Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published September 2011 | Submitted
Journal Article Open

Kalman filtering and smoothing for linear wave equations with model error

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.

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.

Attached Files

Submitted - 1101.5096.pdf

Files

1101.5096.pdf
Files (414.8 kB)
Name Size Download all
md5:8360571e8775e5a58bc2f6ce556fd4be
414.8 kB Preview Download

Additional details

Created:
August 19, 2023
Modified:
March 5, 2024