This paper deals with the state estimation of stochastic models with continuous dynamics. The aim is to incorporate spectral differentiation methods into the solution to the Fokker-Planck equation in grid-based state estimation routine, while taking into account the specifics of the field, such as probability density function (PDF) features, moving grid, zero boundary conditions, etc. The spectral methods, in general, achieve very fast convergence rate of $\mathcal{O}\left(c^{N}\right)(O{\lt}$ $c{\lt}1$) for analytical functions such as the probability density function, where N is the number of grid points. This is significantly better than the standard finite difference method (or midpoint rule used in discrete estimation) typically used in grid-based filter design with convergence rate $\mathcal{O}\left(\frac{1}{N^{2}}\right)$. As consequence, the proposed spectral method based filter provides better state estimation accuracy with lower number of grid points, and thus, with lower computational complexity.

This paper deals with state estimation of stochastic models with linear state dynamics, continuous or discrete in time. The emphasis is laid on a numerical solution to the state prediction by the time-update step of the grid-point-based point-mass filter (PMF), which is the most computationally demanding part of the PMF algorithm. A novel efficient PMF (ePMF) estimator, unifying continuous and discrete, approaches is proposed, designed, and discussed. By numerical illustrations, it is shown, that the proposed ePMF can lead to a time complexity reduction that exceeds 99.9% without compromising accuracy. The MATLAB® code of the ePMF is released with this paper.