学术报告

题      目：Optimization and preconditioning: TPD algorithms for nonlinear PDEs

报  告  人：郭汝驰  博士  (邀请人：钟柳强 )

                                    香港中文大学

时      间：9月16日  16：00-17：00

地     点：数科院西楼二楼会议室

报告人简介:

        郭汝驰，于2019年在弗吉尼亚理工大学取得博士学位，先后于俄亥俄州立大学担任Zassenhaus Assistant Professor，于加州大学尔湾分校担任Visiting Assistant Professor, 现于香港中文大学担任研究助理教授。主要研究领域为科学计算，特别是针对偏微分方程的数值方法，包括界面问题的非匹配网格算法，以及界面反问题的重构算法，包括浸入有限元算法、虚拟元算法，以及反问题的优化算法、直接法和深度学习算法等。在 SIAM J. Numer. Anal., M3AS, SIAM J. Sci. Comput., J. Comput. Phys., IMA J. Numer. Anal., ESAIM:M2AN, J. Sci. Comput., Comput. Methods Appl. Mech. Eng.,等计算数学领域杂志，以及深度学习会议ICLR上发表多篇文章。

摘      要：

       In physics and mathematics, a large class of PDE systems can be formulated as minimizing energy functionals subject to certain constraints. Lagrange multipliers are widely used for solving these problems, which however leads to minmax optimization problems, i.e., saddle point systems. The development of fast solvers for saddle point systems, especially the nonlinear ones, is particularly difficult in the sense that (i) one has to consider the preconditioning in two directions and (ii) the preconditioners have to evolve in iteration due to the nonlinearity.
In this work, we introduce an efficient transformed primal-dual (TPD) algorithm to solve the aforementioned nonlinear saddle point problems. We prove the optimal convergence in terms of the condition number. We apply the algorithm to a nonlinear Maxwell equation and show that it is much more efficient than some traditional fixed point and projected gradient descent algorithms.

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