Abstract：

This talk introduces two efficient numerical methods for solving nonlinear filtering (NLF) problems. We propose the utilization of the proper orthogonal decomposition (POD) method and tensor train decomposition method to solve the forward Kolmogorov equation (FKE) associated with NLF problems. Our approach involves offline and online stages. In the offline stage, we discretize the partial differential operators within the FKE using the finite difference method, while extracting low-dimensional structures in the solution through the POD method or tensor train decomposition method. In the online stage, we leverage precomputed POD basis functions or low-rank approximation tensors to rapidly solve the FKE when provided with new observation data. Consequently, real-time solutions to NLF problems can be obtained. Numerical results are also presented to demonstrate the efficiency and accuracy of our proposed method in solving NLF problems of up to six dimensions.