Stabilized BFGS approximate Kalman filter
Alexander Bibov Heikki Haario Antti Solonen
Inverse Problems & Imaging 2015, 9(4): 1003-1024 doi: 10.3934/ipi.2015.9.1003
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: observation-deficient inversion BFGS update low-memory storage Extended Kalman filter approximate Kalman filter chaotic dynamics.
Reduction and identification of dynamic models. Simple example: Generic receptor model
Heikki Haario Leonid Kalachev Marko Laine
Discrete & Continuous Dynamical Systems - B 2013, 18(2): 417-435 doi: 10.3934/dcdsb.2013.18.417
Identification of biological models is often complicated by the fact that the available experimental data from field measurements is noisy or incomplete. Moreover, models may be complex, and contain a large number of correlated parameters. As a result, the parameters are poorly identified by the data, and the reliability of the model predictions is questionable. We consider a general scheme for reduction and identification of dynamic models using two modern approaches, Markov chain Monte Carlo sampling methods together with asymptotic model reduction techniques. The ideas are illustrated using a simple example related to bio-medical applications: a model of a generic receptor. In this paper we want to point out what the researchers working in biological, medical, etc., fields should look for in order to identify such problematic situations in modelling, and how to overcome these problems.
keywords: Model identification Markov chain Monte Carlo (MCMC). model reduction asymptotic methods boundary function method

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