This article addresses sequential Bayesian filtering for nonlinear and stochastic dynamical systems. We extend a Galerkin-approach that was previously used for the prediction of non-Gaussian probability density functions, to incorporate linear and non-linear measurement updates. The proposed method results in a linear pipeline of prediction and update steps, which are computed as sparse matrix operations on the finite-dimensional coefficient vector. The performance of our approach is demonstrated in numerical experiments for nonlinear dynamical 2D- and 4D-systems, using results of a standard particle filter as reference, both in terms of accuracy and computational expenses.

This article addresses the computation of stationary and transient solutions of the Fokker-Planck equation for nonlinear stochastic processes. We extend a Galerkin-method, which was previously used to compute stationary solutions for nonlinear mechanical systems, by a generalized formulation of the test function space, including the original approach as a limit. The use of weighted test functions improves the performance in the stationary setting and enables the computation of transient solutions. The properties of the resulting linear system of equations are discussed, and results for stationary and transient probability density functions for nonlinear 1D-, 2D- and 4D-systems are presented.