Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations

Handle URI:
http://hdl.handle.net/10754/626565
Title:
Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations
Authors:
Giraldi, Loic; Nouy, Anthony
Abstract:
This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
arXiv
Issue Date:
30-Jun-2017
ARXIV:
arXiv:1706.10221
Type:
Preprint
Additional Links:
http://arxiv.org/abs/1706.10221v1; http://arxiv.org/pdf/1706.10221v1
Appears in Collections:
Other/General Submission; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGiraldi, Loicen
dc.contributor.authorNouy, Anthonyen
dc.date.accessioned2017-12-28T07:32:16Z-
dc.date.available2017-12-28T07:32:16Z-
dc.date.issued2017-06-30en
dc.identifier.urihttp://hdl.handle.net/10754/626565-
dc.description.abstractThis paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.en
dc.publisherarXiven
dc.relation.urlhttp://arxiv.org/abs/1706.10221v1en
dc.relation.urlhttp://arxiv.org/pdf/1706.10221v1en
dc.rightsArchived with thanks to arXiven
dc.titleWeakly intrusive low-rank approximation method for nonlinear parameter-dependent equationsen
dc.typePreprinten
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.eprint.versionPre-printen
dc.contributor.institutionDepartment of Computer Science and Mathematics, Ecole Centrale de Nantes, Nantes, France.en
dc.identifier.arxividarXiv:1706.10221en
kaust.authorGiraldi, Loicen
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