On Large-Scale Diagonalization Techniques for the Anderson Model of Localization

• Published in: SIAM review Vol. 50; no. 1; pp. 91 - 112
• Main Authors: Schenk, Olaf, Bollhöfer, Matthias, Römer, Rudolf A.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.03.2008
• Subjects: Algebra; Algebraic geometry; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer memory; Computer science; control theory; systems; Diagonal arguments; Eigenvalues; Eigenvectors; Exact sciences and technology; Factorization; Linear systems; Localization; Mathematics; Matrices; Matrices (mathematics); Matrix; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Position (location); Preconditioning; Quantum theory; Sciences and techniques of general use; Sigest; Studies; Symmetry; Theoretical computing; Diagonalization; Interior eigenvalue; Eigenvalue; Implementation; Lanczos method; 65F50; Maximum; Accuracy; Cullum-Willoughby implementation; 65F10; Eigenvalue problem; symmetric indefinite matrix; Localization; Anderson model of localization; Shift; Jacobi method; 82B44; Eigenvector; 05C85; maximum weighted matching 65F15; Jacobi-Davidson algorithm; multilevel preconditioning; Algorithm; large-scale eigenvalue problem; Physics; Lanczos algorithm; Linear system; Scientific computation; Applied mathematics; Symmetric matrix; Symmetric system; Preconditioning; 65F05
• ISSN: 0036-1445; 1095-7200
• DOI: 10.1137/070707002

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 large-scale 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 algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.