Dimensional reduction in nonlinear filtering: A homogenization approach

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filter with rate $\sqrt{\varepsilon}$. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.Comment: Published in at http://dx.doi.org/10.1214/12-AAP901 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Homogenization Approach to Multiscale Filtering

AbstractWe present a homogenized nonlinear filter for multi-timescale systems, which allows the reduction of the dimension of filtering equation. We prove that the actual nonlinear filter converges to our homogenized filter. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and probabilistically representing the correction terms with the help of backward doubly-stochastic differential equations. This homogenized filter provides a rigorous mathematical basis for the development of reduced-dimension nonlinear filters for multiscale systems. A filtering scheme, based on the homogenized filtering equation and the technique of importance sampling, is applied to a chaotic multiscale system in Lingala et al. [1]

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

Photocatalytic Restoration of Liquid Effluent from Oil Palm Agroindustry in Malaysia using Tungsten Oxides Catalyst

In the current work, the photocatalytic treatment of palm oil mill effluent over tungsten oxides photocatalyst under ultraviolet-irradiation was evaluated. Characterization of fresh and used tungsten oxides photocatalyst was accomplished via X-Ray Diffraction, Scanning Electron Microscopy with Energy Dispersive X-Ray Analysis, Ultraviolet–Visible Light Diffuse Reflectance Spectroscopy, and Fourier Transformed-Infrared Spectroscopy. Photocatalytic treatment of palm oil mill effluent was conducted to determine the effects of catalyst loading, longevity, and recyclability of the tungsten oxides photocatalyst, as well as the effect of pH alteration on palm oil mill effluent. During the photocatalytic reaction, the collected liquid sample was tested for chemical oxygen demand, pH, and colour intensity while the gaseous sample was analyzed via gas chromatography. The optimum catalyst loading was 0.5 g/L, corresponds to highest photocatalytic degradation (51.15%) and decolourization (96.21%). The pH alteration on palm oil mill effluent has negligible effect on its photocatalytic degradation with UV/WO3 system. For longevity study, the optimum reaction time was 16 h, which achieved 84.70% photocatalytic degradation and 98.28% photocatalytic decolourization. From the recyclability study, it can be concluded that the tungsten oxides photocatalyst is suitable for photocatalytic decolourization of palm oil mill effluent, but not suitable for photocatalytic degradation. In addition, analysis of the gaseous product showed that the photocatalytic treatment has successfully degraded the organic pollutants in the liquid effluent into methane and carbon dioxide

Spacecraft Attitude Control: A Consideration of Thrust Uncertainty

© AIAATo account for torque disturbances, and control trajectory error, a model of spacecraft attitude system is presented which replicates uncertainty in the class of continuous low-thrust systems. The generated uncertainty from each thruster is modeled as a Gaussian white noise process, multiplicative in control. An optimal stochastic control law is derived for precision pointing and three-axis stabilization. To derive the optimal control, a Hamilton-Jacobi-Bellman equation is formulated, and a power series-based method is employed to approximate the optimal control. The derived nonlinear control minimizes the objective function of the Lagrange problem in an infinite horizon setting. Stability and existence conditions of control are provided. The nonlinear stochastic optimal controller is compared to its deterministic counterpart for a 6U CubeSat model.The authors acknowledge the support of the AFOSR under Grant Number FA9550-17-1-0001

Facile synthesis of CaFe2O4 for visible light driven treatment of polluting palm oil mill effluent: Photokinetic and scavenging study

In this paper, a facile synthesis method for CaFe2O4 is introduced that produces a catalyst capable of significant photocatalytic degradation of POME under visible light irradiation. The co-precipitation method was used to produce two catalysts at calcination temperatures of 550 °C and 700 °C dubbed CP550 and CP700. CP550 demonstrated the maximum COD removal of 69.0% at 0.75 g/L catalyst loading after 8 h of visible light irradiation which dropped to 61.0% after three consecutive cycles. SEM images indicated that the higher calcination temperature of CP700 led to annealing which reduced the pore volume (0.025 cm3/g) and pore diameter (10.3 nm) while simultaneously creating a smoother and more spherical surface with lower SBET (9.73 m2/g). In comparison, CP550 had a rough hair-like surface with higher SBET (27.28 m2/g) and pore volume (0.077 cm3/g) as evidenced by BET analysis. XRD data indicated the presence of CaFe5O7 in the CP550 composition which was not present in CP700. The presence of Wustite-like FeO structures in CaFe5O7 are likely the cause for lower photoluminescence intensity profile and hence better charge separation of CP550 as these structures in CaFe2O4 have been known to increase resistivity and electron localization. The COD removal of CP550 dropped from 69.0% to just 7.0% upon adding a small quantity of isopropanol into the reaction mixture indicating hydroxyl radicals as the primary reactive oxidative species