On posterior consistency of data assimilation with Gaussian process priors: the 2D Navier-Stokes equations

TL;DR

This paper proves that Bayesian data assimilation for the 2D Navier-Stokes equations with Gaussian priors leads to posterior distributions that concentrate near the true solution with increasing measurements, with convergence rates depending on initial conditions.

Contribution

It establishes posterior consistency and convergence rates for data assimilation in 2D Navier-Stokes equations using Gaussian process priors, including explicit estimates for backward uniqueness.

Findings

Posterior distribution concentrates near the true solution with enough measurements.

Convergence rate is generally at most inverse logarithmic in sample size.

Faster convergence rates are possible under specific initial condition assumptions.

Abstract

We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier-Stokes equations.