In this talk, we discuss the proper orthogonal decomposition (POD) method to numerically solve the forward Kolmogorov equation (FKE). This method aims to explore the low-dimensional structures in the solution space of the FKE and to develop efficient numerical methods. As our primary motivation to study the POD method to FKE, we use it to solve the nonlinear filtering (NLF) problems combined with the real-time algorithm. This algorithm consists of off-line and on-line stages. In the off-line stage, we construct a small number of POD basis functions that capture the dynamics of the system and compute propagation of the POD basis functions under the FKE operator. In the on-line stage, we synchronize the coming observations in a real-time manner. Its convergence analysis has also been discussed. Some numerical experiments of the NLF problems are performed to illustrate the feasibility of our algorithm and to verify the convergence rate.

This is joint work with Z. Wang, S. S.-T. Yau and Z. Zhang.