随机偏微分方程多辛几何算法及不确定性量化

In this project, we focus on the investigation of multi-symplectic geometry algorithms and uncertainty qualification for stochastic partial differential equations. The main content of this project includes: the construction of the multi-symplectic geometry algorithms for stochastic Hamiltonian partial differential equations and the analysis of the strong (weak) order of convergence, stability, backward error analysis and so on；the study of the efficient stochastic multi-symplectic geometry algorithms for specific stochastic partial differential equations which come from quantum physics and radio statistical physics; the design and analysis of the self-adaptive stochastic collocation methods. The last but not the least is, based on the methods of uncertainty qualification, to study the numerical efficiency of stochastic multi-symplectic geometry algorithms, and to establish the connection between each other. This project will not only promote the further development of the algorithms of stochastic partial differential equations, but also boost the development of the corresponding science and technology.

本项目聚焦于随机偏微分方程多辛几何算法及不确定量化研究。主要研究：随机哈密尔顿偏微分方程多辛几何算法的构造和强(弱)收敛阶、稳定性等数值分析理论；量子物理和无线电统计物理中几类高维随机偏微分方程高效随机多辛几何算法；随机偏微分方程不确定量化中的自适应随机配置方法构造和数值分析；基于不确定性量化方法提高随机多辛几何算法等相关数值方法的数值计算效率。本项目的研究将完善、深化进而升华已取得关于随机偏微分方程多辛几何算法及不确定性量化的研究成果，促进随机偏微分方程数值算法的进一步创新发展。

在本项目研究中，我们提出了无穷维随机哈密尔顿系统的随机辛几何算法，发展了随机哈密尔顿偏微分方程随机多辛几何算法，并分别应用于光纤通讯中的随机薛定谔方程和统计无线电物理中的随机麦克斯韦方程数值计算；给出了随机薛定谔方程时间半离散的基本收敛性定理，以及Theta类时间半离散格式的数值分析理论结果；提出了辛间断伽略金全离散数值方法，并给出其均方收敛阶；基于随机耗散薛定谔方程数值遍历性研究，首次给出遍历性数值格式弱收敛性和数值不变测度收敛性的结果；对于具有守恒量的随机微分方程，提出了保持守恒量的平均向量场方法和投影算法；给出若干不确定性量化理论与数值结果。