勷勤数学•专家报告

题      目：Second-order flow approach for solving variational problems

报  告  人：谢资清 教授  (邀请人：钟柳强)

                                   湖南师范大学

时      间：4月10日  14：30-15：30

地     点：数科院西楼二楼会议室

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，湘江实验室副主任，十三届全国人大代表，十四届全国政协委员。主要从事科学计算与应用数学的研究工作。现任中国高等教育学会教师教育分会副理事长、中国数学会数学教育分会常务理事兼拔尖创新人才培养小组副组长，《数学物理学报》中英文刊常务编委。博士毕业于中国科学院应用数学研究所。分别以第一完成人身份获湖南省自然科学奖一等奖和湖南省教学成果奖一等奖。入选教育部新世纪优秀人才支持计划，并获批为享受国务院政府特殊津贴专家。在SIAM J. Sci. Comput., SIAM J. Numer. Anal., Math. Comput.等期刊发表论文80余篇。主持国家自然科学基金项目9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，并在全球华人数学家大会、中德计算与应用数学会议等重要国际会议上做邀请报告。

摘      要：

       In this talk, we introduce a so-called second-order flow approach, a novel computational framework based on dissipative second-order hyperbolic partial differential equations (PDEs) designed to tackle variational problems. Our focus lies on scenarios where energy functionals are nonconvex and may entail nonconvex constraints. This motivation stems from practical applications such as finding stationary points of Ginzburg-Landau energy in phase-field modeling, Landau-de Gennes energy of the Q-tensor model for liquid crystals, as well as simulating ground states for Bose-Einstein condensates. We explore both the analytical and numerical aspects of this novel framework, showing how discretizing the PDEs leads to original numerical methodologies for addressing variational problems. Analytically, for a class of unconstrained nonconvex variational problems, we demonstrate the convergence of second-order flows to stationary points and establish the well-posedness of the second-order flow equations. Our numerical findings underscore the superiority of second-order flow methods over gradient flow methods across all discussed application scenarios.

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