"Solution of large nonlinear eigenvalue problems in
electronic structure calculations"
Yousef Saad
Department of Computer Science and Engineering
University of Minnesota
Density Functional Theory (DFT) is a successful technique used to
determine the electronic structure of matter that is based on a number
of approximations. It converts the original n-particle problem into
an effective one-electron system, resulting in a coupled one-electron
Schr\"odinger equation and a Poisson equation. This coupling is
nonlinear and rather complex. It involves a charge density $\rho$
which can be computed from the eigenfunctions $\psi_i$, for all
occupied states. These eigenfunctions are solutions of a nonlinear
eigenvalue problem resulting from a Schr\"odinger equation whose
potential depends nonlinearly on the charge density, which in turn
depends on the eigenfunctions. This problem is solved
`self-consistently' by an iterative procedure. The challenge comes
from the large number of eigenfunctions to be computed for realistic
systems with, say, thousands of electrons. We will desribe a parallel
implementation with a finite difference approach for this problem with
an emphasis on diagonalization, illustrating as needed with our
in-house code called PARSEC. PARSEC evolved over more than a decade
from a multi-disciplinary collaboration as many features were
progressively added to it and the diagonalization routine, which
accounts for the biggest part of a typical execution time, was
upgraded a few times. We found that it is important to consider the
problem as one of computing an invariant subspace in the non-linear
context of the Kohn-Sham equations. This viewpoint leads to
considerable savings as it de-emphasizes the accurate computation of
individual eigenvectors and focuses instead on the subspace which they
span.
Biography of Yousef Saad
Yousef Saad is an Institute of Technology (I.T.) distinguished
professor with the department of computer science and engineering at
the University of Minnesota. He received the "Doctorat d'Etat" from
the university of Grenoble (France) in 1983. He joined the university
of Minnesota in 1990 as a Professor of computer science and a Fellow
of the Minnesota Supercomputer Institute. He was head of the
department of Computer Science and Engineering from January 1997 to
June 2000, and became an IT distinguished professor in May 2005. From
1981 to 1990, he held positions at the University of California at
Berkeley, Yale, the University of Illinois, and the Research Institute
for Advanced Computer Science (RIACS). His current research interests
include: numerical linear algebra, sparse matrix computations,
iterative methods, parallel computing, numerical methods for
electronic structure, and data analysis. He is the author of two
(single authored) books and over 100 journal articles. He is also the
developer or co-developer of several software packages for solving
sparse linear systems including SPARSKIT, pARMS, and ITSOL.