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.

翻译：几何复杂域上随机场的高效表示对于工程与机器学习中的贝叶斯建模至关重要。当前主流的随机场表示方法要么适用于无界域，要么在可表达的场属性方面过于受限。由于这些局限性，基于随机偏微分方程（SPDE）与随机场之间历史悠久的联系的技术正引起广泛关注。SPDE表示法尤其适用于已有有限元离散化以求解物理守恒方程的工程应用。与随机场的稠密协方差矩阵相比，其逆矩阵（即精度矩阵）通常是稀疏的，并等于椭圆型SPDE的刚度矩阵。我们利用SPDE表示法，针对复杂几何域上的大规模统计有限元分析与高斯过程（GP）回归开发了一个可扩展框架。由Girolami等人（2022）提出的统计有限元方法（statFEM）是一种融合测量数据与有限元模型的新方法。在statFEM和GP回归中，我们均采用SPDE公式以获取具有稀疏精度矩阵的相关先验概率密度。先验属性由SPDE的参数（可能包含分数阶）控制，从而能够在有界域和流形上建模具有任意光滑度的各向异性、非平稳随机场。statFEM和GP回归的观测模型使得后验概率密度为具有闭式均值与精度的高斯分布，其相应的均值向量和精度矩阵仅需通过稀疏矩阵运算即可求得。我们通过泊松和薄壳算例展示了所提框架的多功能性及其收敛特性。