  In this paper, we propose an efficient numerical method to solve
high-dimensional nonlinear filtering (NLF) problems. Specifically, we use the
tensor train decomposition method to solve the forward Kolmogorov equation
(FKE) arising from the NLF problem. Our method consists of offline and online
stages. In the offline stage, we use the finite difference method to discretize
the partial differential operators involved in the FKE and extract
low-dimensional structures in the solution space using the tensor train
decomposition method. In addition, we approximate the evolution of the FKE
operator using the tensor train decomposition method. In the online stage using
the pre-computed low-rank approximation tensors, we can quickly solve the FKE
given new observation data. Therefore, we can solve the NLF roblem in a
real-time manner. Under some mild assumptions, we provide convergence analysis
for the proposed method. Finally, we present numerical results to show the
efficiency and accuracy of the proposed method in solving high-dimensional NLF
problems.
