Markovian Whittle-Matérn 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, \[ (κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}. \] 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 Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $α, κ$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(α, κ)$-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 Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

翻译：马尔可夫Whittle-Matérn场已通过稀疏精度矩阵的离散高斯马尔可夫随机场(GMRF)得到收敛近似，该方法采用两参数族\( (κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N} \)的随机偏微分方程有限元逼近。利用离散外微分(DEC)分析的最新进展，我们提出一种不同但密切相关的收敛GMRF近似，用于在完全无边界黎曼流形上离散为良好中心单纯复形的这些Matérn场。这种收敛方法：(i) 对\( α, κ \)参数不可知，从而能为整个\( (α, κ) \)-族GMRF的精度和协方差矩阵提供通用近似方案，使其可通过推断而非猜测获得；(ii) 天然建模随机场的点测量和分段平滑测量，并对两者给出同等良好的近似；(iii) 计算上与所用插值函数无关——若在同一网格上将一种收敛插值函数替换为另一种合适插值函数，则不会产生额外开销。此外，我们证明，在精确意义下良好连接且体积集中的离散化上，精度矩阵是图拉普拉斯算子的谱函数。我们为这类Matérn GMRF族提供低秩近似器，并提及一个应用场景：通过压缩感知减少建模该GMRF所需的测量数量。