In recent years, there has been a surge of interest in the development of probabilistic approaches to problems that might appear to be purely deterministic. One example of this is the solving of partial differential equations. Since numerical solvers require some approximation of the infinite-dimensional solution space, there is an inherent uncertainty to the solution that is obtained. In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. Due to intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. The prior distribution assumed over the fine discretization is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean close to the deterministic fine-scale solution that is endowed with an uncertainty measure. The prior distribution over the solution space is defined implicitly by assigning a white noise distribution to the right-hand side. This allows for a sparse representation of the prior distribution, and guarantees that the prior samples have the appropriate level of smoothness for the problem at hand. Special attention is paid to inhomogeneous Dirichlet and Neumann boundary conditions, and how these can be used to enhance this white noise prior distribution. For various problems, we demonstrate how regions of large discretization error are captured in the structure of the posterior standard deviation. The effects of the hyperparameters and observation noise on the quality of the posterior mean and standard deviation are investigated in detail.

翻译：近年来，针对看似纯粹确定性问题发展概率方法的研究兴趣显著增长。求解偏微分方程便是其中一例。由于数值求解器需要对无穷维解空间进行近似，所获得的解天然存在不确定性。本研究遵循贝叶斯范式对有限元离散化误差相关的不确定性进行建模。首先推导出连续形式，其中基于有限元离散化的观测数据更新解空间上的高斯过程先验。由于存在难以处理的积分，引入第二个更精细的离散化，假设其足够密集以表示真实解场。随后基于粗离散化的观测数据更新精细离散化上的先验分布，从而得到均值接近确定性细尺度解且具有不确定性度量的后验分布。通过为右端项赋予白噪声分布隐式定义解空间上的先验分布，这既实现了先验分布的稀疏表示，又能保证先验样本具有适用于所研究问题的适当光滑性。特别关注非齐次狄利克雷和诺伊曼边界条件，以及如何利用这些条件增强白噪声先验分布。针对多个问题，我们展示了后验标准差结构如何捕捉离散化误差较大的区域。详细研究了超参数和观测噪声对后验均值与标准差质量的影响。