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From ODE Solvers to Accelerated Optimization Methods

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:陈龙(美国University of California at Irvine)
:2023-06-23 10:00
:海韵园实验楼105报告厅

报告人:陈龙美国University of California at Irvine

 间:202362310:00

 点:海韵园实验楼105报告厅

内容摘要:

The convergence analysis of accelerated first-order methods for convex optimization problems is presented from the point of view of ordinary differential equation (ODE) solvers. We first take another look at the acceleration phenomenon via A-stability theory for ODE solvers and explain it by transforming the spectrum to the complex plane. After that, we present the Lyapunov framework for dynamical systems and introduce the strong Lyapunov condition. This framework addresses and analyzes many existing continuous convex optimization models, such as gradient flow, heavy ball system, Nesterov accelerated gradient flow, and dynamical inertial Newton system, among others. In the second half of this talk, we extend the acceleration to a class of non-linear saddle point problems. We first develop a transformed primal-dual (TPD) flow in which the flow for the dual variable contains a Schur complement that is strongly convex. We obtain exponential stability of the TPD flow by demonstrating the strong Lyapunov property. Then, we derive TPD iterations using ODE solvers. We propose an accelerated transformed primal-dual (ATPD) method and prove the accelerated linear rates with optimal lower iteration complexity. Furthermore, we extend the accelerated gradient flow and develop accelerated gradient and skew-symmetric splitting (AGSS) methods for a more general class of monotone operator equations. This is a joint work with Hao Luo (Chongqing Normal University) and Jingrong Wei (UCI).

人简介

陈龙,加州大学欧文分校(UCI)的数学教授。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,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,在Math. Comp., SINUM, Numer. Math., SISC等国际知名期刊发表学术论文70余篇,担任多个SCI期刊编委。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。

 

联系人:陈黄鑫