6,550 research outputs found
On large-scale diagonalization techniques for the Anderson model of localization
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 JacobiāDavidson 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
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
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