Data assimilation with the 2D Navier-Stokes equations: Optimal Gaussian asymptotics for the posterior measure

TL;DR

This paper proves a Bernstein-von Mises theorem for Bayesian data assimilation in 2D Navier-Stokes equations, showing Gaussian approximation of the posterior and optimal uncertainty quantification.

Contribution

It establishes the asymptotic Gaussianity of the posterior measure in a nonparametric Navier-Stokes data assimilation setting, with implications for prediction and uncertainty quantification.

Findings

Posterior measure approximates a Gaussian random field in supremum norm.

Root(N)-consistent estimators enable accurate future state prediction.

Bayesian algorithm attains the lower bound in local asymptotic minimax sense.

Abstract

A functional Bernstein - von Mises theorem is proved for posterior measures arising in a data assimilation problem with the two-dimensional Navier-Stokes equation where a Gaussian process prior is assigned to the initial condition of the system. The posterior measure, which provides the update in the space of all trajectories arising from a discrete sample of the (deterministic) dynamics, is shown to be approximated by a Gaussian random vector field arising from the solution to a linear parabolic PDE with Gaussian initial condition. The approximation holds in the strong sense of the supremum norm on the regression functions, showing that predicting future states of Navier-Stokes systems admits root(N)-consistent estimators even for commonly used nonparametric models. Consequences for coverage of credible bands and uncertainty quantification are discussed. A local asymptotic minimax theorem is derived that describes the lower bound for estimating the state of the nonlinear system, which is shown to be attained by the Bayesian data assimilation algorithm.