A flexible extended Krylov subspace method for large Lyapunov equations

作者:sds_poster 发布时间:2019-07-03

Speaker: Fei Xue, Clemson University, USA
Host: Meiyue Shao, School of Data Science, Fudan University
Time: 10:30-11:30, July 5, 2019
Location: Zibin N102, Fudan University
Abstract: Rational Krylov subspace methods have been extensively investigated and used in recent years for solving a wide range of large-scale numerical linear algebra problems, including eigenvalue problems, approximating actions of functions of matrices, matrix equations and more. The current state of art for solving large Lyapunov equations includes the extended Krylov subspace method (EKSM) and adaptive rational Krylov subspace method (RKSM). We propose a flexible variant of EKSM (F-EKSM) which develops a rational Krylov subspace using a single fixed optimal pole. This requires computing and using only one sparse LU factorization. For symmetric and positive definite matrices, we show the existence and uniqueness of such a pole, and the faster convergence rate of F-EKSM than EKSM. For non-symmetric matrices, we propose a heuristic to compute a near-optimal pole. Numerical experiments show that F-EKSM with a properly selected pole outperforms EKSM and adaptive RKSM in runtime, and it also needs smaller spaces than EKSM.
Bio: Fei Xue is an Associate Professor of Mathematical and Statistical Sciences at Clemson University, USA. He obtained his PhD at University of Maryland College Park in 2009. His research interests lie in scientific computing in general, and large-scale numerical linear algebra in particular. He has been working on iterative methods for linear systems of equations, linear and nonlinear eigenvalue problems, functions of matrices and matrix equations.