Parallel sampling of GMRFs and geostatistical GMRF models

In this report the main focuses are geostatistical Gaussian Markov random field (GMRF) models and parallel exact sampling of GMRFs. There are also brief overviews of parallel computing and Markov chain Monte Carlo (MCMC) methods, and a literature review of parallel MCMC. The geostatistical GMRF models are constructed by discretising the domain region using a lattice. Instead of giving this lattice a Gaussian random field prior, that corresponds to a Gaussian process, a GMRF that is an approximation to the GRF is chosen. More computational benefits are achieved through the nice parallelisation possibilities of GMRF sampling end evaluation. The computationally expensive part of GMRF sampling is Choleskey decomposition of the precision matrix. Parallelisation is done with parallel algorithms from linear algebra for sparse symmetric positive definite matrices. The parallel GMRF sampler is tested for graphs and lattices, and gives both good speed-up and good scalability. A parallel one-block updating scheme Metropolis-Hastings sampler for latent GMRF models is constructed using a GMRF approximation to π(x|y, θ) as proposal for the latent field. It is used for a geostatistical GMRF model with binomial likelihood, and shows good mixing for both the latent field and the hyper-parameters, as well as good speed-up from the parallelisation.