{"id":"https://openalex.org/W2000700114","doi":"https://doi.org/10.1137/s0363012900369599","title":"Consistent Approximations and Approximate Functions and Gradients in Optimal Control","display_name":"Consistent Approximations and Approximate Functions and Gradients in Optimal Control","publication_year":2002,"publication_date":"2002-01-01","ids":{"openalex":"https://openalex.org/W2000700114","doi":"https://doi.org/10.1137/s0363012900369599","mag":"2000700114"},"language":"en","primary_location":{"id":"doi:10.1137/s0363012900369599","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0363012900369599","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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 Control and Optimization","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/A5060781558","display_name":"Olivier Pironneau","orcid":"https://orcid.org/0000-0003-2185-9909"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Olivier Pironneau","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5111492194","display_name":"E. Polak","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Elijah Polak","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.6551,"has_fulltext":false,"cited_by_count":30,"citation_normalized_percentile":{"value":0.92274613,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"41","issue":"2","first_page":"487","last_page":"510"},"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":0.9995999932289124,"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":0.9995999932289124,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9968000054359436,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9955999851226807,"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/discretization","display_name":"Discretization","score":0.8855651617050171},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7374723553657532},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.606513261795044},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.5774490237236023},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5588995814323425},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5276393294334412},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.5245385766029358},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.44177156686782837},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.4306217432022095},{"id":"https://openalex.org/keywords/discretization-of-continuous-features","display_name":"Discretization of continuous features","score":0.41582173109054565},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41147711873054504},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.21037349104881287},{"id":"https://openalex.org/keywords/discretization-error","display_name":"Discretization error","score":0.10594785213470459}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.8855651617050171},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7374723553657532},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.606513261795044},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.5774490237236023},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5588995814323425},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5276393294334412},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.5245385766029358},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.44177156686782837},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.4306217432022095},{"id":"https://openalex.org/C105427703","wikidata":"https://www.wikidata.org/wiki/Q665193","display_name":"Discretization of continuous features","level":4,"score":0.41582173109054565},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41147711873054504},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.21037349104881287},{"id":"https://openalex.org/C126148662","wikidata":"https://www.wikidata.org/wiki/Q3890355","display_name":"Discretization error","level":3,"score":0.10594785213470459},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s0363012900369599","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0363012900369599","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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 Control and Optimization","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":20,"referenced_works":["https://openalex.org/W146697093","https://openalex.org/W639940401","https://openalex.org/W1500747596","https://openalex.org/W1537335979","https://openalex.org/W1976008995","https://openalex.org/W1984605095","https://openalex.org/W2001611717","https://openalex.org/W2018333764","https://openalex.org/W2024702694","https://openalex.org/W2037128090","https://openalex.org/W2043728417","https://openalex.org/W2051373638","https://openalex.org/W2060279602","https://openalex.org/W2070630896","https://openalex.org/W2083365247","https://openalex.org/W2093581986","https://openalex.org/W2109911863","https://openalex.org/W2159336103","https://openalex.org/W2165002617","https://openalex.org/W4249513058"],"related_works":["https://openalex.org/W2100670034","https://openalex.org/W2046637502","https://openalex.org/W3005204046","https://openalex.org/W2065595881","https://openalex.org/W2974753373","https://openalex.org/W2787514335","https://openalex.org/W2008621392","https://openalex.org/W2952649113","https://openalex.org/W4289097782","https://openalex.org/W2012186120"],"abstract_inverted_index":{"Because":[0],"of":[1,5,18,70,82,85,91,104,106,145,164,177,197,207,229,234,240,266,298,316,380],"the":[2,15,36,55,71,80,89,99,102,107,137,175,184,188,198,222,227,238,264,272,276,314,326,335,345,378],"unavoidable":[3],"use":[4],"numerical":[6,16,195,363],"integration":[7,37,42,215,269],"methods,":[8],"such":[9,34],"as":[10,35,143,295],"Runge--Kutta":[11],"or":[12,25],"finite":[13,208],"elements,":[14],"solution":[17,185,196,315],"optimal":[19],"control":[20,341],"problems,":[21],"with":[22,133,256,278,284],"either":[23],"ODE":[24,329],"PDE":[26],"dynamics,":[27],"is":[28,94,140,149,192],"governed":[29],"by":[30,79,96,161],"a":[31,86,122,162,205,285,296,319,339,351],"discretization":[32,90,242,258],"parameter":[33],"mesh-size.":[38],"Usually,":[39],"when":[40,213],"explicit":[41,268],"techniques":[43],"are":[44,180,217,288,331,348,374],"used,":[45],"function":[46,66],"and":[47,67,101,115,271,281,338,373],"derivative":[48,68],"values":[49,69],"can":[50,157,253,292,356],"be":[51,77,158,254,293,357,387],"computed":[52],"exactly":[53],"for":[54,124,182,226,313],"discretized":[56,73,327,346],"problems.":[57,169],"Recently,":[58],"we":[59,120,247],"have":[60],"come":[61],"across":[62],"some":[63,390],"examples":[64],"where":[65],"explicitly":[72],"problems":[74,93,127],"had":[75],"to":[76,110,142,377,386,389],"approximated":[78,160],"outcome":[81],"N":[83],"iterations":[84,105],"solver.":[87,108],"Consequently,":[88],"these":[92,235,303,367],"controlled":[95],"two":[97,134,241,257,317],"parameters:":[98],"mesh-size":[100,265,287],"number":[103],"Referring":[109],"[E.":[111],"Polak,":[112],"Optimization:":[113],"Algorithms":[114],"Consistent":[116],"Approximations,":[117],"Springer-Verlag,":[118],"1997],":[119],"find":[121],"theory":[123,223],"solving":[125],"optimization":[126,155,168],"that":[128,144,151,194,252,366],"require":[129],"discretization.":[130],"It":[131],"deals":[132],"situations.":[135],"In":[136,170,187,244],"first,":[138],"which":[139,279,325,344,355],"referred":[141],"consistent":[146,210],"approximations,":[147],"it":[148,191],"assumed":[150,193],"an":[152,267,308],"infinite":[153],"dimensional":[154,167,209],"problem":[156,323,342],"suitably":[159],"family":[163,206],"progressively":[165],"higher":[166],"this":[171,220,245],"case,":[172,221],"strategies,":[173],"in":[174,204,324,343],"form":[176],"algorithm":[178,250,305,354],"models,":[179],"presented":[181],"\"diagonalizing\"":[183],"process.":[186],"second":[189,273],"situation,":[190],"dynamic":[199],"equations":[200],"does":[201],"not":[202],"result":[203,291],"approximations":[211],"(e.g.,":[212],"implicit":[214],"methods":[216],"used).":[218],"For":[219],"provides":[224],"models":[225,251,306],"implementation":[228],"conceptual":[230],"algorithms.":[231],"Unfortunately,":[232],"neither":[233],"situations":[236],"envisions":[237],"possibility":[239],"parameters.":[243,259,382],"paper,":[246],"present":[248],"new":[249,304,368],"used":[255],"The":[260,290],"first":[261],"one":[262,274],"controls":[263,275],"scheme,":[270],"precision":[277],"functions":[280],"gradients":[282],"associated":[283],"fixed":[286],"computed.":[289],"seen":[294],"framework":[297],"quasi-consistent":[299],"approximations.":[300],"We":[301],"implemented":[302,358],"using":[307,334,350],"approximate":[309],"steepest":[310],"descent":[311],"method":[312,337],"problems:":[318],"two-point":[320],"boundary":[321],"value":[322],"linear":[328],"dynamics":[330,347],"solved":[332,349],"approximately":[333],"Gauss--Seidel":[336],"distributed":[340],"domain":[352],"decomposition":[353],"on":[359],"parallelized":[360],"computers.":[361],"Our":[362],"results":[364],"show":[365],"algorithms":[369],"perform":[370],"quite":[371],"well":[372],"fairly":[375],"insensitive":[376],"selection":[379],"user-set":[381],"Also,":[383],"they":[384],"appear":[385],"superior":[388],"alternative,":[391],"ad":[392],"hoc":[393],"schemes.":[394]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
