Splitting-up Spectral Method for Nonlinear Filtering Problems with Correlation Noises

• Published in: Journal of scientific computing Vol. 93; no. 1; p. 25
• Main Authors: Zhang, Fengshan, Zou, Yongkui, Chai, Shimin, Zhang, Ran, Cao, Yanzhao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2022; Springer Nature B.V; Springer
• Subjects: Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Discretization; Dynamical systems; Filtration; Finite difference method; Galerkin method; Iterative methods; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Noise; Partial differential equations; Signal processing; Spectral methods; Splitting; Stochastic models; Theoretical; Nonlinear filtering problem; Splitting-up technique; Error analysis; Zakai equation
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-01994-6

In this paper, we study nonlinear filtering problems via solving their corresponding Zakai equations. Using the splitting-up technique, we approximate the Zakai equation with two equations consisting of a first-order stochastic partial differential equation and a deterministic second-order partial differential equation. For the splitting-up equations, we use a spectral Galerkin method for the spatial discretization and a finite difference scheme for the temporal discretization. The main results are an error estimate for the semi-discretized scheme with respect to the spatial variable, and an error estimate for the full discretized scheme. To improve the numerical performance, we apply an adaptive technique to accurately locate the support domain of the solution in each time iteration. Finally, we present numerical experiments to demonstrate our theoretical analysis.