An integration-free approach for particle flow filtering

Log-homotopy particle flow filters realize nonlinear Bayesian estimation by continuously migrating samples from the prior to the posterior distribution. This transport is governed by a pseudo-time ordinary differential equation (ODE). A major practical challenge of these filters is the need for numerical integration, which suffers from high computational cost and susceptibility to stiffness. This paper develops an exact, integration-free closed-form solution for the exact Daum--Huang deterministic particle flow under vector linear Gaussian measurements. By transforming the ODE into a specific eigenspace, we derive closed-form algebraic expressions for both the homogeneous state transition matrix and the inhomogeneous forcing term. We prove that this analytic solution is equivalent to the exact Kalman measurement update. We embed this closed-form evaluation within an $N$-step piecewise method for nonlinear measurement models. We further propose a constant contraction rate substep schedule that equalizes the per-step contraction along the eigendirection of $D$ associated with the largest eigenvalue $\alpha_{\max}$. The result is a stiffness-mitigating, integration-free particle update for highly nonlinear measurement models. On a bearings-only tracking benchmark, it achieves the lowest error among the compared filters, at a per-update cost comparable to deterministic particle flow baselines and substantially lower than stochastic flows.