ON POSTERIOR CONSISTENCY OF DATA ASSIMILATION WITH GAUSSIAN PROCESS PRIORS: THE 2D-NAVIER-STOKES EQUATIONS

• 成果类型: Article
• 署名作者: Nickl, Richard; Titi, Edriss s.
• 署名单位: University of Cambridge; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2427
• 发表日期: 2024
• 页码: 1825-1844
• 关键词: Bayesian Inverse Problems; inference; mcmc

摘要

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.