Continuous Data Assimilation with Stochastically Noisy Data

TL;DR

This paper studies a data-assimilation method for the 2D Navier-Stokes equations with noisy observations, providing explicit bounds on long-term error based on measurement noise variance and observation density.

Contribution

It offers explicit conditions on observation resolution that guarantee asymptotic error bounds for stochastic data assimilation in fluid dynamics.

Findings

Explicit bounds on the asymptotic error in $L^2$ and $H^1$ norms.

Conditions on observation density ensuring error control.

Analysis of average time error in mean.

Abstract

We analyze the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimension incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the the limit supremum, as the time tends to infinity, of the expected value of the $L^2$-norm and of the $H^1$ Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.