The SPDE Approach to Matérn Fields: Graph Representations

Abstract: This paper investigates Gaussian Markov random field approximations to nonstationary Gaussian fields using graph representations of stochastic partial differential equations. We establish approximation error guarantees building on the theory of spectral convergence of graph Laplacians. The proposed graph representations provide a generalization of the Mat\'ern model to unstructured point clouds, and facilitate inference and sampling using linear algebra methods for sparse matrices. In addition, they bridge and unify several models in Bayesian inverse problems, spatial statistics and graph-based machine learning. We demonstrate through examples in these three disciplines that the unity revealed by graph representations facilitates the exchange of ideas across them.