Efficient Spectral Differentiation in Grid-Based Continuous State Estimation

TL;DR

This paper introduces a spectral differentiation method for grid-based continuous state estimation that significantly improves convergence speed and accuracy over traditional finite difference methods, reducing computational complexity.

Contribution

It integrates spectral differentiation into the Fokker-Planck equation solution, enhancing efficiency and accuracy in grid-based state estimation for stochastic models.

Findings

Spectral methods achieve exponential convergence rate O(c^N).

Proposed method improves estimation accuracy with fewer grid points.

Lower computational complexity compared to finite difference approaches.

Abstract

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 O(c^N )(O < c < 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 O( 1 / N^2 ). 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.