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

Blömker, D. and Law, K. and Stuart, A. M. and Zygalakis, K. C. (2013) Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation. Nonlinearity, 26 (8). pp. 2193-2219. ISSN 0951-7715. doi:10.1088/0951-7715/26/8/2193.

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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.

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Additional Information:©2013 IOP Publishing Ltd. & London Mathematical Society. Received 4 October 2012, in final form 21 May 2013 Published 2 July 2013 Online at Recommended by A L Bertozzi.
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ107
Issue or Number:8
Classification Code:Mathematics Subject Classification: 34A45, 34A55, 65K10, 65L09, 76D05, 93E24, 60H10, 60H15
Record Number:CaltechAUTHORS:20160726-141854847
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69230
Deposited By: Linda Taddeo
Deposited On:26 Jul 2016 22:41
Last Modified:11 Nov 2021 04:11

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