報告題目：Normalized Wolfe-Powell-type Local Minimax Method for Finding Multiple Unstable Solutions of Nonlinear Elliptic PDEs
報告人：謝資清 教授 （湖南師範大學）
邀請人：沈曉芹 教授 （理學院數學係）
報告時間：2023年4月13日下午3:00-4:30
報告地點：教九樓理學院會議室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.
報告人簡介：謝資清，教授、博士生導師，“計算與隨機數學”教育部重點實驗室主任，湖南師範大學副校長，第十三屆全國人大代表，第十四屆全國政協委員。主要從事計算數學與應用數學研究。現任中國數學會理事、中國工業與應用數學會理事、中國數學教育學會常務理事、湖南省數學會副理事長。博士畢業於中國科學院應用數學研究所。曾分別以第一完成人獲湖南省自然科學獎一等獎和湖南省教學成果獎一等獎。曾入選教育部新世紀優秀人才支持計劃，並享受國務院政府特殊津貼。在SIAM J. Sci. Comput.、SIAM J. Numer. Anal.、Math. Comput.、J. Comp. Phys.等期刊發表論文80餘篇。主持國家自然科學基金項目9項，其中重大研究計劃項目1項。曾多次應邀訪問美國、瑞典、挪威、德國、日本、新加坡、香港等國家和地區的知名大學。
