{"id":"https://openalex.org/W2126906098","doi":"https://doi.org/10.1137/s106482750241565x","title":"Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver","display_name":"Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2126906098","doi":"https://doi.org/10.1137/s106482750241565x","mag":"2126906098"},"language":"en","primary_location":{"id":"doi:10.1137/s106482750241565x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s106482750241565x","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","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/A5044137409","display_name":"George Biros","orcid":"https://orcid.org/0000-0002-0033-3994"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"George Biros","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5049331711","display_name":"Omar Ghattas","orcid":"https://orcid.org/0000-0001-7742-2509"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Omar Ghattas","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5044137409"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":20.8727,"has_fulltext":false,"cited_by_count":271,"citation_normalized_percentile":{"value":0.99755184,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"27","issue":"2","first_page":"687","last_page":"713"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9995999932289124,"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":0.9995999932289124,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9991000294685364,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9973000288009644,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/karush\u2013kuhn\u2013tucker-conditions","display_name":"Karush\u2013Kuhn\u2013Tucker conditions","score":0.7766760587692261},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.7565306425094604},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.7172366380691528},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6408011317253113},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.5829599499702454},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5132052898406982},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.5104195475578308},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5014326572418213},{"id":"https://openalex.org/keywords/krylov-subspace","display_name":"Krylov subspace","score":0.4900931119918823},{"id":"https://openalex.org/keywords/augmented-lagrangian-method","display_name":"Augmented Lagrangian method","score":0.4606582820415497},{"id":"https://openalex.org/keywords/sequential-quadratic-programming","display_name":"Sequential quadratic programming","score":0.44085171818733215},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.43429166078567505},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.3095978796482086},{"id":"https://openalex.org/keywords/quadratic-programming","display_name":"Quadratic programming","score":0.27203965187072754},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1472330093383789},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.07848858833312988}],"concepts":[{"id":"https://openalex.org/C50454189","wikidata":"https://www.wikidata.org/wiki/Q2075125","display_name":"Karush\u2013Kuhn\u2013Tucker conditions","level":2,"score":0.7766760587692261},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.7565306425094604},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.7172366380691528},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6408011317253113},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.5829599499702454},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5132052898406982},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.5104195475578308},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5014326572418213},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.4900931119918823},{"id":"https://openalex.org/C150452318","wikidata":"https://www.wikidata.org/wiki/Q4820432","display_name":"Augmented Lagrangian method","level":2,"score":0.4606582820415497},{"id":"https://openalex.org/C198927703","wikidata":"https://www.wikidata.org/wiki/Q4373881","display_name":"Sequential quadratic programming","level":3,"score":0.44085171818733215},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.43429166078567505},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.3095978796482086},{"id":"https://openalex.org/C81845259","wikidata":"https://www.wikidata.org/wiki/Q290117","display_name":"Quadratic programming","level":2,"score":0.27203965187072754},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1472330093383789},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.07848858833312988},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/s106482750241565x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s106482750241565x","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.106.9837","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.106.9837","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.seas.upenn.edu/~biros/papers/lnks/paper.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.113.7104","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.7104","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.aladdin.cs.cmu.edu/papers/pdfs/y2000/new_lagr_kryl.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.118.8316","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.118.8316","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.seas.upenn.edu/~biros/papers/lnks/paperns.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.142.3546","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.142.3546","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-2.cs.cmu.edu/~oghattas/papers/lnks/lnks2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.65.9942","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.9942","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-2.cs.cmu.edu/~oghattas/papers/lnks/lnks1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.68.4636","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.68.4636","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.aladdin.cs.cmu.edu/papers/pdfs/y2000/siag.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.71.8012","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.71.8012","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.aladdin.cs.cmu.edu/papers/pdfs/y2000/mesh_lagr.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6600000262260437,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":44,"referenced_works":["https://openalex.org/W41522164","https://openalex.org/W90568776","https://openalex.org/W192530076","https://openalex.org/W1506342804","https://openalex.org/W1510510199","https://openalex.org/W1546004968","https://openalex.org/W1560263223","https://openalex.org/W1569090332","https://openalex.org/W1972015017","https://openalex.org/W1976685335","https://openalex.org/W1981745143","https://openalex.org/W1987336415","https://openalex.org/W1987375043","https://openalex.org/W1988357492","https://openalex.org/W1993553082","https://openalex.org/W2001511715","https://openalex.org/W2002352627","https://openalex.org/W2024541916","https://openalex.org/W2024949119","https://openalex.org/W2028299074","https://openalex.org/W2033238921","https://openalex.org/W2034691351","https://openalex.org/W2038977026","https://openalex.org/W2050361884","https://openalex.org/W2068696746","https://openalex.org/W2075194837","https://openalex.org/W2077658674","https://openalex.org/W2079361963","https://openalex.org/W2087150501","https://openalex.org/W2088116381","https://openalex.org/W2091515296","https://openalex.org/W2115215129","https://openalex.org/W2121565597","https://openalex.org/W2121866348","https://openalex.org/W2138241465","https://openalex.org/W2149416133","https://openalex.org/W2165622925","https://openalex.org/W2238405919","https://openalex.org/W2798909945","https://openalex.org/W3029645440","https://openalex.org/W4205624883","https://openalex.org/W4212997115","https://openalex.org/W4251180451","https://openalex.org/W4293775970"],"related_works":["https://openalex.org/W1981158318","https://openalex.org/W4386916233","https://openalex.org/W2615021531","https://openalex.org/W4367186753","https://openalex.org/W4298347182","https://openalex.org/W4288092242","https://openalex.org/W2408672909","https://openalex.org/W2155865164","https://openalex.org/W2783836771","https://openalex.org/W2980744749"],"abstract_inverted_index":{"Large-scale":[0],"optimization":[1],"of":[2,35,77,96,102,110,116,134,189,206,214,282,302,318],"systems":[3,121],"governed":[4],"by":[5,150,163,168,236],"partial":[6],"differential":[7],"equations":[8],"(PDEs)":[9],"is":[10,47,183,199,243],"a":[11,66,79,128,164,169,251],"frontier":[12],"problem":[13,51],"in":[14,229],"scientific":[15],"computation.":[16],"Reduced":[17],"quasi-Newton":[18,90],"sequential":[19],"quadratic":[20],"programming":[21],"(SQP)":[22],"methods":[23,31],"are":[24,99,234,258],"state-of-the-art":[25],"approaches":[26],"for":[27,49,69,193,291],"such":[28],"problems.":[29],"These":[30],"take":[32],"full":[33,80],"advantage":[34],"existing":[36],"PDE":[37,148,195],"solver":[38,83],"technology":[39],"and":[40,154,198,271,279,286,308,311],"parallelize":[41],"well.":[42],"However,":[43],"their":[44,151],"algorithmic":[45],"scalability":[46,278],"questionable;":[48],"certain":[50],"classes":[52],"they":[53],"can":[54],"be":[55],"very":[56],"slow":[57],"to":[58,244,266,293,314],"converge.":[59],"In":[60,203,299],"this":[61,207],"two-part":[62,208],"article":[63],"we":[64,174,210,305],"propose":[65],"new":[67,177],"method":[68,98,178,223],"steady-state":[70,238],"PDE-constrained":[71],"optimization,":[72],"based":[73],"on":[74,224,264,272],"the":[75,97,103,111,117,135,146,155,176,187,194,212,215,231,237,248,255,259,283,303,315,319],"idea":[76],"using":[78,127,138],"space":[81,89,157],"Newton":[82,100,125],"combined":[84],"with":[85,288],"an":[86,139,273],"approximate":[87,140],"reduced":[88,161,289],"SQP":[91,290],"preconditioner.":[92],"The":[93,241],"basic":[94],"components":[95],"solution":[101,115],"first-order":[104],"optimality":[105],"conditions":[106],"that":[107,144],"characterize":[108],"stationarity":[109],"Lagrangian":[112],"function;":[113],"Krylov":[114],"Karush--Kuhn--Tucker":[118],"(KKT)":[119],"linear":[120,217],"arising":[122],"at":[123],"each":[124],"iteration":[126],"symmetric":[129],"quasi-minimum":[130],"residual":[131],"method;":[132],"preconditioning":[133],"KKT":[136,216],"system":[137,218],"state/decision":[141],"variable":[142],"decomposition":[143],"replaces":[145],"forward":[147,196],"Jacobians":[149],"own":[152],"preconditioners,":[153],"decision":[156,297],"Schur":[158],"complement":[159],"(the":[160],"Hessian)":[162],"BFGS":[165],"approximation":[166],"initialized":[167],"two-step":[170],"stationary":[171],"method.":[172],"Accordingly,":[173],"term":[175],"{\\it":[179],"Lagrange--Newton--Krylov--Schur}":[180],"(LNKS).":[181],"It":[182],"fully":[184],"parallelizable,":[185],"exploits":[186],"structure":[188],"available":[190],"parallel":[191],"algorithms":[192],"problem,":[197],"locally":[200],"quadratically":[201],"convergent.":[202],"part":[204,300],"I":[205],"article,":[209,304],"investigate":[211],"effectiveness":[213],"solver.":[219],"We":[220],"test":[221],"our":[222],"two":[225],"optimal":[226,316],"control":[227,256,317],"problems":[228],"which":[230],"state":[232,295],"constraints":[233],"described":[235],"Stokes":[239],"equations.":[240,323],"objective":[242],"minimize":[245],"dissipation":[246],"or":[247],"deviation":[249],"from":[250],"given":[252],"velocity":[253],"field;":[254],"variables":[257],"boundary":[260],"velocities.":[261],"Numerical":[262],"experiments":[263],"up":[265,292],"256":[267],"Cray":[268],"T3E":[269],"processors":[270],"SGI":[274],"Origin":[275],"2000":[276],"include":[277],"performance":[280],"assessment":[281],"LNKS":[284,313],"algorithm":[285],"comparisons":[287],"$1,000,000$":[294],"and50,000":[296],"variables.":[298],"II":[301],"address":[306],"globalization":[307],"inexactness":[309],"issues,":[310],"apply":[312],"steady":[320],"incompressible":[321],"Navier--Stokes":[322]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":12},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":10},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":12},{"year":2018,"cited_by_count":16},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":13},{"year":2015,"cited_by_count":13},{"year":2014,"cited_by_count":17},{"year":2013,"cited_by_count":13},{"year":2012,"cited_by_count":14}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}