  We consider a fast approximation method for a solution of a certain
stochastic non-local pseudodifferential equation. This equation defines a
Mat\'ern class random field. The approximation method is based on the spectral
compactness of the solution. We approximate the pseudodifferential operator
with a Taylor expansion. By truncating the expansion, we can construct an
approximation with Gaussian Markov random fields. We show that the solution of
the truncated version can be constructed with an over-determined system of
stochastic matrix equations with sparse matrices. We solve the system of
equations with a sparse Cholesky decomposition. We consider the convergence of
the discrete approximation of the solution to the continuous one. Finally
numerical examples are given.
