{"id":"https://openalex.org/W1967370978","doi":"https://doi.org/10.1137/100786502","title":"Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems","display_name":"Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W1967370978","doi":"https://doi.org/10.1137/100786502","mag":"1967370978"},"language":"en","primary_location":{"id":"doi:10.1137/100786502","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100786502","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1004.0382","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016752483","display_name":"Andrei Dr\u0103g\u0103nescu","orcid":"https://orcid.org/0009-0001-1425-8374"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Andrei Dr\u0103g\u0103nescu","raw_affiliation_strings":["draga@umbc.edu#TAB#"],"affiliations":[{"raw_affiliation_string":"draga@umbc.edu#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007669325","display_name":"Cosmin G. Petra","orcid":"https://orcid.org/0000-0002-1050-3221"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cosmin Petra","raw_affiliation_strings":["petra@mcs.anl.gov#TAB#"],"affiliations":[{"raw_affiliation_string":"petra@mcs.anl.gov#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5016752483"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.05849047,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"50","issue":"1","first_page":"328","last_page":"353"},"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":0.9998999834060669,"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":0.9998999834060669,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9972000122070312,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.6780869960784912},{"id":"https://openalex.org/keywords/tikhonov-regularization","display_name":"Tikhonov regularization","score":0.6553575396537781},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.633994460105896},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.5760776400566101},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5450667142868042},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.5432748198509216},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.4956885576248169},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4936105012893677},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.47997212409973145},{"id":"https://openalex.org/keywords/interior-point-method","display_name":"Interior point method","score":0.45425134897232056},{"id":"https://openalex.org/keywords/coefficient-matrix","display_name":"Coefficient matrix","score":0.4412859380245209},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.41848015785217285},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.41562598943710327},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4134540557861328},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.28592348098754883},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.26337766647338867},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.14639872312545776},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.12297052145004272},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.11806941032409668},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1108875572681427},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10564759373664856}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.6780869960784912},{"id":"https://openalex.org/C152442038","wikidata":"https://www.wikidata.org/wiki/Q2778212","display_name":"Tikhonov regularization","level":3,"score":0.6553575396537781},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.633994460105896},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.5760776400566101},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5450667142868042},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.5432748198509216},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.4956885576248169},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4936105012893677},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.47997212409973145},{"id":"https://openalex.org/C155253501","wikidata":"https://www.wikidata.org/wiki/Q461992","display_name":"Interior point method","level":2,"score":0.45425134897232056},{"id":"https://openalex.org/C60866291","wikidata":"https://www.wikidata.org/wiki/Q5140577","display_name":"Coefficient matrix","level":3,"score":0.4412859380245209},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.41848015785217285},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.41562598943710327},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4134540557861328},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28592348098754883},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.26337766647338867},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.14639872312545776},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.12297052145004272},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.11806941032409668},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1108875572681427},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10564759373664856},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"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/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/100786502","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100786502","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1004.0382","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1004.0382","pdf_url":"https://arxiv.org/pdf/1004.0382","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:osti.gov:1493159","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/1493159","pdf_url":"https://www.osti.gov/servlets/purl/1493159","source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},{"id":"mag:1967370978","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1004.0382v2","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1004.0382","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1004.0382","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1004.0382","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1004.0382","pdf_url":"https://arxiv.org/pdf/1004.0382","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2294326009","display_name":"Interior Point Methods for  Complementarity Problems","funder_award_id":"0728878","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2372726881","display_name":null,"funder_award_id":"72887","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3207428304","display_name":null,"funder_award_id":"DMS-0821311","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3349248314","display_name":null,"funder_award_id":"DMS-1016177","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5061334006","display_name":"Multilevel methods in PDE constrained optimization","funder_award_id":"1016177","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7129481701","display_name":null,"funder_award_id":"0821311","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1530872699","https://openalex.org/W1570089119","https://openalex.org/W1577199596","https://openalex.org/W1966766345","https://openalex.org/W1971270535","https://openalex.org/W1974815257","https://openalex.org/W1992056335","https://openalex.org/W2023884611","https://openalex.org/W2026365342","https://openalex.org/W2029170505","https://openalex.org/W2037116002","https://openalex.org/W2039513266","https://openalex.org/W2054437396","https://openalex.org/W2055183575","https://openalex.org/W2055360045","https://openalex.org/W2055822950","https://openalex.org/W2059228862","https://openalex.org/W2059991864","https://openalex.org/W2061045704","https://openalex.org/W2070584352","https://openalex.org/W2071485538","https://openalex.org/W2078197553","https://openalex.org/W2079482881","https://openalex.org/W2080514141","https://openalex.org/W2086867325","https://openalex.org/W2115178538","https://openalex.org/W2133248904","https://openalex.org/W2158258156"],"related_works":["https://openalex.org/W2134414648","https://openalex.org/W2560679848","https://openalex.org/W2079018274","https://openalex.org/W2556411650","https://openalex.org/W2970453762","https://openalex.org/W2117071079","https://openalex.org/W2118431207","https://openalex.org/W1996982555","https://openalex.org/W2071700284","https://openalex.org/W2155298633","https://openalex.org/W1970267095","https://openalex.org/W3111433034","https://openalex.org/W2046360167","https://openalex.org/W1984354517","https://openalex.org/W3157487560","https://openalex.org/W2181983165","https://openalex.org/W1602201142","https://openalex.org/W76896619","https://openalex.org/W2153619784","https://openalex.org/W2003086485"],"abstract_inverted_index":{"In":[0,143],"this":[1,123],"article":[2],"we":[3],"construct":[4],"and":[5,31,76,89,136],"analyze":[6],"multigrid":[7],"preconditioners":[8,104],"for":[9,84,92,182],"discretizations":[10],"of":[11,13,101,118,152,164],"operators":[12],"the":[14,23,48,61,71,85,99,102,116,134,150,161,175],"form":[15],"${\\mathcal":[16,32],"D}_{\\lambda}+{\\mathcal":[17],"K}^*{\\mathcal":[18],"K}$,":[19],"where":[20],"$D_{\\lambda}$":[21],"is":[22,34,67,90,105,147,168],"multiplication":[24],"with":[25,56,131,170],"a":[26,35,126,138],"relatively":[27],"smooth":[28],"function":[29],"$\\lambda>0$":[30],"K}$":[33],"compact":[36],"linear":[37,153],"operator.":[38],"These":[39],"systems":[40],"arise":[41],"when":[42],"applying":[43],"interior":[44],"point":[45],"methods":[46],"to":[47,70,107,177],"minimization":[49],"problem":[50,88],"$\\min_{u}":[51],"\\frac{1}{2}(|\\!|{\\mathcal":[52],"K}":[53],"u-f|\\!|^2":[54],"+\\beta|\\!|u|\\!|^2)$":[55],"box-constraints":[57,132],"$\\underline{u}\\leqslant":[58],"u\\leqslant\\overline{u}$":[59],"on":[60,125,133,137],"controls.":[62],"The":[63],"presented":[64],"preconditioning":[65],"technique":[66],"closely":[68],"related":[69],"one":[72],"developed":[73],"by":[74],"Dr\u0103g\u0103nescu":[75],"Dupont":[77],"[Math.":[78],"Comp.,":[79],"77":[80],"(2008),":[81],"pp.":[82],"2001\u20132038]":[83],"associated":[86],"unconstrained":[87],"intended":[91],"large-scale":[93,184],"problems.":[94,185],"As":[95],"in":[96],"that":[97,149],"work,":[98],"quality":[100],"resulting":[103],"shown":[106,148],"increase":[108],"as":[109,115,158,160],"$h\\downarrow":[110],"0$,":[111],"but":[112],"it":[113,146],"decreases":[114],"smoothness":[117],"$\\lambda$":[119],"declines.":[120],"We":[121],"test":[122],"algorithm":[124],"Tikhonov-regularized":[127],"backward":[128],"parabolic":[129],"equation":[130],"control":[135],"standard":[139],"elliptic-constrained":[140],"optimization":[141,156],"problem.":[142],"both":[144],"cases":[145],"number":[151,163],"iterations":[154],"per":[155],"step,":[157],"well":[159],"total":[162],"finest-scale":[165],"matrix-vector":[166],"multiplications,":[167],"decreasing":[169],"increasing":[171],"resolution,":[172],"thus":[173],"showing":[174],"method":[176],"be":[178],"potentially":[179],"very":[180],"efficient":[181],"truly":[183]},"counts_by_year":[],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
