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HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification

​A. Litvinenko, HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification, https://arxiv.org/abs/1709.08625, 2017
Alexander Litvinenko
HLIBPro, HLIBCov, parallel hierarchical matrices, approximating large covariance matrice, parameter estimation, Gaussian likelihood
2017

​The main goal of this article is to introduce the parallel hierarchical matrix library HLIBpro to the statistical community. We describe the HLIBCov package, which is an extension of the HLIBpro library for approximating large covariance matrices and maximizing likelihood functions. We show that an approximate Cholesky factorization of a dense matrix of size 2M×2M can be computed on a modern multi-core desktop in few minutes. Further, HLIBCov is used for estimating the unknown parameters such as the covariance length, variance and smoothness parameter of a Mat\'ern covariance function by maximizing the joint Gaussian log-likelihood function. The computational bottleneck here is expensive linear algebra arithmetics due to large and dense covariance matrices. Therefore covariance matrices are approximated in the hierarchical (H-) matrix format with computational cost O(k^2nlog^2n) and storage O(knlogn), where the rank k is a small integer (typically k<25), n the number of locations on a fairly general mesh. We demonstrate a synthetic example, where the true values of known parameters are known. For reproducibility we provide the C++ code, the documentation, and the synthetic data.