报告题目：计算非线性偏微分方程多解的标准化Wolfe- Powel I型局部极小极大方法

报 告 人：刘伟（华南师范大学博士后）

报告时间：2021年9月1日  10:00-11:30

报告地点：腾讯会议（838574316）

报告摘要：

The local minimax method (LMM) proposed in[Y. Li and J. Zhou, SIAM J. Sci. Comput. 23(3), 840-- -865 (2001)] and[Y. Li and J. Zhou, SIAM J. Sci. Comput. 24(3), 865- - - -885 (2002)] is an eficient method to solve nonlinear eliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in the above work and playing a significant role in the performance and convergence analysis of traditional LMs. In this talk, a new algorithm framework. of the LMMs is established based on general descent directions and normalized Wolfe- -Powell-type step- -size search rules. The feasibility and global convergence of the corresponding algorithm, named as the normalized Wolfe-Powell-type LMM (NWP-LMM)， are rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient- type (CG- type) descent directions are utilized to speed up the LMM algrithms. Finally, extensive numerical results for several semilinear eliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM's family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP- _LMM combined with the CG- -type direction indeed performs much better among its LMM companions .

报告人简介：

刘伟，男，华南师范大学博士后。2010-2020年就读于湖南师范大学，分别于2014年和2020年获得学士学位和博士学位。2020年秋进入华南师范大学从事博士后研究工作。曾访问新加坡国立大学数学科学研究所、北京计算科学研究中心等国内外知名大学或科研机构。主要研究领域为计算与应用数学，研究工作涉及非线性偏微分方程多解计算、计算量子物理学、非线性色散方程的数值方法等。主要成果发表在SIAM J. Sci. Comput.、 J. Comput. Phys.、 Commun. Math. Sci.、《中国科学：数学》等期刊上。2021年获批国家自然科学基金青年科学基金项目和中国博士后国际交流计划项目各1项。