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Kalman filtering and smoothing for linear wave equations with model error

Lee, Wonjung and McDougall, D. and Stuart, A. M. (2011) Kalman filtering and smoothing for linear wave equations with model error. Inverse Problems, 27 (9). Art. No. 095008. ISSN 0266-5611. https://resolver.caltech.edu/CaltechAUTHORS:20160801-175538072

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Abstract

Filtering is a widely used methodology for the incorporation of observed data into time-evolving systems. It provides an online approach to state estimation inverse problems when data are acquired sequentially. The Kalman filter plays a central role in many applications because it is exact for linear systems subject to Gaussian noise, and because it forms the basis for many approximate filters which are used in high-dimensional systems. The aim of this paper is to study the effect of model error on the Kalman filter, in the context of linear wave propagation problems. A consistency result is proved when no model error is present, showing recovery of the true signal in the large data limit. This result, however, is not robust: it is also proved that arbitrarily small model error can lead to inconsistent recovery of the signal in the large data limit. If the model error is in the form of a constant shift to the velocity, the filtering and smoothing distributions only recover a partial Fourier expansion, a phenomenon related to aliasing. On the other hand, for a class of wave velocity model errors which are time dependent, it is possible to recover the filtering distribution exactly, but not the smoothing distribution. Numerical results are presented which corroborate the theory, and also propose a computational approach which overcomes the inconsistency in the presence of model error, by relaxing the model.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1088/0266-5611/27/9/095008DOIArticle
http://iopscience.iop.org/article/10.1088/0266-5611/27/9/095008/metaPublisherArticle
https://arxiv.org/abs/1101.5096arXivDiscussion Paper
Additional Information:© 2011 IOP. Received 19 January 2011, in final form 29 June 2011. Published 16 August 2011. The authors would like to thank the following institutions for financial support: NERC, EPSRC, ERC and ONR. The authors also thank the Mathematics Institute and Centre for Scientific Computing at Warwick University for supplying valuable computation time.
Funders:
Funding AgencyGrant Number
Natural Environment Research Council (NERC)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ88
Issue or Number:9
Record Number:CaltechAUTHORS:20160801-175538072
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160801-175538072
Official Citation:Wonjung Lee et al 2011 Inverse Problems 27 095008
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69369
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:02 Aug 2016 23:18
Last Modified:03 Oct 2019 10:21

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