We apply parallel hierarchical matrix technique to identify unknown parameters and do predictions

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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Photo by Markus Spiske on Unsplash

Photo by Markus Spiske on Unsplash

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).

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