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Preprint 15-2005

On Large Scale Diagonalization Techniques For The Anderson Model Of Localization

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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

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