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 by Yau and Yau combined with the POD method. This algorithm is referred as POD algorithm in this paper. Our POD algorithm consists of offline and online stages. In the offline 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 online 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 methods.

中文翻译：

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中高维非线性滤波问题的合适正交分解方法

在本文中，我们研究了适当的正交分解 (POD) 方法来数值求解正向 Kolmogorov 方程 (FKE)。我们的方法旨在探索 FKE 解空间中的低维结构并开发有效的数值方法。作为研究 POD 方法到 FKE 的重要应用和主要动机，我们使用 Yau 和 Yau 提出的实时算法结合 POD 方法解决非线性滤波 (NLF) 问题。该算法在本文中称为POD算法。我们的 POD 算法由离线和在线阶段组成。在离线阶段，我们构建了少量的 POD 基函数来捕捉系统的动态并计算 FKE 算子下 POD 基函数的传播。在线上阶段，我们以实时方式同步即将到来的观察结果。也讨论了它的收敛性分析。对 NLF 问题进行了一些数值实验，以说明我们算法的可行性并验证收敛速度。我们的数值结果表明，与现有数值方法相比，POD 算法提供了可观的计算节省。