报告题目：Normalized Wolfe-Powell-type Local Minimax Method for Finding Multiple Unstable Solutions of Nonlinear Elliptic PDEs
报告人：谢资清 教授 （湖南师范大学）
邀请人：沈晓芹 教授 （hbs02红宝石线路数学系）
报告时间：2023年4月13日下午3:00-4:30
报告地点：教九楼hbs02红宝石线路会议室9-320
摘要: The major ingredients of classical local minimax methods (LMMs) are to characterize multiple unstable solutions of nonlinear PDEs as stable solutions of corresponding local minimization problems on submanifolds, and solve them by the steepest descent iteration with some appropriate step-size search strategies. In this talk, a new algorithm framework of the LMM, named as the normalized Wolfe-Powell-type LMM (NWP-LMM), is proposed for finding multiple unstable solutions of nonlinear elliptic partial differential equations (PDEs) with certain variational structures based on a Wolfe-Powell-type search rule and general descent directions. It provides the possibility to speed up the convergence of LMMs by choosing an appropriate descent direction. The feasibility of the NWP-LMM is justified and the global convergence of the corresponding algorithm is rigorously established under certain assumptions on general descent directions. Then, two different types of directions: preconditioned steepest descent (PSD) directions and conjugate gradient-type (CG-type) descent directions, are analyzed and utilized to implement our NWP-LMM algorithm. Especially, the global convergence of the NWP-LMM algorithm combined with the PSD direction is also verified. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction performs much better among its LMM companions.
报告人简介：谢资清，教授、博士生导师，“计算与随机数学”教育部重点实验室主任，湖南师范大学副校长，第十三届全国人大代表，第十四届全国政协委员。主要从事计算数学与应用数学研究。现任中国数学会理事、中国工业与应用数学会理事、中国数学教育学会常务理事、湖南省数学会副理事长。博士毕业于中国科学院应用数学研究所。曾分别以第一完成人获湖南省自然科学奖一等奖和湖南省教学成果奖一等奖。曾入选教育部新世纪优秀人才支持计划，并享受国务院政府特殊津贴。在SIAM J. Sci. Comput.、SIAM J. Numer. Anal.、Math. Comput.、J. Comp. Phys.等期刊发表论文80余篇。主持国家自然科学基金项目9项，其中重大研究计划项目1项。曾多次应邀访问美国、瑞典、挪威、德国、日本、新加坡、香港等国家和地区的知名大学。
