Quasi-Monte Carlo filtering in nonlinear dynamic systems

We develop a new framework for Bayesian filtering in general nonlinear dynamic systems based on the quasi-Monte Carlo (QMC) numerical techniques. We first propose a general approach to deterministic filtering called the quasi-Monte Carlo Kalman filter (QMC-KF), which unifies several existing advanced filtering methods in the literature, such as the unscented Kalman filter (UKF) and the quadrature Kalman filter (QKF). The computationally expensive step of calculating the Jacobian matrix involved in the extended Kalman filter (EKF) is avoided in the proposed QMC-KF approach. We also propose sequential quasi-Monte Carlo (SQMC) filtering techniques which is analogous to the sequential Monte Carlo (SMC) or particle filtering methods in the literature. We show in particular how to effectively combine deterministic filtering and adaptive importance sampling schemes, which lead to powerful SQMC filtering strategies. The properties of the proposed SQMC and SQMC/IS methods in terms of almost sure convergence and numerical error propagation behavior are analyzed. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed new QMC-based filtering algorithms.