Kernel-based collocation methods for Zakai equations

  • Yumiharu NakanoEmail author


We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the approximation error is bounded by the order of the square root of the time step and the error that comes from a single step interpolation. Our theorem is well consistent with the results of numerical experiments.


Zakai equations Kernel-based interpolation Stochastic partial differential equations Radial basis functions 

Mathematics Subject Classification

60H15 65M70 93E11 



The author is thankful to the anonymous reviewers for their careful reading of previous versions of this paper and their constructive comments. This study is partially supported by JSPS KAKENHI Grant No. JP17K05359.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Computing Science, School of ComputingTokyo Institute of TechnologyOokayama, Meguro-kuJapan

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