Communications in Information and Systems

Volume 14 (2014)

Number 2

An efficient algorithm of Yau-Yau method for solving nonlinear filtering problems

Pages: 111 – 134

DOI: https://dx.doi.org/10.4310/CIS.2014.v14.n2.a3

Authors

Mei-Heng Yueh (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Wen-Wei Lin (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

It is well known that the nonlinear filter has important applications in military, engineering and commercial industries. In this paper, we propose efficient and accurate numerical algorithms for the realization of the Yau-Yau method for solving nonlinear filtering problems by using finite difference schemes. The Yau-Yau method reduces the nonlinear filtering problem to the initial-value problem of Kolmogorov equations. We first solve this problem by the implicit Euler method, which is stable in most cases, but costly. Then, we propose a quasi-implicit Euler method which is feasible for acceleration by fast Fourier transformations. Furthermore, we propose a superposition technique which enables us to deal with the nonlinear filtering problem in an off-time process and thus, save a large amount of computational cost. Next, we prove that the numerical solutions of Kolmogorov equations by our schemes are always nonnegative in each iteration. Consequently, our iterative process preserves the probability density functions. In addition, we prove convergence of our schemes under some mild conditions. Numerical results show that the proposed algorithms are efficient and promising.

Keywords

nonlinear filtering, Kolmogorov equations

2010 Mathematics Subject Classification

Primary 60G35, 62M20, 93E11. Secondary 65M06, 65M12.

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Published 31 October 2014