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