Inverse Problems and Imaging (IPI)

Stabilized BFGS approximate Kalman filter

Pages: 1003 - 1024, Volume 9, Issue 4, November 2015      doi:10.3934/ipi.2015.9.1003

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Alexander Bibov - LUT Mafy - Department of Mathematics and Physics, Lappeenranta University Of Technology, P.O. Box 20 FI-53851, Finland (email)
Heikki Haario - Department of Mathematics and Physics, Lappeenranta University of Technology, P.O.Box 20, FIN-53851 Lappeenranta, Finland (email)
Antti Solonen - Lappeenranta University of Technology, Department of Mathematics and Physics, Lappeenranta, P.O. Box 20 FI-53851, Finland (email)

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 $n\times n$ and $n\times m$ matrices and inversion of $m\times m$ 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.

Keywords:  Extended Kalman filter, approximate Kalman filter, low-memory storage, BFGS update, observation-deficient inversion, chaotic dynamics.
Mathematics Subject Classification:  Primary: 60G35, 93E11; Secondary: 62M20.

Received: August 2014;      Revised: May 2015;      Available Online: October 2015.