Lee, Wonjung, McDougall, D. (Damon) and Stuart, A. M.. (2011) Kalman filtering and smoothing for linear wave equations with model error. Inverse Problems, Vol.27 (No.9). Article no. 095008. ISSN 0266-5611

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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:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Kalman filtering, Wave equation
Journal or Publication Title:	Inverse Problems
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	0266-5611
Date:	2011
Volume:	Vol.27
Number:	No.9
Page Range:	Article no. 095008
Identification Number:	10.1088/0266-5611/27/9/095008
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/38484

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