Proper orthogonal decomposition method to nonlinear filtering problems in medium-high dimension

Abstract

In this paper, we investigate the proper orthogonal decomposition (POD) method to numerically solve the forward Kolmogorov equation (FKE). Our method aims to explore the low-dimensional structures in the solution space of the FKE and to develop efficient numerical methods. As an important application and our primary motivation to study the POD method to FKE, we solve the nonlinear filtering (NLF) problems with a real-time algorithm proposed in cite{YauYau:2008} combined with the POD method. This algorithm is referred as POD algorithm in this paper. Our POD 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. Our numerical results show that the POD algorithm provides considerable computational savings over existing numerical