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

Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation
Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation
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.
0951-7715
2193
Blömker, D.
8a16a87c-3447-4a10-9378-9d946977d639
Law, K.
f5316293-e949-4051-9dbe-670c68e715e8
Stuart, A.M.
0b611978-6d94-488c-b92b-09cfc1a2ceb2
Zygalakis, K.C.
a330d719-2ccb-49bd-8cd8-d06b1e6daca6
Blömker, D.
8a16a87c-3447-4a10-9378-9d946977d639
Law, K.
f5316293-e949-4051-9dbe-670c68e715e8
Stuart, A.M.
0b611978-6d94-488c-b92b-09cfc1a2ceb2
Zygalakis, K.C.
a330d719-2ccb-49bd-8cd8-d06b1e6daca6

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, 26 (8), 2193. (doi:10.1088/0951-7715/26/8/2193).

Record type: Article

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.

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More information

e-pub ahead of print date: 2 July 2013
Published date: August 2013
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 355727
URI: https://eprints.soton.ac.uk/id/eprint/355727
ISSN: 0951-7715
PURE UUID: e24dff99-b613-4bec-9650-1420a3f6a60f

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Date deposited: 04 Sep 2013 10:07
Last modified: 18 Jul 2017 03:45

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