  In this paper, we introduce an adaptive kernel method for solving the optimal
filtering problem. The computational framework that we adopt is the Bayesian
filter, in which we recursively generate an optimal estimate for the state of a
target stochastic dynamical system based on partial noisy observational data.
The mathematical model that we use to formulate the propagation of the state
dynamics is the Fokker-Planck equation, and we introduce an operator
decomposition method to efficiently solve the Fokker-Planck equation. An
adaptive kernel method is introduced to adaptively construct Gaussian kernels
to approximate the probability distribution of the target state. Bayesian
inference is applied to incorporate the observational data into the state model
simulation. Numerical experiments have been carried out to validate the
performance of our kernel method.
