Preconditioned Eigenvalue Solvers in Electronic Structure Calculations

Andrew Knyazev

Abstract

We start with a short and informal survey of eigenvalue problems
in electronic structure calculations in real and plane wave spaces.
We briefly describe a number of known software packages, namely,
Vienna Ab-initio Simulation Package (VASP), ABINIT, and
Multigrid Instead of the K-spAce (MIKA),
and review some eigenvalue solvers used in these packages
to calculate the electronic groundstate:
\begin{itemize}
\item Simple Davidson-block iteration scheme
·\item Single band, steepest descent scheme
\item Conjugate gradient optimization
\item Residual minimization scheme, direct inversion in the iterative subspace (RMM-DIIS)
\end{itemize}
We present, as a possible alternative,
the locally optimal block preconditioned conjugate gradient (LOBPCG)
method in the framework of VASP and ABINIT.
LOBPCG method can be interpreted as a specific version of conjugate gradient optimization,
different from those used in VASP. It can also be viewed as the steepest descent scheme,
augmented with extra vectors in the basis set, namely with the wavefunctions
from the previous iteration step, not with the residuals as implemented in VASP.
Finally, it can be treated
as simplified specially restarted block Davidson method [1].
The relative simplicity, reasonable robustness,
fast convergence and an easy possibility of blocking make the LOBPCG method
  a strong competitor in electronic structure calculations both in real and plane wave spaces.
We describe the LOBPCG as being developed in [2-4] and
compare it informally to VASP and ABINIT algorithms.
We discuss a recent LOBPCG implementation in ABINIT and
highlight the possibilities to make the LOBPCG even faster by
taking into account the specificity of
the electronic structure calculations eigenvalue problems.

References:
[1]. C. Murray, S. C. Racine, E. R. Davidson,
Improved algorithms for the lowest few eigenvalues and associated eigenvectors of large matrices.
J. Comput. Phys. 103 (1992), no. 2, 382--389.

[2] A. V. Knyazev,
A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace.
Numerical treatment of eigenvalue problems, Vol. 5 (Oberwolfach, 1990), 143--154, Internat. Ser. Numer. Math., 96,
Birkhäuser, Basel, 1991.

[3] A. V. Knyazev, Preconditioned eigensolvers---an oxymoron?
Electron. Trans. Numer. Anal. 7 (1998), 104--123 (electronic).

[3] A. V. Knyazev, Preconditioned eigensolvers.
In Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Editors: Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk Van der Vorst, SIAM, pp. 337-368, 2000.

[4] A. V. Knyazev, Toward the optimal preconditioned eigensolver:
locally optimal block preconditioned conjugate gradient method.
SIAM J. Sci. Comput. 23 (2001), no. 2, 517--541 (electronic).