A Bayesian Approach to Modeling Finite Element Discretization Error

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.