An explicit link between Gaussian fields and Gaussian Markov random fields: The SPDE approach

Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorising dense matrices is cubic in the dimension. Although the computational power today is at an all-time-high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the involved precision matrix sparse which enables the use of numerical algorithms for sparse matrices, that for fields in