# American Institute of Mathematical Sciences

September  2021, 3(3): 615-645. doi: 10.3934/fods.2021023

## Analysis of the feedback particle filter with diffusion map based approximation of the gain

 1 University of Potsdam, Institute of Mathematics, Karl-Liebknecht Str. 24/25, D-14476 Potsdam, Germany 2 TU Berlin, Institute of Mathematics, Str. des 17. Juni 136, D-10623 Berlin, Germany 3 Bernstein Center for Computational Neuroscience, Philippstr. 13, 10115 Berlin, Germany

* Corresponding author: Sahani Pathiraja

Received  March 2021 Revised  July 2021 Published  September 2021 Early access  September 2021

Fund Project: This research has been partially funded by the Deutsche Forschungsgemeinschaft (DFG)-Project-ID 318763901 - SFB1294.

Control-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite $N$ system to its mean field limit. A number of possible future research directions are also discussed.

Citation: Sahani Pathiraja, Wilhelm Stannat. Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science, 2021, 3 (3) : 615-645. doi: 10.3934/fods.2021023
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