Solving Nonlinear Filtering Problems Using a Tensor Train Decomposition Method

In this article, we propose an efficient numerical method to solve 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 tensor using the tensor train decomposition method. In the online stage using the precomputed low-rank approximation tensors, we can quickly solve the FKE given new observation data. Therefore, we can solve the NLF problem in a real-time manner. Finally, we present numerical results to show the efficiency and accuracy of the proposed method in solving up to six-dimensional NLF problems.