Quasi-Monte Carlo particle filters: the JV filter

Abstract

We describe a new particle filter that uses quasi-Monte Carlo (QMC) sampling with product measures rather than boring old Monte Carlo sampling or QMC with or without randomization. The product measures for QMC were recently invented by M. Junk and G. Venkiteswaran, and therefore we call this new nonlinear filter the "JV filter". Standard particle filters use boring old Monte Carlo sampling and suffer from the curse of dimensionality, and they converge at the sluggish rate of c(d)/√N in which N is the number of particles, and c(d) depends strongly on dimension of the state vector (d). Oh's theory and numerical experiments (by us) show that for good proposal densities, c(d) grows as d³, whereas for poor proposal densities c(d) grows exponentially with d. In contrast, for certain problems, QMC converges much faster than MC with N. In particular, QMC converges as k(d)/N, in which k(d) is logarithmic in N and its dependence on d is an interesting story.