Sequential Bayesian Inference for Dynamical Systems Using the Finite Volume Method

Abstract

Optimal Bayesian sequential inference, or filtering, for the state of a deterministic dynamical system requires simulation of the Frobenius-Perron operator, that can be formulated as the solution of an initial value problem in the continuity equation on filtering distributions. For low-dimensional, smooth systems the finitevolume method is an effective solver that conserves probability and gives estimates that converge to the optimal continuous-time values. A Courant–Friedrichs–Lewy condition assures that intermediate discretized solutions remain positive density functions. We demonstrate this finite-volume filter (FVF) in a simulated example of filtering for the state of a pendulum, including a case where rank-deficient observations lead to multi-modal probability distributions.