Stabilized BFGS approximate Kalman filter
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Bibov_2015_Stabilized BFGs.pdf
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Author(s)
Bibov, Alexander
Haario, Heikki
Solonen, Antti
Date Issued
October 2015
Journal
Inverse Problems and Imaging
Publisher
American Institute of Mathematical Sciences (AIMS)
Citation
Bibov, Alexander, Heikki Haario, and Antti Solonen. “Stabilized BFGS Approximate Kalman Filter.” IPI 9, no. 4 (October 2015): 1003–1024. © 2015 American Institute of Mathematical Sciences
Version
Final published version
Abstract
The Kalman filter (KF) and Extended Kalman filter (EKF) are well-known tools for assimilating data and model predictions. The filters require storage and multiplication of n × n and n × m matrices and inversion of m × m matrices, where n is the dimension of the state space and m is dimension of the observation space. Therefore, implementation of KF or EKF becomes impractical when dimensions increase. The earlier works provide optimization-based approximative low-memory approaches that enable filtering in high dimensions. However, these versions ignore numerical issues that deteriorate performance of the approximations: accumulating errors may cause the covariance approximations to lose non-negative definiteness, and approximative inversion of large close-to-singular covariances gets tedious. Here we introduce a formulation that avoids these problems. We employ L-BFGS formula to get low-memory representations of the large matrices that appear in EKF, but inject a stabilizing correction to ensure that the resulting approximative representations remain non-negative definite. The correction applies to any symmetric covariance approximation, and can be seen as a generalization of the Joseph covariance update.
We prove that the stabilizing correction enhances convergence rate of the covariance approximations. Moreover, we generalize the idea by the means of Newton-Schultz matrix inversion formulae, which allows to employ them and their generalizations as stabilizing corrections.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.3934/ipi.2015.9.1003
