Explicit Construction of Gmrf Approximations to Generalised Matérn Fields on Irregular Grids

Models based on stationary covariance functions, such as the Matérn family, are commonly used in spatial statistics. For large grid resolutions, the computational effort can be substantial, as well as the memory requirements. A solution to this problem is to approximate the stationary covariances with aMarkov random field model on the grid. This report contains the theoretical basis for one such approximation method, using the finite element method to explicitly construct sparse precision matrices for the solutions to the same stochastic partial differential equations that generate the Matérn covariances. The method can be applied not only for planar regular grids, but also on irregular triangulations of curved manifolds.