Extended Direct Method: A Time-Varying Kolmogorov Equation Approach for Infinite-Dimensional Optimal Filtering Problems

One of the most important breakthroughs was the propose of the Kalman filter in the 1960 s, which motivated a large number of studies on nonlinear filtering. Among these efforts, a special class of systems known as Yau filters has attracted attention. For the finite-dimensional case, the Direct Method (DM) was proposed to address systems with quadratic potential quantity [1, 2]. To generalize the DM, we shall focus on a class of infinite-dimensional filtering systems in a statistical sense. Our approach is capable of handling the potential quantity of arbitrary polynomial degree of filtering systems, meanwhile accommodating systems with nonlinear observation terms expressed as arbitrary-degree polynomials. The core idea is to apply several transformations to simplify the fundamental kernel corresponding to the filtering system. By leveraging a series expansion, we derive an explicit series solution from which optimal state estimates are obtained via convolution. Building on this series solution, we introduce a truncation technique and develop the Extended Direct Method (EDM) filtering algorithm which is supported by strong theoretical guarantees of uniqueness and existence through Euler operator theory. Numerical experiments demonstrate that our method not only generalizes the classical Direct Method to a wider class of systems but also enables fast, real-time implementation. It achieves near-optimal performance in challenging filtering scenarios compared to the Galerkin-Spectral method with less computational cost time, significantly outperforming conventional algorithms, including the Particle Filter, Extended Kalman Filter etc.