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Published June 13, 2017 | Submitted
Report Open

Stability of Filters for the Navier-Stokes Equation


Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in a on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise filtering can be performed exactly using the Kalman filter. For nonlinear systems it can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the properties of these ad hoc filters, working in the context of the 2D incompressible Navier-Stokes equation. By working in this infinite dimensional setting we provide an analysis which is useful for understanding high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. Numerical results are given which illustrate the theory. In a simplified scenario we also derive, and study numerically, a stochastic PDE which determines filter stability in the limit of frequent observations, subject to large observational noise. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters perform poorly in reproducing statistical variation about the true signal.

Additional Information

Submitted on 11 Oct 2011. AMS would like to thank the following institutions for financial support: EPSRC, ERC and ONR; KJHL was supported by EPSRC and ONR; and CEAB, KFL, DSM and MRS were supported EPSRC, through the MASDOC Graduate Training Centre at Warwick University. The authors also thank The Mathematics Institute and Centre for Scientific Computing. at Warwick University for supplying valuable computation time. Finally, the authors thank Masoumeh Dashti for valuable input.

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