讲座主题：A structure-preserving reduced-order finite element method for semilinear stochastic partial differential equations with multiplicative noise

专家姓名：李焕荣

工作单位：云南师范大学

讲座时间：2026年11月11日 10:00-11:00

讲座地点：腾讯会议：482-684-051

主办单位：烟台大学数学与信息科学学院

内容摘要：

Semilinear stochastic partial differential equations (SPDEs) with multiplicative noise are widely used to describe complex dynamical systems subject to random perturbations. A major computational and theoretical challenge arises from the fact that full-order finite element (FE) simulations incur prohibitive computational costs and often struggle to preserve key physical structures over long-time integration. This paper develops a novel structure-preserving reduced-order finite element (ROFE) method by integrating proper orthogonal decomposition (POD) with standard FE discretization. Exploiting the inherent boundedness of solutions to three representative benchmark SPDEs, we convert the locally Lipschitz nonlinearity into a globally Lipschitz one on the effective solution domain, thereby establishing a rigorous foundation for the subsequent analysis of second- and higher-order moment stability, as well as optimal a priori error estimates for both full-order and reduced-order schemes. For the stochastic Allen-Cahn equation in particular, we employ a corrected Ito energy identity that incorporates the covariance operator of the Q-Wiener process to derive a discrete perturbed energy inequality, from which we prove the uniform boundedness of the mean energy without imposing unrealistic strict dissipation constraints. Numerical experiments on three models (Allen-Cahn, Fisher and Fitzhugh-Nagumo equations) demonstrate that the proposed ROFE method reduces computational runtime by approximately 85%-90% while maintaining robust accuracy and structure-preserving properties. The developed method offers a rigorous, efficient, and structure-preserving numerical framework for large-scale uncertainty quantification of semilinear SPDEs, and can be extended to a broader class of stochastic dynamical systems.

主讲人介绍：

李焕荣，山东泰安人，博士，教授，博士/硕士生导师，云南师范大学高层次引进人才。曾先后访问美国密苏里大学、美国田纳西大学和中国科学院大气物理研究所。目前主要从事随机偏微分方程、整数和分数阶偏微分方程的降阶数值模拟算法的理论分析及其应用、降阶深度学习方法设计及其应用等。在（SIAM）Multiscale Modeling ＆ Simulation、Journal of Computational Mathematics等国际知名期刊发表学术论文近40篇。主持国家自然科学基金项目2项、国家重点实验室开放课题2项、省自然科学基金项目和省教育厅项目等若干，获重庆市自然科学奖三等奖1项。