Posterior Contraction Rates of Computational Methods for Bayesian Data Assimilation

In this paper, we analyze posterior consistency of a Bayesian data assimilation problem under discretization. We prove convergence rates for the discrete posterior to the ground truth solution under both conforming discretization and finite element discretization (usually nonconforming). The analysis is based on the coupling of asymptotics between the number of samples and the dimension of discrete spaces. In the finite element discretization, tailor-made discrete priors, instead of the discretization of continuous priors, are used to generate posterior contraction.