Organizer: 邓权灵

Abstract:

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.

Personal Profile:

Ruchi Guo earned his Ph.D. from Virginia Tech in 2019. Following his graduation, he served as a Zassenhaus Assistant Professor at Ohio State University, as a Visiting Assistant Professor at the University of California, Irvine, subsequently as the Research Assistant Professor at the Chinese University of Hong Kong. He is currently is a research scientist at the Sichuan University. Dr. Guo's research primarily focuses on numerical methods and scientific computing, with an emphasis on interface problems, unfitted mesh finite element methods, and inverse problems. Recently, he has expanded his interests to include deep learning and data science. His research has received funding from the NSF. Dr. Guo has published articles in journals including SIAM J. Numer. Anal., M3AS, SIAM J. Sci. Comput., J. Comput. Phys., IMA J. Numer. Anal., ESAIM: M2AN, J. Sci. Comput., etc.