The study of random dynamical systems involves understanding the evolution of state variables that contain uncertainties and that are usually hidden, or not directly observable. Therefore, state variables have to be estimated and updated based on system models using information from observational data, which themselves are noisy, in the sense that they contain uncertainties and disturbances due to imperfections in observational devices and disturbances in the environment within which data are being collected. The development of efficient data assimilation methods for integrating observational data in predicting the evolution of random state variables is thus an important aspect in the study of random dynamical systems. In this paper, we consider a particle filtering approach to nonlinear filtering in multiscale dynamical systems. Particle filtering methods [1–3] utilizes ensembles of particles to represent the conditional density of state variables using particle positions, distributed over a sample space. The distribution of an ensemble of particles is updated using observational data to obtain the best representation of the conditional density of the state variables of interest. On the other hand, homogenization theory [4, 5], allows us to estimate the coarse-grained (slow) dynamics of a multiscale system on a larger timescale without having to explicitly study the fast variable evolution on a small timescale. The results of filter convergence presented in [6] shows the convergence of the filter of the actual state variable to a homogenized solution to the original multiscale system, and thus we develop a particle filtering scheme for multiscale random dynamical systems that utilizes this convergence result. This particle filtering method is called the Homogenized Hybird Particle Filter, and it incorporates a multiscale computation scheme, the Heterogeneous Multiscale Method developed in [7], with the novel branching particle filter described in [8–10]. By incorporating a multiscale scheme based on homogenization of the original system, estimation of the coarse-grained dynamics using observational data is performed over a larger timescale, thus resulting in computational time and cost reduction in terms of the evolution of the state variables as well as functional evaluations for the filtering aspect. We describe the theory behind this combined scheme and its general algorithm, concluded with an application to the Lorenz-96 [11] atmospheric model that mimics midlatitude geophysical dynamics with microscopic convective processes.
Lattice reduction in two dimensions: analyses under realistic probabilistic models
