'Society for Industrial & Applied Mathematics (SIAM)'
Publication date
01/01/2005
Field of study
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
We discuss an approach for solving sparse or dense banded linear systems
Ax=b on a Graphics Processing Unit (GPU) card. The
matrix AāRNĆN is possibly nonsymmetric and
moderately large; i.e., 10000ā¤Nā¤500000. The ${\it split\ and\
parallelize}({\tt SaP})approachseekstopartitionthematrix{\bf A}intodiagonalsubāblocks{\bf A}_i,i=1,\ldots,P,whichareindependentlyfactoredinparallel.Thesolutionmaychoosetoconsiderortoignorethematricesthatcouplethediagonalsubāblocks{\bf A}_i.Thisapproach,alongwiththeKrylovsubspaceābasediterativemethodthatitpreconditions,areimplementedinasolvercalled{\tt SaP::GPU},whichiscomparedintermsofefficiencywiththreecommonlyusedsparsedirectsolvers:{\tt PARDISO},{\tt SuperLU},and{\tt MUMPS}.{\tt SaP::GPU},whichrunsentirelyontheGPUexceptseveralstagesinvolvedinpreliminaryrowācolumnpermutations,isrobustandcompareswellintermsofefficiencywiththeaforementioneddirectsolvers.InacomparisonagainstIntelā²s{\tt MKL},{\tt SaP::GPU}alsofareswellwhenusedtosolvedensebandedsystemsthatareclosetobeingdiagonallydominant.{\tt SaP::GPU}$ is publicly available and distributed as
open source under a permissive BSD3 license.Comment: 38 page