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The efficient representation of random fields on geometrically complexdomains is crucial for Bayesian modelling in engineering and machine learning.Today's prevalent random field representations are either intended forunbounded domains or are too restrictive in terms of possible field properties.Because of these limitations, techniques leveraging the historicallyestablished link between stochastic PDEs (SPDEs) and random fields have beengaining interest. The SPDE representation is especially appealing forengineering applications which already have a finite element discretisation forsolving the physical conservation equations. In contrast to the densecovariance matrix of a random field, its inverse, the precision matrix, isusually sparse and equal to the stiffness matrix of an elliptic SPDE. We usethe SPDE representation to develop a scalable framework for large-scalestatistical finite element analysis and Gaussian process (GP) regression oncomplex geometries. The statistical finite element method (statFEM) introducedby Girolami et al. (2022) is a novel approach for synthesising measurement dataand finite element models. In both statFEM and GP regression, we use the SPDEformulation to obtain the relevant prior probability densities with a sparseprecision matrix. The properties of the priors are governed by the parametersand possibly fractional order of the SPDE so that we can model on boundeddomains and manifolds anisotropic, non-stationary random fields with arbitrarysmoothness. The observation models for statFEM and GP regression are such thatthe posterior probability densities are Gaussians with a closed-form mean andprecision. The respective mean vector and precision matrix and can be evaluatedusing only sparse matrix operations. We demonstrate the versatility of theproposed framework and its convergence properties with Poisson and thin-shellexamples.

Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis