constant vectors. The algorithm assumes that H(x) is “low-rank” and is designed for properly clustered X so that the matrix A has “low-rank” blocks and “low-dimensional” features. We demonstrate the divide-and-conquer idea when A is a symmetric positive definite tridiagonal matrix and present the necessary building blocks and rigorous potential theory–based algorithm analysis when A is given by the exponential covariance model. The algorithm overall complexity is O(N) for N-dimensional problems, with a prefactor determined by the rank of the off-diagonal matrix blocks and number of effective variables. Very high accuracy results for N as large as 2048 are obtained on a desktop computer with 16G memory using the fast Fourier transform (FFT) and non-uniform FFT to validate the analysis. The current paper focuses on the ideas using the simple yet representative examples where the off-diagonal matrix blocks are rank 1 and the number of effective variables is bounded by 2, to allow concise notations and easier explanation. In a subsequent paper, we discuss the generalization of current scheme using the sparse grid technique for higher rank problems and demonstrate how all the moments of k_th order or less (a total of O(N^k) integrals) can be computed using O(N^k) operations for k ≥ 2 and \(O(N \log N)\) operations for k = 1.

在本文中，我们提出了用于计算N维积分的分层算法的基本原理\(\phi (\mathbf {a}, \mathbf {b}; A) = {\int \limits }_{\mathbf { a}}^{\mathbf {b}} H(\mathbf {x}) f(\mathbf {x} | A) \text {d} \mathbf {x}\)表示函数H ( X ) 其中f ( x | A ) 是均值为零的截断多元正态 (TMVN) 分布，x是N维随机向量X的积分变量向量，A是协方差矩阵Σ的逆矩阵，并且一种和b是常数向量。该算法假设H ( x ) 是“低秩”并且是为正确聚类X设计的，因此矩阵A具有“低秩”块和“低维”特征。我们证明了当A是对称正定三对角矩阵时的分而治之的思想，并在A由指数协方差模型给出时，提出了必要的构建块和严格的基于潜在理论的算法分析。算法整体复杂度为O ( N ) for N维问题，其前置因子由非对角矩阵块的秩和有效变量的数量决定。使用快速傅立叶变换 (FFT) 和非均匀 FFT 来验证分析，在具有 16G 内存的台式计算机上获得了高达 2048 的N 的非常高精度的结果。当前的论文使用简单而具有代表性的例子来关注这些想法，其中非对角矩阵块为 1 级，有效变量的数量以 2 为界，以允许简洁的符号和更容易的解释。在随后的论文中，我们使用稀疏网格技术更高的秩的问题讨论当前方案的概括和说明如何所有的时刻ķ_吨ħ顺序或更小（共ö( N ^k ) 积分) 可以使用对k ≥ 2 的O ( N ^k ) 操作和对k = 1 的\(O(N \log N)\)操作来计算。