On Large Scale Diagonalization Techniques For The Anderson Model Of Localization

dc.contributor.authorSchenk, Olaf
dc.contributor.authorBollhöfer, Matthias
dc.contributor.authorRömer, Rudolf A.
dc.date.accessioned2022-05-11T12:11:32Z
dc.date.available2022-05-11T12:11:32Z
dc.date.issued2005-06-10
dc.description.abstractWe 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 the largest 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 sparse direct and algebraic multilevel preconditioning solvers can accelerative the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.en
dc.identifier.issn2197-8085
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/16874
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-15652
dc.language.isoen
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/
dc.subject.ddc510 Mathematiken
dc.subject.otherAnderson model of localizationen
dc.subject.otherlarge–scale eigenvalue problemen
dc.subject.otherLanczos algorithmen
dc.subject.otherJacobi–Davidson algorithmen
dc.subject.otherCullum–Willoughby implementationen
dc.subject.othersymmetric indefinite matrixen
dc.subject.othermultilevel--preconditioningen
dc.subject.othermaximum weighted matchingen
dc.titleOn Large Scale Diagonalization Techniques For The Anderson Model Of Localizationen
dc.typeResearch Paperen
dc.type.versionsubmittedVersionen
tub.accessrights.dnbfree
tub.affiliationFak. 2 Mathematik und Naturwissenschaften::Inst. Mathematikde
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische Universität Berlinen
tub.series.issuenumber2005, 15en
tub.series.namePreprint-Reihe des Instituts für Mathematik, Technische Universität Berlinen
tub.subject.msc200065F15 Eigenvalues, eigenvectorsen
tub.subject.msc200065F50 Sparse matricesen
tub.subject.msc200082B44 Disordered systemsen
tub.subject.msc200065F10 Iterative methods for linear systemsen
tub.subject.msc200065F05 Direct methods for linear systems and matrix inversionen
tub.subject.msc200005C85 Graph algorithmsen

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