Journal
SIAM REVIEW
Volume 50, Issue 1, Pages 91-112Publisher
SIAM PUBLICATIONS
DOI: 10.1137/070707002
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
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Funding
- Engineering and Physical Sciences Research Council [EP/C007042/1] Funding Source: researchfish
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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 ill 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 LDLT 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.
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