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    An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution I: fundamentals

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    Huang_HDInteg_Part1_2021_Final.pdf
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    Type
    Article
    Authors
    Huang, Jingfang cc
    Cao, Jian cc
    Fang, Fuhui
    Genton, Marc G. cc
    Keyes, David E. cc
    Turkiyyah, George
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
    Extreme Computing Research Center
    Office of the President
    Spatio-Temporal Statistics and Data Analysis Group
    Statistics Program
    Date
    2021-09-01
    Online Publication Date
    2021-09-01
    Print Publication Date
    2021-10
    Embargo End Date
    2022-09-01
    Submitted Date
    2020-12-29
    Permanent link to this record
    http://hdl.handle.net/10754/671102
    
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    Abstract
    In this paper, we present the fundamentals of a hierarchical algorithm for computing the N-dimensional integral ϕ(a,b;A)=∫abH(x)f(x|A)dx 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 kth order or less (a total of O(Nk) integrals) can be computed using O(Nk) operations for k ≥ 2 and O(NlogN) operations for k = 1.
    Citation
    Huang, J., Cao, J., Fang, F., Genton, M. G., Keyes, D. E., & Turkiyyah, G. (2021). An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution I: fundamentals. Advances in Computational Mathematics, 47(5). doi:10.1007/s10444-021-09888-1
    Sponsors
    J. Huang was supported by the NSF grant DMS1821093, and the work was finished while he was a visiting professor at the King Abdullah University of Science and Technology, National Center for Theoretical Sciences (NCTS) in Taiwan, Mathematical Center for Interdisciplinary Research of Soochow University, and Institute for Mathematical Sciences of the National University of Singapore.
    Publisher
    Springer Science and Business Media LLC
    Journal
    Advances in Computational Mathematics
    DOI
    10.1007/s10444-021-09888-1
    Additional Links
    https://link.springer.com/10.1007/s10444-021-09888-1
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10444-021-09888-1
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Extreme Computing Research Center; Statistics Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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