A PDE-based Adaptive Kernel Method for Solving Optimal Filtering Problems

TL;DR

This paper presents an innovative PDE-based adaptive kernel approach for solving optimal filtering problems, leveraging the Fokker-Planck equation and Bayesian inference to improve state estimation accuracy.

Contribution

The paper introduces a novel adaptive kernel method combined with operator decomposition to efficiently solve the Fokker-Planck equation in optimal filtering.

Findings

Effective approximation of probability distributions using adaptive Gaussian kernels

Enhanced filtering accuracy demonstrated through numerical experiments

Efficient solution of Fokker-Planck equation via operator decomposition

Abstract

In this paper, we introduce an adaptive kernel method for solving the optimal filtering problem. The computational framework that we adopt is the Bayesian filter, in which we recursively generate an optimal estimate for the state of a target stochastic dynamical system based on partial noisy observational data. The mathematical model that we use to formulate the propagation of the state dynamics is the Fokker-Planck equation, and we introduce an operator decomposition method to efficiently solve the Fokker-Planck equation. An adaptive kernel method is introduced to adaptively construct Gaussian kernels to approximate the probability distribution of the target state. Bayesian inference is applied to incorporate the observational data into the state model simulation. Numerical experiments have been carried out to validate the performance of our kernel method.