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Accuracy and stability of filters for dissipative PDEs

Brett, C. E. A. and Lam, K. F. and Law, K. J. H. and McCormick, D. S. and Scott, M. R. and Stuart, A. M. (2013) Accuracy and stability of filters for dissipative PDEs. Physica D: Nonlinear Phenomena, 245 (1). pp. 34-35. ISSN 0167-2789. doi:10.1016/j.physd.2012.11.005.

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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 an 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 filtering 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 oceanography and weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the accuracy and stability properties of these ad hoc filters. We work in the context of the 2D incompressible Navier–Stokes equation, although the ideas readily generalize to a range of dissipative partial differential equations (PDEs). By working in this infinite dimensional setting we provide an analysis which is useful for the understanding of 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 perform accurately in tracking the signal itself (filter accuracy), 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. The tuning corresponds to what is known as variance inflation in the applied literature. Numerical results are given which illustrate the theory. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters can perform poorly in reproducing statistical variation about the true signal.

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Additional Information:© 2012 Elsevier. Received 27 March 2012; Received in revised form 26 July 2012; Accepted 21 November 2012; Available online 28 November 2012; Communicated by J. Garnier. 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 by 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.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Subject Keywords:Data assimilation, Dissipative PDE, Inverse problems, Cycled 3DVAR, Tikhonov–Phillips regularization, Navier–Stokes
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Andrew StuartJ98
Issue or Number:1
Record Number:CaltechAUTHORS:20160727-180601953
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69266
Deposited By: Linda Taddeo
Deposited On:29 Jul 2016 00:22
Last Modified:11 Nov 2021 04:11

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