B-splines chaos and Kalman Filters for solving a stochastic differential equation

Abstract

A methodology is proposed to solve stochastic differential equations (SDEs) using B-spline chaos and Kalman Filters. The Fokker–Planck equation is employed to model physical processes involving nonlinear structures, non-Gaussian distributions, and multimodal distributions. B-spline chaos is used to approximate the solution of the SDE, while state estimation is achieved using Ensemble and Unscented Kalman Filter algorithms. To validate the approach, a time series of temperature and humidity data collected from manufacturing sensors is used, demonstrating the accuracy of the methodology in reconstructing the true system states. The proposed method is specifically designed for solving SDEs related to known physical processes. Additionally, numerical comparisons with existing approaches are presented to highlight the advantages in terms of performance and accuracy.