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In this paper, we study the $N$-dimensional integral $\phi(a,b; A) =\int_{a}^{b} H(x) f(x | A) \text{d} x$ representing the expectation of afunction $H(X)$ 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$\Sigma$, and $a$ and $b$ are constant vectors. We present a new hierarchicalalgorithm which can evaluate $\phi(a,b; A)$ using asymptotically optimal $O(N)$operations when $A$ has "low-rank" blocks with "low-dimensional" features and$H(x)$ is "low-rank". We demonstrate the divide-and-conquer idea when $A$ is asymmetric positive definite tridiagonal matrix, and present the necessarybuilding blocks and rigorous potential theory based algorithm analysis when $A$is given by the exponential covariance model. Numerical results are presentedto demonstrate the algorithm accuracy and efficiency for these two cases. Wealso briefly discuss how the algorithm can be generalized to a wider class ofcovariance models and its limitations.

O(N) Hierarchical algorithm for computing the expectations of truncated multi-variate normal distributions in N dimensions