Stiffness Mitigation in Stochastic Particle Flow Filters

TL;DR

This paper explores alternative homotopy forms in particle flow filters, deriving conditions for stability and proposing an optimal control approach to mitigate stiffness, demonstrated through numerical example.

Contribution

It introduces a novel method for stiffness mitigation in particle flow filters by formulating it as an optimal control problem and deriving an efficient solution.

Findings

The proposed approach improves stability of particle flows.

Optimal homotopy reduces stiffness more effectively than traditional methods.

Numerical example confirms the method's effectiveness.

Abstract

The linear convex log-homotopy has been used in the derivation of particle flow filters. One natural question is whether it is beneficial to consider other forms of homotopy. We revisit this question by considering a general linear form of log-homotopy for which we derive particle flow filters, validate the distribution of flows, and obtain conditions for the stability of particle flows. We then formulate the problem of stiffness mitigation as an optimal control problem by minimizing the condition number of the Hessian matrix of the posterior density function. The optimal homotopy can be efficiently obtained by solving a one-dimensional second order two-point boundary value problem. Compared with traditional matrix analysis based approaches to condition number improvements such as scaling, this novel approach explicitly exploits the special structure of the stochastic differential equations in particle flow filters. The effectiveness of the proposed approach is demonstrated by a numerical example.