Posterior contraction rates of computational methods for Bayesian data assimilation

TL;DR

This paper investigates how quickly Bayesian data assimilation methods converge to the true solution as discretization becomes finer, providing theoretical convergence rates for different discretization techniques.

Contribution

It establishes convergence rates for the discrete posterior in Bayesian data assimilation, including novel analysis for finite element discretizations with tailored priors.

Findings

Convergence rates are proven for conforming discretization.

Finite element discretization achieves optimal rates with tailored priors.

Coupling of sample size and discretization dimension is key to analysis.

Abstract

In this paper, we analyze posterior consistency of a Bayesian data assimilation problem under discretization. We prove convergence rates for the discrete posterior to ground truth solution under both conforming discretization and finite element discretization (usually non-conforming). 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 an optimal convergence rate.