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 Matérn 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.
DSA is thankful for the support of NSF and NGA through the grant DMS-2027056. The work of DSA was also partially supported by the NSF Grant DMS-1912818/1912802.
Daniel Sanz-Alonso. Ruiyi Yang. "The SPDE Approach to Matérn Fields: Graph Representations." Statist. Sci. 37 (4) 519 - 540, November 2022. https://doi.org/10.1214/21-STS838