Particle filtering in high-dimensional chaotic systems

We present an efficient particle filtering algorithm for multiscale systems, that is adapted for simple atmospheric dynamics models which are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. The purpose of the present paper is to show that the homogenization method developed in Imkeller et al. (2011), which is applicable to high dimensional multi-scale filtering problems, along with important sampling and control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.Comment: 28 pages, 12 figure

Numerical Fitting-based Likelihood Calculation to Speed up the Particle Filter

The likelihood calculation of a vast number of particles is the computational bottleneck for the particle filter in applications where the observation information is rich. For fast computing the likelihood of particles, a numerical fitting approach is proposed to construct the Likelihood Probability Density Function (Li-PDF) by using a comparably small number of so-called fulcrums. The likelihood of particles is thereby analytically inferred, explicitly or implicitly, based on the Li-PDF instead of directly computed by utilizing the observation, which can significantly reduce the computation and enables real time filtering. The proposed approach guarantees the estimation quality when an appropriate fitting function and properly distributed fulcrums are used. The details for construction of the fitting function and fulcrums are addressed respectively in detail. In particular, to deal with multivariate fitting, the nonparametric kernel density estimator is presented which is flexible and convenient for implicit Li-PDF implementation. Simulation comparison with a variety of existing approaches on a benchmark 1-dimensional model and multi-dimensional robot localization and visual tracking demonstrate the validity of our approach.Comment: 42 pages, 17 figures, 4 tables and 1 appendix. This paper is a draft/preprint of one paper submitted to the IEEE Transaction

Laboratory Studies on Granular Filters and Their Relationship to Geotextiles for Stormwater Pollutant Reduction

Applications of geotextiles within tertiary stormwater treatment systems and for stormwater infiltration can provide a substrate for biofilm formation, enabling biological treatment of contaminants. Geotextiles can serve as an efficient part of stormwater filtration within the urban water environment. The project assessed the applications of three experimental granular filters as a sustainable urban drainage system (SUDS) for the decomposition of organic pollutant loading present in stormwater. The three filter rigs were packed with alternating layers of filter media consisting of gravel, pea gravel, sand and either a single, double or no layer of geotextile membrane. A nonwoven geotextile was layered within the filter media. The hydraulic loading capacity for the three filters matched that commonly used with conventional sand filters systems. Water quality parameters were quantified by measuring suspended solids, chemical oxygen demand, dissolved oxygen, pH, nitrate-nitrogen, and phosphate concentrations. It was found that Filter Rig No. 3 (upper and lower geotextile membrane) and Filter Rig No. 2 (single geotextile membrane) had a significant statistical difference in treatment performance from Filter Rig No. 1 (no geotextile membrane)