Optimal nonlinear filtering using the finite-volume method

Optimal 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 a continuity equation. For low-dimensional, smooth systems, the finite-volume numerical method provides a solution that conserves probability and gives estimates that converge to the optimal continuous-time values, while a Courant-Friedrichs-Lewy-type condition assures that intermediate discretized solutions remain positive density functions. This method is demonstrated in an example of nonlinear filtering for the state of a simple pendulum, with comparison to results using the unscented Kalman filter, and for a case where rank-deficient observations lead to multimodal probability distributions.

Figure 1

Filter results for ${\theta }_0=0.2\pi$ with 30 observations per $2\pi$ time, with ${\sigma }_z=0.2$. Each UKF (upper pair) and FVF (lower pair) shows a true and filtered angular displacement $\theta$ and angular velocity $\omega$. Predicted mean (blue solid), 95% region (shaded), observations (red crosses), and true solution (black dashed).

Figure 2

Filter results for ${\theta }_0=0.9\pi$ with five observations per $2\pi$ time, having ${\sigma }_z=0.2$. Equivalent plots to Fig. 1.

Figure 3

Filter results for ${\theta }_0=0$ with 50 observations per $2\pi$ time, having ${\sigma }_z=0.4$. Equivalent plots to Fig. 1.

Figure 4

Initial ($t=0$) and filtered pdf's in phase space after measurements at times $t=\pi /4$, $\pi$, and $3\pi$ (left to right, top to bottom).