报告人：陈龙 教授 University of California at Irvine
报告时间：2024年9月12日14:00-15:00
报告地点：红瓦楼726
报告内容简介：The heavy-ball momentum method has gained widespread popularity for accelerating gradient descent by incorporating a momentum term. Recent studies have conclusively shown that the heavy-ball method cannot achieve an accelerated convergence rate for general smooth strongly convex optimization problems.
We first revisit the acceleration phenomenon through A-stability theory for ODE solvers, explaining it by transforming the spectrum to the complex plane. We present the heavy-ball flow as an example of this transformation. Additionally, we introduce the Lyapunov framework for dynamical systems and the strong Lyapunov condition.
We then introduce the Accelerated Over-Relaxation Heavy-Ball (AOR-HB) method, a novel approach that is the first heavy-ball method to demonstrate provable global and accelerated convergence for smooth strongly convex optimization. The key innovation of the AOR-HB method lies in applying an over-relaxation technique to the gradient term.
This novel approach enables the method to be applied to min-max problems and meet optimal lower complexity bounds. We extend the acceleration to a class of non-linear saddle point problems, presenting an accelerated transformed primal-dual (ATPD) method for saddle point systems and accelerated gradient and skew-symmetric splitting (AGSS) methods for a broader class of monotone operator equations.
Numerical experiments validate the effectiveness of the proposed algorithms, with their performance matching that of other leading first-order optimization methods.
This is joint work with Dr. Jingrong Wei.
报告人简介：陈龙(Long Chen) 任职于加州大学欧文分校（UCI）数学系。1997年毕业于南京大学，2000年获北京大学硕士学位，2005年获宾夕法尼亚州立大学博士学位，博士生导师为许进超教授。2005年至2007年在加州大学圣地亚哥分校和马里兰大学帕克分校从事博士后研究。 2007年起在UCI工作，2011年获得终身教职，2015年晋升为正教授。
陈教授的研究领域是偏微分方程的数值解，尤其是有限元方法的设计与分析。陈教授开发了iFEM有限元软件包，为有限元方法的教学和研究提供了极大的便利。陈教授在国际知名期刊发表学术论文80余篇，担任Journal of Computational Mathematics、Computers and Mathematics with Applications、Multiscale Modeling and Simulation、Advances in Computational Mathematics和Advances in Applied Mathematics and Mechanics等多个SCI期刊编委。从他开始工作到现在，陈教授一直得到美国国家科学基金会的持续支持。另外，他还创立了微信公众号《CAM 传习录》(CAMtips)，分享关于计算和应用数学的学习和研究方法。