Particle flow for nonlinear filters with log-homotopy
by
Fred Daum
Raytheon
Coauthors: Jim Huang

We derive and test a new nonlinear filter that uses log-homotopy to implement Bayes’ rule with an ODE rather than a point wise multiplication of two functions. Homotopy allows us to migrate the particles smoothly in time, thereby avoiding the fundamental problem with particle filters, namely “particle collapse” as a result of Bayes’ rule. The initial condition of this ODE is the unnormalized conditional density prior to the kth measurement, and the final solution of the ODE is the unnormalized conditional density after the kth measurement. It turns out that a homotopy of the density itself does not work, owing to singularity of the initial condition, whereas a homotopy of the log of the density removes this singularity and works well. We compare the computational complexity of our new algorithm with a carefully designed particle filter for an interesting class of smooth nonlinear filter problems of increasing dimension (d = 1 to 20) for optimal estimation accuracy. The computational complexity of the log-homotopy filter is many orders of magnitude less than the classic particle filter for optimal estimation accuracy for d = 10 and above.

Date received: August 20, 2007