Past Event: Oden Institute Seminar

Solving Large Scale Linear Response Eigenvalue Problems

Ren-Cang Li, Dept. of Mathematics, UT Arlington

Abstract

The linear response (LR) eigenvalue problems arise from linear response perturbation analysis in the time-dependent density functional theory. It is for calculating excitation states (energies) of physical systems in the study of collective motion of many particle systems, ranging from silicon nanoparticles and nanoscale materials to the analysis of interstellar clouds. Although the LR eigenvalue problem is a nonsymmetric eigenvalue problem, it has many of the symmetric eigenvalue problem's characteristics. In this talk, we'll first briefly discuss how the LR problem is derived and then present a minimization principle for the sum of the first few smallest positive eigenvalues, Rayleigh quotient-like projection matrices, and Cauchy-like interlacing inequalities. Subsequently, we'll develop the best approximation theorem for these smallest positive eigenvalues via a structure-preserving subspace projection. Finally, we'll present conjugate gradient-like algorithms for simultaneously computing the first few smallest positive eigenvalues and associated eigenvectors. Numerical examples will be provided to demonstrate the effectiveness of our proposed methods. This is a joint work with Zhaojun Bai (University of California at Davis). Biography: Ren-Cang Li is a Professor of Mathematics at UT Arlington. Ren-Cang received his BS in Computational Mathematics from Xiamen University in 1985 and his MS also in Computational Mathematics from Chinese Academy of Science in 1988 and his Ph.D. in Applied Mathematics from UC Berkeley in 1995. He joined ORNL in 1995 as a Householder Fellow and moved to University of Kentucky in 1996 as an assistant and then associate. He joined the University of Texas at Arlington as a professor in 2006. He was awarded a Friedman memorial prize in Applied Mathematics from UC Berkeley in 1996, and a CAREER award from NSF in 1999. He helped HP in developing its libm library for HP Itanium computers in 2001. His research interest includes floating-point support for scientific computing, numerical linear algebra, reduced order modeling, nonlinear manifold learning, large scale eigenvalue computations in electronic structure calculations, and unconventional schemes for ordinary differential equations.