Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation

Abstract

The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates study of the problem of accuracy and stability of 3DVAR filters for the Navier-Stokes equation.We work in the limit of high frequency observations and derive continuous time filters. This leads to a stochastic partial differential equation (SPDE) for state estimation, in the form of a damped-driven Navier-Stokes equation, with mean-reversion to the signal, and spatially-correlated time-white noise. Both forward and pullback accuracy and stability results are proved for this SPDE, showing in particular that when enough low Fourier modes are observed, and when the model uncertainty is larger than the data uncertainty in these modes (variance inflation), then the filter can lock on to a small neighbourhood of the true signal, recovering from order one initial error, if the error in the observations modes is small. Numerical examples are given to illustrate the theory.

Long-Time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems

“…Let δ 1 j and δ 2 j be two realizations of the H-valued random variable δ j and set 2 for all x ∈ H and some θ > 0. 2.…”

Optimization Based Methods for Partially Observed Chaotic Systems

“…After some simple algebra, the smoothing distribution for an initial point u and the filtering distribution for the current position u(T ) can be expressed as 6) where C sm k is a normalising constant independent of v (but depending on (Z j ) j≥0 ). In the following sections, we will present our main results for the smoother and the filter.…”

“…In the context of infinite-dimensional models, MAP estimators are non-trivial to define in a mathematically precise way on the infinitedimensional function space, but several definitions of MAP estimators, various weak consistency results under the small noise limit, and posterior contraction rates have been shown in recent years, see, for example, [10,16,19,23,25,34,36,49]. Some other important works on similar models and/or associated filtering/smoothing algorithms include [6,22,28]. These results are very interesting from a mathematical and statistical point of view; however, the intuitive meaning of some of the necessary conditions, and their algorithmic implications are difficult to grasp.…”

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

“…See, e.g., [69] for an overview of nudging methods. We note that a method that resembles the AOT method in some respects was introduced in [14] in the context of stochastic differential equations. Much recent literature has built upon the AOT algorithm and its ideas; see, e.g., [1, 2, 11-13, 20, 23, 39-43, 47, 48, 51-53, 57, 59-61, 70, 72, 85-87, 94, 96].…”