Hermite Spectral Method to 1D Forward Kolmogorov Equation and its Application to Nonlinear Filtering Problems

Xue Luo Beihang University Stephen S.-T. Yau Tsinghua University

Optimization and Control mathscidoc:1701.27014

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 58, (10), 2495-2507, 2013.10
In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problems with a real-time algorithm developed by S.-T. Yau and the second author in 2008. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filters as a real-time solver to NLF.
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@inproceedings{xue2013hermite,
  title={Hermite Spectral Method to 1D Forward Kolmogorov Equation and its Application to Nonlinear Filtering Problems},
  author={Xue Luo, and Stephen S.-T. Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170129110957786085133},
  booktitle={IEEE TRANSACTIONS ON AUTOMATIC CONTROL},
  volume={58},
  number={10},
  pages={2495-2507},
  year={2013},
}
Xue Luo, and Stephen S.-T. Yau. Hermite Spectral Method to 1D Forward Kolmogorov Equation and its Application to Nonlinear Filtering Problems. 2013. Vol. 58. In IEEE TRANSACTIONS ON AUTOMATIC CONTROL. pp.2495-2507. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170129110957786085133.
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