On Large Scale Diagonalization Techniques for the Anderson Model of Localization

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 $LD{L}^T$ 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 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.

9 .eps figures, submitted to SIAM J. Sci. Comp., SIAM LaTeX styles (included), high quality figures can be obtained from http://eprints.csc.warwick.ac.uk/archive/00000110/

References in corpus (2)

• On Large Scale Diagonalization Techniques for the Anderson Model of Localization
• The three-dimensional Anderson model of localization with binary random potential

Cited by in corpus (10)

• Multifractal finite-size-scaling and universality at the Anderson transition
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model II: Symmetry relation under ensemble averaging
• On Large Scale Diagonalization Techniques for the Anderson Model of Localization
• The general approach to the critical phase with coupled quasiperiodic chains
• Random Walks and Anderson Localisation in a Three-Dimensional Class C Network Model
• Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class
• Localization-delocalization transition for disordered cubic harmonic lattices
• Resolution of the exponent puzzle for the Anderson transition in doped semiconductors
• Anderson localization: a density matrix approach