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### Open Access Journals

IPI

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.

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.

DCDS-B

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.

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