Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

Markovian Whittle-Mat'ern fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, [ (kappa2 - Delta)α/2 u = W, ;; kappa ∈ ℝ, ; α ∈ mathbbN. ] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Mat'ern fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to α, kappa and thus allows a universal approximation scheme for the precision and covariance matrices of the entire (α, kappa)-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Mat'ern GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.