On large-scale diagonalization techniques for the Anderson model of localization
Schenk, Olaf, Bollhoefer, Matthias and Roemer, Rudolf A.. (2008) On large-scale diagonalization techniques for the Anderson model of localization. SIAM Review, Vol.50 (No.1). pp. 91-112. ISSN 0036-1445Full text not available from this repository.
Official URL: http://dx.doi.org/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 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.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics
Q Science > QC Physics
|Divisions:||Faculty of Science > Physics
Faculty of Science > Centre for Scientific Computing
|Library of Congress Subject Headings (LCSH):||Anderson model, Eigenvalues, Algorithms, Matrices|
|Journal or Publication Title:||SIAM Review|
|Official Date:||March 2008|
|Number of Pages:||22|
|Page Range:||pp. 91-112|
|Version or Related Resource:||Schenk, O., et al. (2007). On large-scale diagonalization techniques for the Anderson model of localization. Proceedings in Applied Mathematics and Mechanics, 7(1), pp. 1021003-1021004 ; Schenk, O., Bollhöfer, M. and Römer R.A. (2006). On large-scale diagonalization techniques for the Anderson model of localization. SIAM Journal on Scientific Computing, 28(3), pp.963-983|
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