Locally optimal block preconditioned conjugate gradient method

for electronic structure calculations.

Andrew Knyazev

Center for Computational Mathematics
University of Colorado Denver
Campus Box 170, P.O. Box 173364
Denver, Colorado 80217-3364

We describe the locally optimal block preconditioned conjugate gradient (LOBPCG) method  in the framework of a well known and widely used Vienna Ab-initio Simulation Package (VASP). Several methods are available in VASP to calculate the electronic groundstate:

 

·  Simple Davidson-block iteration scheme

·  Single band, steepest descent scheme

·  Conjugate gradient optimization

·  Residual minimization scheme, direct inversion in the iterative subspace (RMM-DIIS) 

 

LOBPCG 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 seen as simplified specially restarted block Davidson method [1].

 

We described the LOBPCG as being developed in [2-4] and compare it informally to VASP algorithms. We discuss a possibility to use LOBPCG instead of RMM-DIIS.

 

[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).