Kronecker Products on Preconditioning

Handle URI:
http://hdl.handle.net/10754/303766
Title:
Kronecker Products on Preconditioning
Authors:
Gao, Longfei
Abstract:
Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.
Advisors:
Calo, Victor M. ( 0000-0002-1805-4045 )
Committee Member:
Efendiev, Yalchin; Keyes, David E. ( 0000-0002-4052-7224 ) ; Sun, Shuyu ( 0000-0002-3078-864X )
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
Aug-2013
Type:
Dissertation
Appears in Collections:
Applied Mathematics and Computational Science Program; Dissertations; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.advisorCalo, Victor M.en
dc.contributor.authorGao, Longfeien
dc.date.accessioned2013-10-22T06:47:39Z-
dc.date.available2013-10-22T06:47:39Z-
dc.date.issued2013-08en
dc.identifier.urihttp://hdl.handle.net/10754/303766en
dc.description.abstractNumerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.en
dc.language.isoenen
dc.subjectKronecker Productsen
dc.subjectPreconditioningen
dc.subjectIsogeometric Analysisen
dc.subjectmass matrixen
dc.subjectAlternating Direction Impliciten
dc.subjectHybrid Preconditioneren
dc.titleKronecker Products on Preconditioningen
dc.typeDissertationen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen_GB
dc.contributor.committeememberEfendiev, Yalchinen
dc.contributor.committeememberKeyes, David E.en
dc.contributor.committeememberSun, Shuyuen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameDoctor of Philosophyen
dc.person.id113331en
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