Hermite spectral method to 1D forward Kolmogorov equation and its application to nonlinear filtering problems

TL;DR

This paper develops a Hermite spectral method for solving the forward Kolmogorov equation, improving resolution and convergence, and applies it to nonlinear filtering problems, demonstrating real-time efficiency surpassing particle filters.

Contribution

It introduces a scaled Hermite spectral method with a guideline for parameter choice and applies it as an off-line component in a real-time nonlinear filtering algorithm.

Findings

Enhanced resolution of HSM with proper scaling

Verified convergence rate through analysis and simulation

Outperforms particle filters in real-time nonlinear filtering

Abstract

In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problem with a real-time algorithm developed in [17]. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filter as a real-time solver to NLF.