BIMSA > Seminar on Control Theory and Nonlinear Filtering A Convergent Scheme for the Bayesian Filtering Problem Based on the Fokker-Planck Equation and Deep Splitting
A Convergent Scheme for the Bayesian Filtering Problem Based on the Fokker-Planck Equation and Deep Splitting
Organizer
Shing Toung Yau
Speaker
Zeju Sun
Time
Thursday, January 30, 2025 9:00 PM - 9:30 PM
Venue
Online
Abstract
A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hormander condition, and empirically in two numerical examples. For the prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker–Planck equation with a deep splitting scheme, combined with an exact update through Bayes’ formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman–Kac approach, designed to mitigate the curse of dimensionality. The convergence proof relies on stochastic integration by parts from the Malliavin calculus. As a corollary we obtain the convergence rate for the approximation of the Fokker–Planck equation alone, disconnected from the filtering problem.