What is it about?

We describe the method and provide implementation details. 1. We use parallel hierarchical (H-) matrices to approximate large covariance matrices of the Matern type. 2. Then we compute the Cholesky factorisation, matrix-determinant, MV product, quadratic form, the inverse, the joint Gaussian likelihood. Everything in the H-matrix format. 3. We search the maximum of the joint Gaussian log-likelihood. This is a non-linear function, depending in a non-linear way on 3 parameters of the Matern covariance. 4. We research how the H-matrix approximation accuracy influences the confidence intervals for parameter estimates. See also https://www.hlibpro.com/doc/2.7/TUTORLoglikelyhood.html Reproducible updated code https://github.com/litvinen/large_random_fields

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Why is it important?

Estimation of unknown parameters is fundamental engineering, medical, chemical, geological, weather prediction problem. The maximum likelihood estimation (MLE) method is a well known and very often used method. The drawback is that it requires very expensive matrix arithmetic, with O(n^3) computing cost and storage. The usage of the H-matrix technique allows us to avoid this drawback. The new storage and complexity are O(n log n).


Since we provide very professional, fast and very well documented c++ code, we strongly believe that our colleagues from statistics and engineering science will use it for estimation of unknown parameters. One can solve problems with matrices of size 2.000.000 x 2.000.000 in few minutes on a good desktop computer. To profit from the parallel version, we recommend to use shared memory computers with multiple parallel cores (good one costs ~5.000- 10.000 USD).

Dr. Alexander Litvinenko
Rheinisch Westfalische Technische Hochschule Aachen

Read the Original

This page is a summary of: HLIBCov: Parallel hierarchical matrix approximation of large covariance matrices and likelihoods with applications in parameter identification, MethodsX, July 2019, Elsevier, DOI: 10.1016/j.mex.2019.07.001.
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