# MathSciDoc: An Archive for Mathematician ∫

#### Distinguished Paper Award in 2017

Communications in Information and Systems, 14, (2), 111--134, 2014
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.
nonlinear filtering, Kolmogorov equations
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```@inproceedings{mei-heng2014an,
title={An efficient algorithm of Yau-Yau method for solving nonlinear filtering problems},
author={Mei-Heng Yueh, Wen-Wei Lin, and Shing-Tung Yau},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822155536465359379},
booktitle={Communications in Information and Systems},
volume={14},
number={2},
pages={111--134},
year={2014},
}
```
Mei-Heng Yueh, Wen-Wei Lin, and Shing-Tung Yau. An efficient algorithm of Yau-Yau method for solving nonlinear filtering problems. 2014. Vol. 14. In Communications in Information and Systems. pp.111--134. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822155536465359379.
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