报告题目: Fast algorithms for 3D Maxwell equations in electromagnetism
报告人：黄聪明 教授
报告人单位：台湾师范大学
报告时间：2018年10月27日,下午14:00-15:00
报告地点：数学学院第一报告厅
报告摘要:
Metamaterials with periodic structures are building blocks of various photonic and electronic materials. Numerical simulations, which based on the solutions of three dimensional Maxwell’s equations, play an important role to explore and design these novel artificial materials. To solve the governing equations, Yee’s finite difference scheme has been widely used to discretize the Maxwell equations. The discretizations lead to large-scale generalized eigenvalue problems (GEPs) for photonic crystals and complex media, and nonlinear eigenvalue problems for dispersive metallic photonic crystals. Due to a high dimensional subspace associated with the eigenvalue zeros, it is very challenging to solve such eigenvalue problems. To tackle these challenging problems, we derive a singular value decomposition of the discrete single-curl operator and propose a nullspace-free method to transform the GEPs into a null-space free standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEP. Therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton- type method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that our proposed methods successfully solve each of a set of 5.184 million dimension eigenvalue problems on a workstation.
报告人介绍：
黄聪明教授于1994年获得台湾清华大学应用数学博士。现任台湾师范大学数学系教授，其研究专长领域是科学计算与数值分析。黄教授在SIAM系列刊物，J. Comp. Physics等国际知名学术期刊已发布学术论文50多篇，着重于矩阵方程的保结构算法和大规模矩阵特征值问题的快数求解等方面的研究。