"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.