{"id":"https://openalex.org/W2019434176","doi":"https://doi.org/10.1137/08072084x","title":"A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation","display_name":"A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2019434176","doi":"https://doi.org/10.1137/08072084x","mag":"2019434176"},"language":"en","primary_location":{"id":"doi:10.1137/08072084x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/08072084x","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020006925","display_name":"Jinyu Tang","orcid":"https://orcid.org/0000-0002-1877-7665"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"J. M. Tang","raw_affiliation_strings":["jtang@cs.umn.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"jtang@cs.umn.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027208204","display_name":"Scott MacLachlan","orcid":"https://orcid.org/0000-0002-6364-0684"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"S. P. MacLachlan","raw_affiliation_strings":["scott.maclachlan@tufts.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"scott.maclachlan@tufts.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082240312","display_name":"Reinhard Nabben","orcid":"https://orcid.org/0000-0001-5076-9895"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"R. Nabben","raw_affiliation_strings":["nabben@math.tu-berlin.de#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"nabben@math.tu-berlin.de#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080989974","display_name":"C. Vuik","orcid":"https://orcid.org/0000-0002-5988-9153"},"institutions":[{"id":"https://openalex.org/I98358874","display_name":"Delft University of Technology","ror":"https://ror.org/02e2c7k09","country_code":"NL","type":"education","lineage":["https://openalex.org/I98358874"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"C. Vuik","raw_affiliation_strings":["c.vuik@tudelft.nl#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"c.vuik@tudelft.nl#TAB#","institution_ids":["https://openalex.org/I98358874"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.0315,"has_fulltext":false,"cited_by_count":46,"citation_normalized_percentile":{"value":0.96659268,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"31","issue":"4","first_page":"1715","last_page":"1739"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9923999905586243,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.949284553527832},{"id":"https://openalex.org/keywords/cholesky-decomposition","display_name":"Cholesky decomposition","score":0.8311806917190552},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.7955788373947144},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6580036282539368},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.6386922597885132},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.636969268321991},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5780537128448486},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.5384703874588013},{"id":"https://openalex.org/keywords/coefficient-matrix","display_name":"Coefficient matrix","score":0.5223339796066284},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.5051267743110657},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.47725626826286316},{"id":"https://openalex.org/keywords/additive-schwarz-method","display_name":"Additive Schwarz method","score":0.47233906388282776},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.44038254022598267},{"id":"https://openalex.org/keywords/eigendecomposition-of-a-matrix","display_name":"Eigendecomposition of a matrix","score":0.4356818199157715},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.37344664335250854},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.34208545088768005},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.32902926206588745},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1739346981048584},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.14549851417541504},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.12035408616065979}],"concepts":[{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.949284553527832},{"id":"https://openalex.org/C34727166","wikidata":"https://www.wikidata.org/wiki/Q515375","display_name":"Cholesky decomposition","level":3,"score":0.8311806917190552},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.7955788373947144},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6580036282539368},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.6386922597885132},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.636969268321991},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5780537128448486},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5384703874588013},{"id":"https://openalex.org/C60866291","wikidata":"https://www.wikidata.org/wiki/Q5140577","display_name":"Coefficient matrix","level":3,"score":0.5223339796066284},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.5051267743110657},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.47725626826286316},{"id":"https://openalex.org/C174819683","wikidata":"https://www.wikidata.org/wiki/Q4681340","display_name":"Additive Schwarz method","level":4,"score":0.47233906388282776},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.44038254022598267},{"id":"https://openalex.org/C169756996","wikidata":"https://www.wikidata.org/wiki/Q194919","display_name":"Eigendecomposition of a matrix","level":3,"score":0.4356818199157715},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.37344664335250854},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.34208545088768005},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32902926206588745},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1739346981048584},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.14549851417541504},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.12035408616065979},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/08072084x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/08072084x","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:tudelft.nl:uuid:4d1dfdee-a481-4253-906d-78ea4d741719","is_oa":false,"landing_page_url":"http://resolver.tudelft.nl/uuid:4d1dfdee-a481-4253-906d-78ea4d741719","pdf_url":null,"source":{"id":"https://openalex.org/S4306400906","display_name":"Research Repository (Delft University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I98358874","host_organization_name":"Delft University of Technology","host_organization_lineage":["https://openalex.org/I98358874"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"journal article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1485732691","https://openalex.org/W1491819224","https://openalex.org/W1519509134","https://openalex.org/W1530872699","https://openalex.org/W1548589512","https://openalex.org/W1572639095","https://openalex.org/W1973061863","https://openalex.org/W1974521166","https://openalex.org/W1981745143","https://openalex.org/W1995078768","https://openalex.org/W2007975598","https://openalex.org/W2011389520","https://openalex.org/W2017085690","https://openalex.org/W2028989647","https://openalex.org/W2049644675","https://openalex.org/W2059679865","https://openalex.org/W2064878022","https://openalex.org/W2076229433","https://openalex.org/W2082146198","https://openalex.org/W2082232986","https://openalex.org/W2106088594","https://openalex.org/W2114535880","https://openalex.org/W2130528149","https://openalex.org/W2137008096","https://openalex.org/W2142385993","https://openalex.org/W2157875412","https://openalex.org/W2798909945","https://openalex.org/W3020850290"],"related_works":["https://openalex.org/W2809896915","https://openalex.org/W1592639845","https://openalex.org/W1482750774","https://openalex.org/W2966631139","https://openalex.org/W2146296113","https://openalex.org/W1986812736","https://openalex.org/W1968940950","https://openalex.org/W2963172443","https://openalex.org/W1623009832","https://openalex.org/W2773710110"],"abstract_inverted_index":{"It":[0],"is":[1,122,158,172,199,231,270],"well":[2],"known":[3],"that":[4,236],"two-level":[5,32,71,125,146,168,248,268],"and":[6,19,28,57,93,144,241],"multilevel":[7,64],"preconditioned":[8],"conjugate":[9],"gradient":[10],"(PCG)":[11],"methods":[12,100,127,143,250],"provide":[13],"efficient":[14],"techniques":[15],"for":[16,210,237],"solving":[17],"large":[18,58],"sparse":[20],"linear":[21],"systems":[22],"whose":[23],"coefficient":[24,62],"matrices":[25],"are":[26,83,114,281],"symmetric":[27,129],"positive":[29],"definite.":[30],"A":[31],"PCG":[33,126,249],"method":[34],"combines":[35],"a":[36,45,145],"traditional":[37],"(one-level)":[38],"preconditioner,":[39,240],"such":[40,81],"as":[41,235],"incomplete":[42],"Cholesky,":[43],"with":[44,128],"projection-type":[46],"preconditioner":[47,157,269],"to":[48,102,123,141,161,174,187,283],"get":[49],"rid":[50],"of":[51,54,60,89,111,119,182,213,228,245,266,276],"the":[52,61,70,74,78,87,151,183,195,214,219,224,233,238,242,246,255,263,267,274,277,285],"effect":[53],"both":[55],"small":[56],"eigenvalues":[59],"matrix;":[63],"approaches":[65],"arise":[66],"by":[67,150],"recursively":[68],"applying":[69],"technique":[72],"within":[73],"projection":[75],"step.":[76],"In":[77],"literature,":[79],"various":[80],"preconditioners":[82,132,217,230],"known,":[84],"coming":[85],"from":[86,107,134],"fields":[88],"deflation,":[90],"domain":[91],"decomposition,":[92],"multigrid":[94],"(MG).":[95],"At":[96],"first":[97],"glance,":[98],"these":[99,166,229],"seem":[101],"be":[103,162,175,253],"quite":[104],"distinct;":[105],"however,":[106],"an":[108],"abstract":[109,139],"point":[110],"view,":[112],"they":[113],"closely":[115],"related.":[116],"The":[117,155],"aim":[118],"this":[120,171,204,261],"paper":[121],"relate":[124],"two-grid":[130],"(V(1,1)-cycle)":[131],"(derived":[133],"MG":[135,185,239],"approaches),":[136],"in":[137,191,218,260],"their":[138],"form,":[140],"deflation":[142,220],"domain-decomposition":[147,222],"approach":[148],"inspired":[149],"balancing":[152],"Neumann\u2013Neumann":[153],"method.":[154],"MG-based":[156],"often":[159],"expected":[160],"more":[163,200],"effective":[164],"than":[165,273],"other":[167],"preconditioners,":[169],"but":[170,194],"shown":[173],"not":[176],"always":[177],"true.":[178],"For":[179],"common":[180],"choices":[181,212],"parameters,":[184],"leads":[186],"larger":[188],"error":[189],"reductions":[190],"each":[192],"iteration,":[193],"work":[196,225],"per":[197,226],"iteration":[198,227],"expensive,":[201],"which":[202],"makes":[203],"comparison":[205],"unfair.":[206],"We":[207],"show":[208],"that,":[209,259],"special":[211],"underlying":[215],"one-level":[216],"or":[221],"methods,":[223],"approximately":[232],"same":[234],"convergence":[243],"properties":[244],"resulting":[247],"will":[251],"also":[252],"(approximately)":[254],"same.":[256],"This":[257],"means":[258],"respect,":[262],"particular":[264],"choice":[265,275],"less":[271],"important":[272],"parameters.":[278],"Numerical":[279],"experiments":[280],"presented":[282],"emphasize":[284],"theoretical":[286],"results.":[287]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":5}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}