Sparse Approximations of Fractional Matérn Fields

We consider fast lattice approximation methods for a solution of a certain stochastic non-local pseudodifferential operator equation. This equation defines a Matérn class random field. We approximate the pseudodifferential operator with truncated Taylor expansion, spectral domain error functional minimization and rounding approximations. This allows us to construct Gaussian Markov random field approximations. We construct lattice approximations with finite-difference methods. We show that the solutions can be constructed with overdetermined systems of stochastic matrix equations with sparse matrices, and we solve the system of equations with a sparse Cholesky decomposition. We consider convergence of the truncated Taylor approximation by studying band-limited Matérn fields. We consider the convergence of the discrete approximations to the continuous limits. Finally, we study numerically the accuracy of different approximation methods with an interpolation problem.