# Grid'5000 publication

## 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}, }

*Bibtex parsing powered by http://bibliophile.sourceforge.net*

*Shared by: Desire Nuentsa*

*Last update: 2011-04-01 11:25:32*

*Publication #836*