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 restricted to unbounded domains or are too restrictive in terms of possible field properties. As a result, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields are especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. Unlike the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of a Helmholtz-like SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis (statFEM) and Gaussian process (GP) regression on geometrically complex domains. 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 Helmholtz-like SPDE so that we can model on bounded domains and manifolds anisotropic, non-homogeneous random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. 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 expressions for the mean vector and the precision matrix can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples.

翻译：在几何复杂域上高效表示随机场对于工程和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局限于无界域，要么在可能的场属性方面过于受限。因此，利用随机偏微分方程（SPDE）与随机场之间历史确立的联系的新技术，对于已有用于求解物理守恒方程的有限元离散化的复杂几何工程应用尤其具有吸引力。与随机场稠密的协方差矩阵不同，其逆矩阵（即精度矩阵）通常是稀疏的，且等于类亥姆霍兹SPDE的刚度矩阵。在本文中，我们利用SPDE表示为几何复杂域上的大规模统计有限元分析（statFEM）和高斯过程（GP）回归开发了一个可扩展框架。我们采用SPDE公式获得具有稀疏精度矩阵的相关先验概率密度。先验的性质由类亥姆霍兹SPDE的参数（可能包括分数阶）决定，因此我们能够在有界域和流形上对具有任意光滑度的各向异性、非均匀随机场进行建模。我们使用与求解物理守恒方程相同的有限元网格来组装稀疏精度矩阵。statFEM和GP回归的观测模型设定使得后验概率密度为高斯分布，且具有闭式均值和精度。均值向量和精度矩阵的表达式仅需使用稀疏矩阵运算即可求值。我们通过一维和二维泊松及薄壳示例展示了所提出框架的通用性及其收敛性质。