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Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation
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Blömker, D., Law, K., Stuart, A. M. and Zygalakis, K. C. (2013) Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation. Nonlinearity, Volume 26 (Number 8). pp. 2193-2219. doi:10.1088/0951-7715/26/8/2193 ISSN 0951-7715.
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Official URL: http://dx.doi.org/10.1088/0951-7715/26/8/2193
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 the 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 observed modes is small. Numerical examples are given to illustrate the theory.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Nonlinearity | ||||
Publisher: | Institute of Physics Publishing Ltd. | ||||
ISSN: | 0951-7715 | ||||
Official Date: | August 2013 | ||||
Dates: |
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Volume: | Volume 26 | ||||
Number: | Number 8 | ||||
Page Range: | pp. 2193-2219 | ||||
DOI: | 10.1088/0951-7715/26/8/2193 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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