Inhalt des Dokuments
Preprint 15-2005
On Large Scale Diagonalization Techniques For The Anderson Model Of Localization
Author(s) :
Olaf Schenk
,
Matthias Bollhöfer
,
Rudolf A. Römer
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 15-2005
MSC 2000
- 65F15 Eigenvalues, eigenvectors
-
65F50 Sparse matrices
-
82B44 Disordered systems
-
65F10 Iterative methods for linear systems
-
65F05 Direct methods for linear systems and matrix inversion
-
05C85 Graph algorithms
Abstract :
We propose efficient preconditioning algorithms for an eigenvalue
problem arising in quantum physics, namely the computation of a few
interior eigenvalues and their associated eigenvectors for the largest
sparse real and symmetric indefinite matrices of the Anderson model of
localization. We compare the Lanczos algorithm in the 1987
implementation by Cullum and Willoughby with the shift-and-invert
techniques in the implicitly restarted Lanczos method and in the
Jacobi-Davidson method. Our preconditioning approaches for the shift-and
invert symmetric indefinite linear system are based on maximum weighted
matchings and algebraic multilevel incomplete $LDL^T$ factorizations.
These techniques can be seen as a complement to the alternative idea of
using more complete pivoting techniques for the highly ill-conditioned
symmetric indefinite Anderson matrices. We demonstrate the effectiveness
and the numerical accuracy of these algorithms. Our numerical examples
reveal that recent sparse direct and algebraic multilevel
preconditioning solvers can
accelerative the computation of a large-scale eigenvalue problem
corresponding to the Anderson model of localization by several orders of
magnitude.
Keywords :
Anderson model of localization, large--scale eigenvalue problem,
Lanczos algorithm,
Jacobi--Davidson algorithm, Cullum--Willoughby implementation, symmetric
indefinite matrix, multilevel--preconditioning, maximum weighted
matching
Notes :
SISC, to appear