Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods

Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored i...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	Numerical linear algebra with applications Jg. 29; H. 1
Hauptverfasser:	Buttari, Alfredo, Huber, Markus, Leleux, Philippe, Mary, Theo, Rüde, Ulrich, Wohlmuth, Barbara
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Oxford Wiley Subscription Services, Inc 01.01.2022 Wiley
Schlagworte:	Algorithms Analysis of PDEs Approximation Asymptotic methods block low‐rank Computer Science Distributed, Parallel, and Cluster Computing efficient coarse level solver hierarchical hybrid grids high‐performance computing Iterative methods Mathematics Multigrid methods MUMPS Saddle points Solvers sparse direct solver sparse direct solver block low-rank efficient coarse level solver hierarchical hybrid grids geometric multigrid MUMPS multifrontal high-performance computing
ISSN:	1070-5325, 1099-1506
Online-Zugang:	Volltext
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!

Abstract	Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to 1011 degrees of freedom on a petascale supercomputer using 43,200 processes.
AbstractList	Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to 1011 degrees of freedom on a petascale supercomputer using 43,200 processes. Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to degrees of freedom on a petascale supercomputer using 43,200 processes. Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low-rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes-type saddle point system. We employ a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to 1+e11 degrees of freedom on a petascale supercomputer using 43 200 processes.
Author	Mary, Theo Wohlmuth, Barbara Leleux, Philippe Buttari, Alfredo Huber, Markus Rüde, Ulrich
Author_xml	– sequence: 1 givenname: Alfredo orcidid: 0000-0003-3207-7021 surname: Buttari fullname: Buttari, Alfredo email: alfredo.buttari@irit.fr organization: CNRS – sequence: 2 givenname: Markus surname: Huber fullname: Huber, Markus email: huber@ma.tum.de organization: Technical Universtiy Munich – sequence: 3 givenname: Philippe orcidid: 0000-0002-3760-4698 surname: Leleux fullname: Leleux, Philippe email: leleux@cerfacs.fr organization: CERFACS – sequence: 4 givenname: Theo orcidid: 0000-0001-9949-4634 surname: Mary fullname: Mary, Theo email: theo.mary@lip6.fr organization: Sorbonne Université – sequence: 5 givenname: Ulrich surname: Rüde fullname: Rüde, Ulrich email: Ulrich.ruede@fau.de organization: Friedrich‐Alexander Universität – sequence: 6 givenname: Barbara surname: Wohlmuth fullname: Wohlmuth, Barbara email: wohlmuth@ma.tum.de organization: Technical Universtiy Munich
BackLink	https://hal.science/hal-02528532$$DView record in HAL
BookMark	eNp10M1OAyEQB3BiNPEz8RFIvOhhKwML2z1W41fS6EUPnghlp4rSpcK26s1H8Bl9EqnVi9HTkMlvyMx_k6y2oUVCdoH1gDF-2HrT4yWrVsgGsLouQDK1unhXrJCCy3WymdIDY0zJWmyQ2yMf7CP14fnj7T2a9pEm1955pNOI1iUXWmqDiQnpXXQNTcHPMSY6DpHiSxdxgjRZk_1k5jv3ZSbY3YcmbZO1sfEJd77rFrk5Pbk-Pi-GV2cXx4NhYYWCqsASrColMpAwhga4EE1VQl9gPqepzEiUTd0fcTHC0tjSjpQqjaoqySVY6KPYIgfLf--N19PoJia-6mCcPh8M9aLHuOT9fPocst1b2mkMTzNMnX4Is9jm9TRXrAYloWJZ9ZbKxpBSxLG2rjNdzqKLxnkNTC-i1jlqvYg6D-z_GvjZ4w9aLOmz8_j6r9OXw8GX_wT4to6-
CitedBy_id	crossref_primary_10_1137_21M1402510 crossref_primary_10_1016_j_future_2023_07_024 crossref_primary_10_1109_ACCESS_2023_3288889 crossref_primary_10_1137_24M1653756 crossref_primary_10_1137_23M1578255
Cites_doi	10.1016/0045-7825(86)90025-3 10.1137/18M1182760 10.1137/120903476 10.1016/j.epsl.2012.08.016 10.1016/j.cma.2015.03.014 10.1002/nla.1928 10.1109/IPDPS.2014.119 10.1137/17M1146440 10.1007/978-3-319-32149-3_12 10.1145/2929908.2929913 10.1002/2015GL066237 10.1029/2007GC001743 10.1007/s00211-002-0445-6 10.1007/s00791-014-0231-x 10.1137/16M1077192 10.1093/acprof:oso/9780199678792.001.0001 10.1090/S0025-5718-01-01324-2 10.1016/j.apnum.2017.07.006 10.2172/1090013 10.1137/S0895479899358194 10.1109/SC.2012.91 10.1137/1.9781611970753 10.1007/BF01395886 10.1016/j.jocs.2018.12.006 10.1016/j.jocs.2016.06.006 10.1137/16M1106304 10.1145/2807591.2807675 10.1093/gji/ggs070 10.1007/978-3-540-45209-6_114 10.1145/3242094 10.1002/nla.382 10.4208/nmtma.2015.w10si 10.1007/978-3-642-58312-4_13 10.1007/978-3-662-47324-5
ContentType	Journal Article
Copyright	2021 John Wiley & Sons Ltd. 2022 John Wiley & Sons, Ltd. Distributed under a Creative Commons Attribution 4.0 International License
Copyright_xml	– notice: 2021 John Wiley & Sons Ltd. – notice: 2022 John Wiley & Sons, Ltd. – notice: Distributed under a Creative Commons Attribution 4.0 International License
DBID	AAYXX CITATION 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D 1XC VOOES
DOI	10.1002/nla.2407
DatabaseName	CrossRef Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts Technology Research Database Engineering Research Database ProQuest Computer Science Collection Civil Engineering Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access)
DatabaseTitle	CrossRef Civil Engineering Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Mechanical & Transportation Engineering Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Engineering Research Database Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional
DatabaseTitleList	Civil Engineering Abstracts CrossRef
DeliveryMethod	fulltext_linktorsrc
Discipline	Mathematics Computer Science
EISSN	1099-1506
EndPage	n/a
ExternalDocumentID	oai:HAL:hal-02528532v1 10_1002_nla_2407 NLA2407
Genre	article
GrantInformation_xml	– fundername: Deutsche Forschungsgemeinschaft German Research Foundation through the Priority Programme 1648 ''Software for Exascale Computing'' (SPPEXA) funderid: WO671/11‐1 – fundername: Gauss Centre for Supercomputing
GroupedDBID	-~X .3N .4S .DC .GA .Y3 05W 0R~ 10A 123 1L6 1OB 1OC 1ZS 31~ 33P 3SF 3WU 4.4 50Y 50Z 51W 51X 52M 52N 52O 52P 52S 52T 52U 52W 52X 5VS 66C 702 7PT 8-0 8-1 8-3 8-4 8-5 8UM 930 A03 AAESR AAEVG AAHHS AAHQN AAMNL AANHP AANLZ AAONW AASGY AAXRX AAYCA AAZKR ABCQN ABCUV ABEFU ABEML ABIJN ABPVW ACAHQ ACBWZ ACCFJ ACCZN ACGFS ACPOU ACRPL ACSCC ACXBN ACXQS ACYXJ ADBBV ADEOM ADIZJ ADKYN ADMGS ADNMO ADOZA ADXAS ADZMN AEEZP AEIGN AEIMD AENEX AEQDE AEUQT AEUYR AFBPY AFFPM AFGKR AFPWT AFWVQ AFZJQ AHBTC AITYG AIURR AIWBW AJBDE AJXKR ALAGY ALMA_UNASSIGNED_HOLDINGS ALUQN ALVPJ AMBMR AMYDB ARCSS ASPBG ATUGU AUFTA AVWKF AZBYB AZFZN AZVAB BAFTC BDRZF BFHJK BHBCM BMNLL BMXJE BNHUX BROTX BRXPI BY8 CS3 D-E D-F DCZOG DPXWK DR2 DRFUL DRSTM DU5 EBS EDO EJD F00 F01 F04 FEDTE G-S G.N GBZZK GNP GODZA H.T H.X HBH HF~ HGLYW HHY HVGLF HZ~ IX1 J0M JPC KQQ LATKE LAW LC2 LC3 LEEKS LH4 LITHE LOXES LP6 LP7 LUTES LW6 LYRES M6O MEWTI MK4 MRFUL MRSTM MSFUL MSSTM MXFUL MXSTM N04 N05 N9A NF~ O66 O9- OIG P2P P2W P2X P4D PALCI PQQKQ Q.N Q11 QB0 QRW R.K RIWAO RJQFR ROL RWI RWS RX1 RYL SAMSI SUPJJ TUS UB1 V2E W8V W99 WBKPD WIB WIH WIK WOHZO WQJ WRC WXSBR WYISQ XBAML XG1 XPP XV2 ZZTAW ~IA ~WT AAMMB AAYXX AEFGJ AEYWJ AGHNM AGQPQ AGXDD AGYGG AIDQK AIDYY AIQQE AMVHM CITATION O8X 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D 1XC VOOES
ID	FETCH-LOGICAL-c3617-e41c645e0151f1d1233d74183e100d7ab34d98b23be4ac4cb664a6775251c18e3
IEDL.DBID	DRFUL
ISICitedReferencesCount	7
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000684361800001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	1070-5325
IngestDate	Sat Oct 25 07:27:11 EDT 2025 Fri Jul 25 12:09:19 EDT 2025 Sat Nov 29 05:31:59 EST 2025 Tue Nov 18 20:42:00 EST 2025 Wed Jan 22 16:28:28 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	1
Keywords	sparse direct solver block low-rank efficient coarse level solver hierarchical hybrid grids geometric multigrid MUMPS multifrontal high-performance computing
Language	English
License	Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c3617-e41c645e0151f1d1233d74183e100d7ab34d98b23be4ac4cb664a6775251c18e3
Notes	Funding information Deutsche Forschungsgemeinschaft German Research Foundation through the Priority Programme 1648 ''Software for Exascale Computing'' (SPPEXA), WO671/11‐1; Gauss Centre for Supercomputing ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ORCID	0000-0001-9949-4634 0000-0002-3760-4698 0000-0003-3207-7021
OpenAccessLink	https://hal.science/hal-02528532
PQID	2609165170
PQPubID	2034341
PageCount	15
ParticipantIDs	hal_primary_oai_HAL_hal_02528532v1 proquest_journals_2609165170 crossref_citationtrail_10_1002_nla_2407 crossref_primary_10_1002_nla_2407 wiley_primary_10_1002_nla_2407_NLA2407
PublicationCentury	2000
PublicationDate	January 2022
PublicationDateYYYYMMDD	2022-01-01
PublicationDate_xml	– month: 01 year: 2022 text: January 2022
PublicationDecade	2020
PublicationPlace	Oxford
PublicationPlace_xml	– name: Oxford
PublicationTitle	Numerical linear algebra with applications
PublicationYear	2022
Publisher	Wiley Subscription Services, Inc Wiley
Publisher_xml	– name: Wiley Subscription Services, Inc – name: Wiley
References	2015; 37 2012 2019; 31 2012; 353‐354 2019; 15 1986; 59 2008; 9 2005 2018; 40 2003 1988; 53 2016; 17 2001; 23 2015; 8 2003; 95 2014; 21 2004; 11 2018; 39 2015; 290 2000 2019; 41 2017; 39 2015; 42 2019; 45 2014; 16 2005; 51 2019 2017 2016 2011; 67 2015 2002; 71 2014 2013; 192 2017; 122 Hülsemann F (e_1_2_11_30_1) 2005 e_1_2_11_10_1 e_1_2_11_32_1 e_1_2_11_31_1 e_1_2_11_36_1 e_1_2_11_14_1 e_1_2_11_13_1 e_1_2_11_35_1 e_1_2_11_12_1 e_1_2_11_34_1 e_1_2_11_11_1 e_1_2_11_33_1 e_1_2_11_7_1 e_1_2_11_29_1 e_1_2_11_6_1 e_1_2_11_28_1 e_1_2_11_5_1 e_1_2_11_27_1 e_1_2_11_4_1 e_1_2_11_26_1 e_1_2_11_3_1 e_1_2_11_2_1 Mary T (e_1_2_11_40_1) 2017 e_1_2_11_21_1 e_1_2_11_20_1 e_1_2_11_25_1 e_1_2_11_24_1 e_1_2_11_9_1 e_1_2_11_23_1 e_1_2_11_42_1 e_1_2_11_8_1 e_1_2_11_22_1 e_1_2_11_43_1 e_1_2_11_18_1 e_1_2_11_17_1 e_1_2_11_16_1 e_1_2_11_37_1 e_1_2_11_38_1 e_1_2_11_39_1 Gahvari H (e_1_2_11_15_1) 2014 e_1_2_11_19_1 Higham NJ (e_1_2_11_41_1) 2019
References_xml	– volume: 67 year: 2011 – volume: 11 start-page: 279 year: 2004 end-page: 91 article-title: Hierarchical hybrid grids: data structures and core algorithms for multigrid publication-title: Numer Linear Algebra Appl – volume: 95 start-page: 1 issue: 1 year: 2003 end-page: 28 article-title: Existence of ‐matrix approximants to the inverse FE‐matrix of elliptic operators with ‐coefficients publication-title: Numerische Mathematik – volume: 37 start-page: A1451 issue: 3 year: 2015 end-page: 74 article-title: Improving multifrontal methods by means of block low‐rank representations publication-title: SIAM J Sci Comput – volume: 51 start-page: 165 year: 2005 end-page: 208 – volume: 39 start-page: A1710 issue: 4 year: 2017 end-page: 40 article-title: On the complexity of the block low‐rank multifrontal factorization publication-title: SIAM J Sci Comput – year: 2005 – volume: 290 start-page: 496 year: 2015 end-page: 523 article-title: A scalable, matrix‐free multigrid preconditioner for finite element discretizations of heterogeneous Stokes flow publication-title: Comput Methods Appl Mech Eng – volume: 71 start-page: 479 issue: 238 year: 2002 end-page: 505 article-title: Analysis of iterative methods for saddle point problems: a unified approach publication-title: Math Comput – year: 2003 – volume: 42 start-page: 9270 issue: 21 year: 2015 end-page: 8 article-title: Evidence for long‐lived subduction of an ancient tectonic plate beneath the southern Indian Ocean publication-title: Geophys Res Lett – volume: 353‐354 start-page: 253 year: 2012 end-page: 69 article-title: Reconciling dynamic and seismic models of earth's lower mantle: he dominant role of thermal heterogeneity publication-title: Earth Planet Sci Lett – volume: 39 start-page: 932 issue: 2 year: 2018 end-page: 60 article-title: On the analysis of block smoothers for saddle point problems publication-title: SIAM J Matrix Anal Appl – year: 2016 – year: 2014 – volume: 17 start-page: 509 year: 2016 end-page: 21 article-title: A quantitative performance study for Stokes solvers at the extreme scale publication-title: J Comput Sci – volume: 53 start-page: 225 issue: 1 year: 1988 end-page: 35 article-title: Stabilized mixed methods for the Stokes problem publication-title: Numer Math – year: 2012 – volume: 45 start-page: 2:1 issue: 1 year: 2019 end-page: 2:26 article-title: Performance and scalability of the block low‐rank multifrontal factorization on multicore architectures publication-title: ACM Trans Math Softw – volume: 8 start-page: 22 issue: 1 year: 2015 end-page: 46 article-title: Towards textbook efficiency for parallel multigrid publication-title: Numer Math Theory Methods Appl – volume: 41 start-page: A1414 issue: 3 year: 2019 end-page: 42 article-title: Bridging the gap between flat and hierarchical low‐rank matrix formats: the multilevel block low‐rank format publication-title: SIAM J Sci Comput – volume: 122 start-page: 14 year: 2017 end-page: 38 article-title: A two‐scale approach for efficient on‐the‐fly operator assembly in massively parallel high performance multigrid codes publication-title: Appl Numer Math – volume: 59 start-page: 85 issue: 1 year: 1986 end-page: 99 article-title: A new finite element formulation for computational fluid dynamics: V. circumventing the Babuška‐Brezzi condition: a stable Petrov‐Galerkin formulation of the Stokes problem accommodating equal‐order interpolations publication-title: Comput Methods Appl Mech Eng – start-page: 101 year: 2000 end-page: 7 – volume: 16 start-page: 151 issue: 4 year: 2014 end-page: 64 article-title: A massively parallel geometric multigrid solver on hierarchically distributed grids publication-title: Comput Visual Sci – volume: 31 start-page: 60 year: 2019 end-page: 76 article-title: Large‐scale simulation of mantle convection based on a new matrix‐free approach publication-title: J Comput Sci – volume: 23 start-page: 15 issue: 1 year: 2001 end-page: 41 article-title: A fully asynchronous multifrontal solver using distributed dynamic scheduling publication-title: SIAM J Matrix Anal Appl – volume: 9 start-page: 1525 issue: 4 year: 2008 end-page: 2027 article-title: Age, spreading rates, and spreading asymmetry of the world's ocean crust publication-title: Geochem Geophys Geosyst – volume: 192 start-page: 889 issue: 3 year: 2013 end-page: 906 article-title: Large‐scale adaptive mantle convection simulation publication-title: Geophys J Int – year: 2017 – volume: 40 start-page: C581 issue: 4 year: 2018 end-page: 604 article-title: Scaling structured multigrid to 500K+ cores through coarse‐grid redistribution publication-title: SIAM J Sci Comput – year: 2019 – volume: 15 year: 2019 – year: 2015 – volume: 21 start-page: 275 issue: 2 year: 2014 end-page: 96 article-title: Reducing communication in algebraic multigrid using additive variants publication-title: Numer Linear Algebra Appl – ident: e_1_2_11_27_1 doi: 10.1016/0045-7825(86)90025-3 – ident: e_1_2_11_38_1 doi: 10.1137/18M1182760 – ident: e_1_2_11_22_1 doi: 10.1137/120903476 – ident: e_1_2_11_35_1 doi: 10.1016/j.epsl.2012.08.016 – ident: e_1_2_11_9_1 doi: 10.1016/j.cma.2015.03.014 – volume-title: Solving block low‐rank linear systems by LU factorization is numerically stable year: 2019 ident: e_1_2_11_41_1 – ident: e_1_2_11_14_1 doi: 10.1002/nla.1928 – ident: e_1_2_11_13_1 doi: 10.1109/IPDPS.2014.119 – ident: e_1_2_11_18_1 doi: 10.1137/17M1146440 – ident: e_1_2_11_10_1 – ident: e_1_2_11_11_1 doi: 10.1007/978-3-319-32149-3_12 – start-page: 165 volume-title: Numerical solution of partial differential equations on parallel computers year: 2005 ident: e_1_2_11_30_1 – ident: e_1_2_11_16_1 doi: 10.1145/2929908.2929913 – ident: e_1_2_11_34_1 doi: 10.1002/2015GL066237 – ident: e_1_2_11_24_1 doi: 10.1029/2007GC001743 – ident: e_1_2_11_36_1 doi: 10.1007/s00211-002-0445-6 – ident: e_1_2_11_19_1 doi: 10.1007/s00791-014-0231-x – ident: e_1_2_11_43_1 – ident: e_1_2_11_37_1 doi: 10.1137/16M1077192 – ident: e_1_2_11_25_1 doi: 10.1093/acprof:oso/9780199678792.001.0001 – volume-title: Improving the performance and scalability of algebraic multigrid solvers through applied performance modeling year: 2014 ident: e_1_2_11_15_1 – ident: e_1_2_11_29_1 doi: 10.1090/S0025-5718-01-01324-2 – ident: e_1_2_11_33_1 doi: 10.1016/j.apnum.2017.07.006 – ident: e_1_2_11_42_1 – ident: e_1_2_11_3_1 doi: 10.2172/1090013 – ident: e_1_2_11_20_1 doi: 10.1137/S0895479899358194 – ident: e_1_2_11_12_1 doi: 10.1109/SC.2012.91 – ident: e_1_2_11_2_1 doi: 10.1137/1.9781611970753 – ident: e_1_2_11_26_1 doi: 10.1007/BF01395886 – volume-title: Block low‐rank multifrontal solvers: complexity, performance, and scalability year: 2017 ident: e_1_2_11_40_1 – ident: e_1_2_11_23_1 doi: 10.1016/j.jocs.2018.12.006 – ident: e_1_2_11_6_1 doi: 10.1016/j.jocs.2016.06.006 – ident: e_1_2_11_7_1 doi: 10.1137/16M1106304 – ident: e_1_2_11_8_1 doi: 10.1145/2807591.2807675 – ident: e_1_2_11_28_1 doi: 10.1093/gji/ggs070 – ident: e_1_2_11_4_1 doi: 10.1007/978-3-540-45209-6_114 – ident: e_1_2_11_21_1 doi: 10.1145/3242094 – ident: e_1_2_11_5_1 doi: 10.1002/nla.382 – ident: e_1_2_11_17_1 – ident: e_1_2_11_32_1 doi: 10.4208/nmtma.2015.w10si – ident: e_1_2_11_31_1 doi: 10.1007/978-3-642-58312-4_13 – ident: e_1_2_11_39_1 doi: 10.1007/978-3-662-47324-5
SSID	ssj0006593
Score	2.334851
Snippet	Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to... Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity it is essential to...
SourceID	hal proquest crossref wiley
SourceType	Open Access Repository Aggregation Database Enrichment Source Index Database Publisher
SubjectTerms	Algorithms Analysis of PDEs Approximation Asymptotic methods block low‐rank Computer Science Distributed, Parallel, and Cluster Computing efficient coarse level solver hierarchical hybrid grids high‐performance computing Iterative methods Mathematics Multigrid methods MUMPS Saddle points Solvers sparse direct solver
Title	Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods
URI	https://onlinelibrary.wiley.com/doi/abs/10.1002%2Fnla.2407 https://www.proquest.com/docview/2609165170 https://hal.science/hal-02528532
Volume	29
WOSCitedRecordID	wos000684361800001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVWIB databaseName: Wiley Online Library Full Collection 2020 customDbUrl: eissn: 1099-1506 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0006593 issn: 1070-5325 databaseCode: DRFUL dateStart: 19960101 isFulltext: true titleUrlDefault: https://onlinelibrary.wiley.com providerName: Wiley-Blackwell
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3NbtNAEB41CYdygAKtCIRqQQhOptk__xzT0iiHECFEpHCydtebNmqURHEIVx6BZ-yTdMZ2rCCBVKknS_ZY9u7M7nwzmv0G4H0Su2kWOh1IyU2gpEoC4-MsMIkxsSWPq6ZFs4loNIonk-RrVVVJZ2FKfog64UYro9ivaYEbm5_tkYbOzScKRxrQEmi2ugmtz9_642G9D4cl5S7GN91AS6F31LNdcbZ79y9n1LimUsg9nLmPVgt303_6kB89gicVyGS90iqewYFfPIfHX2qG1vwF_DhHL3bD5stft7__UON2RkmDuWerddV1h7klBr2eXa1nGUMLpfoNhhCX4XZOSUWWo3o9KyoSC5myGXV-DOP-5feLQVC1WQicRPwSeMVdqLRHYMCnPENXJjPitJEeB5BFxkqVJbEV0nplnHI2DJUJo0gjNHI89vIEmovlwr8ExlXmEo0RHfHQ2ciZRHged6UJZWitjtvwcTffqas4yKkVxjwt2ZNFwXFNk9WGt7XkquTd-IfMO1RZ_ZiIsge9YUr3EMkJBCJiy9vQ2Wk0rZZnnmIQh7BY86jbhg-F7v77kXQ07NH11X0FX8OhoCMSRZqmA83N-qd_A4_cdjPL16eVkd4BdnvqDQ
linkProvider	Wiley-Blackwell
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1ZS-RAEC50XFAfvMXxbBdxn6LT6c6FT-MxjBiHZVFwn0Kn06PiMDNMPF79Cf5Gf4lVuRhhFwSfAkmFpLuqu74qqr8C2At83U1c7VhCcGVJIQNLGT-xVKCUH5PHld2s2YTX6fg3N8HvCTgqz8Lk_BBVwo1WRrZf0wKnhPThGGtoTx1QPDIJUxKtCM176vRP6zqsNmI359zFAKdhOcJ2Su7Zhn1YvvvJG03eUS3kGNAch6uZv2nNf-tPF2CugJmsmdvFIkyY_hLMXlYcreky_D1GP_bAeoOX99c3at3OKG3QM2w4KvruMD3AsNew29F9wtBGqYKDIchluKFTWpGlqGDDsprETCZvR52uwHXr7OqkbRWNFiwtEMFYRnLtSscgNOBdnqAzEwmx2giDA0g8FQuZBH5si9hIpaWOXVcq1_McBEea-0asQq0_6Js1YFwmOnAwpiMmutjTKrAN9xtCucKNY8evw69ywiNdsJBTM4xelPMn2xnLNU1WHXYryWHOvPEPmZ-os-oxUWW3m2FE9xDL2QhF7Gdeh81SpVGxQNMIwzgExg73GnXYz5T3349EnbBJ1_WvCu7AdPvqMozC887FBszYdGAiS9psQu1x9GS24Id-frxPR9uFxX4AfKjt_Q
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3bThsxEB2FBCF4aMtNhFJqEIKnJfHae1Of0kJE1RAhBBI8rby2A4goibJcXvsJ_cZ-SWf2piBRCYmnlXZntWuP7Tkzss8B2ItCPTC-9hwhuHKkkJGjbGgcFSkVJhRx5SATmwj6_fDqKjqrwbfyLEzOD1EV3GhmZOs1TXA7MYPWDGvoUB1SPjIHDUkaMnVoHJ13L3vVQuznnLuY4LQdT7heyT3bdlvluy-i0dwt7YWcAZqzcDWLN92P7_rTT_ChgJmsk4-LZajZ0QosnVYcrekqXH_HOHbPhuPnv7__kHQ7o7LB0LLJtNDdYXqMaa9lN9M7w3CM0g4OhiCX4YJOZUWWooMty_YkZja5HHW6Bpfd44sfJ04htOBogQjGsZJrX3oWoQEfcIPBTBhitREWG2AClQhpojBxRWKl0lInvi-VHwQegiPNQyvWoT4aj-wGMC6NjjzM6YiJLgm0ilzLw7ZQvvCTxAubcFB2eKwLFnISwxjGOX-ym7FcU2c1YaeynOTMG6_Y7KLPqsdElX3S6cV0D7Gci1DEfeJN2CpdGhcTNI0xjUNg7PGg3YT9zHn__Ujc73XouvlWw6-wcHbUjXs_-78-w6JL5yWyms0W1B-mj_YLzOunh7t0ul0M2H8iZu14
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Block+low%E2%80%90rank+single+precision+coarse+grid+solvers+for+extreme+scale+multigrid+methods&rft.jtitle=Numerical+linear+algebra+with+applications&rft.au=Buttari%2C+Alfredo&rft.au=Huber%2C+Markus&rft.au=Leleux%2C+Philippe&rft.au=Theo%2C+Mary&rft.date=2022-01-01&rft.pub=Wiley+Subscription+Services%2C+Inc&rft.eissn=1099-1506&rft.volume=29&rft.issue=1&rft_id=info:doi/10.1002%2Fnla.2407&rft.externalDBID=NO_FULL_TEXT
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1070-5325&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1070-5325&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1070-5325&client=summon