Numerical experiments for nonlinear filters with exact particle flow induced by log-homotopy

We show numerical experiments on a new theory of exact particle flow for nonlinear filters. This generalizes our theory of particle flow that was already many orders of magnitude faster than standard particle filters and which is several orders of magnitude more accurate than the extended Kalman filter for difficult nonlinear problems. The new theory generalizes our recent log-homotopy particle flow filters in three ways: (1) the particle flow corresponds to the exact flow of the conditional probability density; (2) roughly speaking, the old theory was based on incompressible flow (like subsonic flight in air), whereas the new theory allows compressible flow (like supersonic flight in air); (3) the old theory suffers from obstruction of particle flow as well as singularities in the equations for flow, whereas the new theory has no obstructions and no singularities. Moreover, our basic filter theory is a radical departure from all other particle filters in three ways: (a) we do not use any proposal density; (b) we never resample; and (c) we compute Bayes' rule by particle flow rather than as a point wise multiplication.