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Filter accuracy for the Lorenz 96 model: fixed versus adaptive observation operators

Law, K. J. H. and Sanz-Alonso, D. and Shukla, A. and Stuart, A. M. (2016) Filter accuracy for the Lorenz 96 model: fixed versus adaptive observation operators. Physica D: Nonlinear Phenomena, 325 . pp. 1-13. ISSN 0167-2789. http://resolver.caltech.edu/CaltechAUTHORS:20160715-152128338

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Abstract

In the context of filtering chaotic dynamical systems it is well-known that partial observations, if sufficiently informative, can be used to control the inherent uncertainty due to chaos. The purpose of this paper is to investigate, both theoretically and numerically, conditions on the observations of chaotic systems under which they can be accurately filtered. In particular, we highlight the advantage of adaptive observation operators over fixed ones. The Lorenz ’96 model is used to exemplify our findings. We consider discrete-time and continuous-time observations in our theoretical developments. We prove that, for fixed observation operator, the 3DVAR filter can recover the system state within a neighbourhood determined by the size of the observational noise. It is required that a sufficiently large proportion of the state vector is observed, and an explicit form for such sufficient fixed observation operator is given. Numerical experiments, where the data is incorporated by use of the 3DVAR and extended Kalman filters, suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction. Adaptive observation operators are then studied numerically; we show that, for carefully chosen adaptive observation operators, the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator. Indeed, we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive Lyapunov exponents of the underlying system.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.physd.2015.12.008DOIArticle
http://www.sciencedirect.com/science/article/pii/S0167278915002766PublisherArticle
https://arxiv.org/abs/1411.3113arXivDiscussion Paper
Alternate Title:Controlling Unpredictability with Observations in the Partially Observed Lorenz '96 Model
Additional Information:© 2016 Elsevier. Received 4 March 2015; Received in revised form 19 October 2015; Accepted 23 December 2015; Available online 23 February 2016; Communicated by Edriss S. Tit. AbS and DSA are supported by the EPSRC-MASDOC graduate training scheme. AMS is supported by EPSRC, ERC and ONR. KJHL is supported by King Abdullah University of Science and Technology, and is a member of the KAUST SRI-UQ Center.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
King Abdullah University of Science and Technology (KAUST)UNSPECIFIED
Subject Keywords:3DVAR; Lorenz ’96; Filter accuracy; Adaptive observations; Extended Kalman filter
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ122
Record Number:CaltechAUTHORS:20160715-152128338
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20160715-152128338
Official Citation:K.J.H. Law, D. Sanz-Alonso, A. Shukla and A.M. Stuart, Filter accuracy for the Lorenz 96 model: fixed versus adaptive observation operators. Physica D: Nonlinear Phenomena 325 (2016) 1-13.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69069
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:18 Jul 2016 23:01
Last Modified:01 Aug 2017 22:56

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