A Split-Step Solution of the Fokker-Planck Equation for the Conditional Density

We have developed and tested a new algorithm, which we call the "wave filter," that solves the nonlinear filtering problem with discrete-time measurements by solving the Fokker-Planck equation for the conditional probability density function using a split-step technique. The wave filter uses fast convolution to compute the effect of process noise at discrete times. Between measurements, the conditional density is propagated by solving a system of ordinary differential equations. Measurement update is carried out via Bayes\´ rule. We propose the "adjoint method" to reduce the computational complexity of the algorithm by adaptively varying the mesh size.