On posterior consistency of data assimilation with Gaussian process priors: The 2D-Navier–Stokes equations
The Annals of Statistics · 2024 · Volume 52 · Issue 4
Abstract
We consider a nonlinear 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.
Keywords
35Q30, 62G20, Bayesian inference for the initial condition, nonlinear dynamical system
Enable JavaScript for citation tools and additional paper interactions.