{"id":"https://openalex.org/W2033362846","doi":"https://doi.org/10.1137/060651161","title":"On the Convergence of a Class of Multilevel Methods for Large Sparse Markov Chains","display_name":"On the Convergence of a Class of Multilevel Methods for Large Sparse Markov Chains","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W2033362846","doi":"https://doi.org/10.1137/060651161","mag":"2033362846"},"language":"en","primary_location":{"id":"doi:10.1137/060651161","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060651161","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/A5022119933","display_name":"Peter Buchholz","orcid":"https://orcid.org/0000-0002-9966-7686"},"institutions":[{"id":"https://openalex.org/I200332995","display_name":"TU Dortmund University","ror":"https://ror.org/01k97gp34","country_code":"DE","type":"education","lineage":["https://openalex.org/I200332995"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Peter Buchholz","raw_affiliation_strings":["Informatik IV, Universit\u00e4t Dortmund, D-44221 Dortmund, Germany"],"affiliations":[{"raw_affiliation_string":"Informatik IV, Universit\u00e4t Dortmund, D-44221 Dortmund, Germany","institution_ids":["https://openalex.org/I200332995"]}]},{"author_position":"last","author":{"id":null,"display_name":"Turul Dayar","orcid":null},"institutions":[{"id":"https://openalex.org/I168864056","display_name":"Bilkent University","ror":"https://ror.org/02vh8a032","country_code":"TR","type":"education","lineage":["https://openalex.org/I168864056"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Turul Dayar","raw_affiliation_strings":["Department of Computer Engineering, Bilkent University, TR-06800 Bilkent, Ankara, Turkey"],"affiliations":[{"raw_affiliation_string":"Department of Computer Engineering, Bilkent University, TR-06800 Bilkent, Ankara, Turkey","institution_ids":["https://openalex.org/I168864056"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5022119933"],"corresponding_institution_ids":["https://openalex.org/I200332995"],"apc_list":null,"apc_paid":null,"fwci":2.246,"has_fulltext":false,"cited_by_count":32,"citation_normalized_percentile":{"value":0.87425246,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"29","issue":"3","first_page":"1025","last_page":"1049"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9969000220298767,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9969000220298767,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9843000173568726,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.7843016386032104},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.6499501466751099},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5364584922790527},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.5322788953781128},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5262162685394287},{"id":"https://openalex.org/keywords/sparse-matrix","display_name":"Sparse matrix","score":0.4986414909362793},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.46924638748168945},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4576197862625122},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.43294233083724976},{"id":"https://openalex.org/keywords/markov-model","display_name":"Markov model","score":0.42150717973709106},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42125651240348816},{"id":"https://openalex.org/keywords/multilevel-model","display_name":"Multilevel model","score":0.4137110114097595},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40249165892601013},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.3537944257259369},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3353555202484131},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3299301862716675},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.13215187191963196},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07087123394012451},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.06632617115974426}],"concepts":[{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.7843016386032104},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.6499501466751099},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5364584922790527},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.5322788953781128},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5262162685394287},{"id":"https://openalex.org/C56372850","wikidata":"https://www.wikidata.org/wiki/Q1050404","display_name":"Sparse matrix","level":3,"score":0.4986414909362793},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.46924638748168945},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4576197862625122},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.43294233083724976},{"id":"https://openalex.org/C163836022","wikidata":"https://www.wikidata.org/wiki/Q6771326","display_name":"Markov model","level":3,"score":0.42150717973709106},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42125651240348816},{"id":"https://openalex.org/C53059260","wikidata":"https://www.wikidata.org/wiki/Q374758","display_name":"Multilevel model","level":2,"score":0.4137110114097595},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40249165892601013},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3537944257259369},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3353555202484131},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3299301862716675},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.13215187191963196},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07087123394012451},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.06632617115974426},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","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},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/060651161","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060651161","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:CiteSeerX.psu:10.1.1.530.223","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.530.223","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.bilkent.edu.tr/tech-reports/2006/BU-CE-0601.pdf","raw_type":"text"},{"id":"pmh:oai:repository.bilkent.edu.tr:11693/23304","is_oa":false,"landing_page_url":"http://hdl.handle.net/11693/23304","pdf_url":null,"source":{"id":"https://openalex.org/S4306400079","display_name":"Bilkent University Institutional Repository (Bilkent University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I168864056","host_organization_name":"Bilkent University","host_organization_lineage":["https://openalex.org/I168864056"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.4300000071525574}],"awards":[],"funders":[{"id":"https://openalex.org/F4320308269","display_name":"Alexander von Humboldt-Stiftung","ror":"https://ror.org/012kf4317"},{"id":"https://openalex.org/F4320322628","display_name":"T\u00fcrkiye Bilimler Akademisi","ror":"https://ror.org/00aqt9352"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W105488278","https://openalex.org/W133145907","https://openalex.org/W1487444668","https://openalex.org/W1500921805","https://openalex.org/W1514324734","https://openalex.org/W1540127687","https://openalex.org/W1547524050","https://openalex.org/W1572639095","https://openalex.org/W1784183156","https://openalex.org/W1969620704","https://openalex.org/W1978546706","https://openalex.org/W1988420772","https://openalex.org/W1994609883","https://openalex.org/W1999879011","https://openalex.org/W1999886256","https://openalex.org/W2028717072","https://openalex.org/W2041394080","https://openalex.org/W2041406274","https://openalex.org/W2044047612","https://openalex.org/W2055287110","https://openalex.org/W2061297568","https://openalex.org/W2063646381","https://openalex.org/W2063973166","https://openalex.org/W2073389244","https://openalex.org/W2076759647","https://openalex.org/W2092517171","https://openalex.org/W2107561541","https://openalex.org/W2111424439","https://openalex.org/W2342316458","https://openalex.org/W2563233999","https://openalex.org/W4231964533","https://openalex.org/W4245432260"],"related_works":["https://openalex.org/W2077078424","https://openalex.org/W2182328658","https://openalex.org/W1519938374","https://openalex.org/W84272617","https://openalex.org/W613282331","https://openalex.org/W2092341060","https://openalex.org/W2625595466","https://openalex.org/W2349420447","https://openalex.org/W2058418735","https://openalex.org/W3153509215"],"abstract_inverted_index":{"This":[0],"paper":[1],"investigates":[2],"the":[3,6,37,41,73,95,100,114],"theory":[4],"behind":[5],"steady":[7],"state":[8],"analysis":[9],"of":[10,20,40,43,68,90,102],"large":[11,106],"sparse":[12,107],"Markov":[13,108],"chains":[14,109],"with":[15],"a":[16,49,85],"recently":[17],"proposed":[18],"class":[19,42,101],"multilevel":[21,44,103],"methods":[22,45,104],"using":[23],"concepts":[24],"from":[25],"algebraic":[26],"multigrid":[27,69],"and":[28,46,63,72,87],"iterative":[29],"aggregation-disaggregation.":[30],"The":[31],"motivation":[32],"is":[33,80],"to":[34,47,98],"better":[35],"understand":[36],"convergence":[38],"characteristics":[39],"have":[48],"clearer":[50],"formulation":[51],"that":[52],"will":[53],"aid":[54],"their":[55],"implementation.":[56],"In":[57],"doing":[58],"this,":[59],"restriction":[60],"(or":[61,65,92],"aggregation)":[62],"prolongation":[64],"disaggregation)":[66],"operators":[67],"are":[70],"used,":[71],"Kronecker-based":[74],"approach":[75],"for":[76,105],"hierarchical":[77],"Markovian":[78],"models":[79],"employed,":[81],"since":[82],"it":[83],"suggests":[84],"natural":[86],"compact":[88],"definition":[89],"grids":[91],"levels).":[93],"However,":[94],"formalism":[96],"used":[97],"describe":[99],"has":[110],"no":[111],"influence":[112],"on":[113],"theoretical":[115],"results":[116],"derived.":[117]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":7}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}