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Parallel GMRES with a multiplicative Schwarz preconditioner

Author: {N}uentsa {W}akam, {D}{\'e}sir{\'e} and {A}tenekeng {K}ahou, {G}uy-{A}ntoine
EntryType: techreport
Hal_id: inria-00508277
Url: http://hal.inria.fr/inria-00508277/PDF/RR-7342.pdf
Abstract: {I}n this paper, we present an hybrid solver for linear systems that combines a {K}rylov subspace method as accelerator with some overlapping domain decomposition method as preconditioner. {T}he preconditioner uses an explicit formulation associated to one iteration of the classical multiplicative {S}chwarz method. {T}o avoid communications and synchronizations between subdomains, the {N}ewton-basis {GMRES} implementation is used as accelerator. {T}hus, it is necessary to divide the computation of the orthonormal basis into two steps: the preconditioned {N}ewton basis is computed then it is orthogonalized. {T}he first step is merely a sequence of matrix-vector products and solutions of linear systems associated to subdomains; we describe the fine-grained parallelism that is used in these kernel operations. {T}he second step uses a parallel implementation of dense ${QR}$ factorization on the resulted basis. {A}t each application of the preconditioner operator, local systems associated to the subdomains are solved with some accuracy depending on the global physical problem. {W}e show that this step can be further parallelized with calls to external third-party solvers. {T}o this end, we define two levels of parallelism in the solver: the first level is intended for the computation and the communication across all the subdomains; the second level of parallelism is used inside each subdomain to solve the smaller linear systems induced by the preconditioner. {N}umerical experiments are performed on several problems to demonstrate the benefits of such approaches, mainly in terms of global efficiency and numerical robustness.
Language: {A}nglais
Affiliation: {SAGE} - {INRIA} - {IRISA} - {CNRS} : {UMR}6074 - {INRIA} - {U}niversit{\'e} de {R}ennes {I} - {L}aboratoire de {R}echerche en {I}nformatique - {LRI} - {CNRS} : {UMR}8623 - {U}niversit{\'e} {P}aris {S}ud - {P}aris {XI}
Type: Research Report
Institution: INRIA
Number: {RR}-7342
Day: 23
Month: 08
Year: 2010

Bibtex:
@techreport{NUENTSAWAKAM:2010:INRIA-00508277:2,
    HAL_ID = {inria-00508277},
    URL = {http://hal.inria.fr/inria-00508277/en/},
    title = { {P}arallel {GMRES} with a multiplicative {S}chwarz preconditioner},
    author = {{N}uentsa {W}akam, {D}{\'e}sir{\'e} and {A}tenekeng {K}ahou, {G}uy-{A}ntoine},
    abstract = {{I}n this paper, we present an hybrid solver for linear systems that combines a {K}rylov subspace method as accelerator with some overlapping domain decomposition method as preconditioner. {T}he preconditioner uses an explicit formulation associated to one iteration of the classical multiplicative {S}chwarz method. {T}o avoid communications and synchronizations between subdomains, the {N}ewton-basis {GMRES} implementation is used as accelerator. {T}hus, it is necessary to divide the computation of the orthonormal basis into two steps: the preconditioned {N}ewton basis is computed then it is orthogonalized. {T}he first step is merely a sequence of matrix-vector products and solutions of linear systems associated to subdomains; we describe the fine-grained parallelism that is used in these kernel operations. {T}he second step uses a parallel implementation of dense ${QR}$ factorization on the resulted basis. {A}t each application of the preconditioner operator, local systems associated to the subdomains are solved with some accuracy depending on the global physical problem. {W}e show that this step can be further parallelized with calls to external third-party solvers. {T}o this end, we define two levels of parallelism in the solver: the first level is intended for the computation and the communication across all the subdomains; the second level of parallelism is used inside each subdomain to solve the smaller linear systems induced by the preconditioner. {N}umerical experiments are performed on several problems to demonstrate the benefits of such approaches, mainly in terms of global efficiency and numerical robustness.},
    language = {{A}nglais},
    affiliation = {{SAGE} - {INRIA} - {IRISA} - {CNRS} : {UMR}6074 - {INRIA} - {U}niversit{\'e} de {R}ennes {I} - {L}aboratoire de {R}echerche en {I}nformatique - {LRI} - {CNRS} : {UMR}8623 - {U}niversit{\'e} {P}aris {S}ud - {P}aris {XI} },
    type = {Research Report},
    institution = {INRIA},
    number = {{RR}-7342},
    day = {23},
    month = {08},
    year = {2010},
    URL = {http://hal.inria.fr/inria-00508277/PDF/RR-7342.pdf},
}

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Last update: 2011-04-01 11:25:32
Publication #836

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