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 and generalizing the theory of spectral convergence of graph Laplacians. Graph representations allow inference and sampling with linear algebra methods for sparse matrices, thus reducing the computational cost. 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.
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