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

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Abstract

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.

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  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Fengshan Zhang, Yongkui Zou, Shimin Chai & Ran Zhang

  2. Department of Mathematics and Statistics, Auburn University, AL 36840, Auburn, USA

    Yanzhao Cao

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The research of Yongkui Zou and Ran Zhang is partly supported by the National Key R &D Program (2020YFA0714100, 2020YFA0713601), NSFC (12171199, 11971198), Jilin Provincial Department of science and technology (20210201015GX), and the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education at Jilin University. Research of Yanzhao Cao is partially supported by U.S. Department of Energy under grant number DE-SC0022253.

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Zhang, F., Zou, Y., Chai, S. et al. Splitting-up Spectral Method for Nonlinear Filtering Problems with Correlation Noises. J Sci Comput 93, 25 (2022). https://doi.org/10.1007/s10915-022-01994-6

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