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In this paper, we introduce an adaptive kernel method for solving the optimalfiltering problem. The computational framework that we adopt is the Bayesianfilter, in which we recursively generate an optimal estimate for the state of atarget stochastic dynamical system based on partial noisy observational data.The mathematical model that we use to formulate the propagation of the statedynamics is the Fokker-Planck equation, and we introduce an operatordecomposition method to efficiently solve the Fokker-Planck equation. Anadaptive kernel method is introduced to adaptively construct Gaussian kernelsto approximate the probability distribution of the target state. Bayesianinference is applied to incorporate the observational data into the state modelsimulation. Numerical experiments have been carried out to validate theperformance of our kernel method.

A PDE-based Adaptive Kernel Method for Solving Optimal Filtering Problems