  The efficient representation of random fields on geometrically complex
domains is crucial for Bayesian modelling in engineering and machine learning.
Today's prevalent random field representations are either intended for
unbounded domains or are too restrictive in terms of possible field properties.
Because of these limitations, techniques leveraging the historically
established link between stochastic PDEs (SPDEs) and random fields have been
gaining interest. The SPDE representation is especially appealing for
engineering applications which already have a finite element discretisation for
solving the physical conservation equations. In contrast to the dense
covariance matrix of a random field, its inverse, the precision matrix, is
usually sparse and equal to the stiffness matrix of an elliptic SPDE. We use
the SPDE representation to develop a scalable framework for large-scale
statistical finite element analysis and Gaussian process (GP) regression on
complex geometries. The statistical finite element method (statFEM) introduced
by Girolami et al. (2022) is a novel approach for synthesising measurement data
and finite element models. In both statFEM and GP regression, we use the SPDE
formulation to obtain the relevant prior probability densities with a sparse
precision matrix. The properties of the priors are governed by the parameters
and possibly fractional order of the SPDE so that we can model on bounded
domains and manifolds anisotropic, non-stationary random fields with arbitrary
smoothness. The observation models for statFEM and GP regression are such that
the posterior probability densities are Gaussians with a closed-form mean and
precision. The respective mean vector and precision matrix and can be evaluated
using only sparse matrix operations. We demonstrate the versatility of the
proposed framework and its convergence properties with Poisson and thin-shell
examples.
