{"id":"https://openalex.org/W2048308882","doi":"https://doi.org/10.1137/100790069","title":"Mean Field Games: Numerical Methods for the Planning Problem","display_name":"Mean Field Games: Numerical Methods for the Planning Problem","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2048308882","doi":"https://doi.org/10.1137/100790069","mag":"2048308882"},"language":"en","primary_location":{"id":"doi:10.1137/100790069","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100790069","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/A5010896193","display_name":"Yves Achdou","orcid":"https://orcid.org/0000-0002-0514-3727"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yves Achdou","raw_affiliation_strings":["achdou@math.jussieu.fr#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"achdou@math.jussieu.fr#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018541911","display_name":"Fabio Camilli","orcid":"https://orcid.org/0000-0003-2976-8112"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fabio Camilli","raw_affiliation_strings":["camilli@dmmm.uniroma1.it#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"camilli@dmmm.uniroma1.it#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091067958","display_name":"Italo Capuzzo-Dolcetta","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Italo Capuzzo-Dolcetta","raw_affiliation_strings":["capuzzo@mat.uniroma1.it#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"capuzzo@mat.uniroma1.it#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":15.7842,"has_fulltext":false,"cited_by_count":205,"citation_normalized_percentile":{"value":0.99025671,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"50","issue":"1","first_page":"77","last_page":"109"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9406999945640564,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9114999771118164,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8119585514068604},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.6890888810157776},{"id":"https://openalex.org/keywords/bellman-equation","display_name":"Bellman equation","score":0.5588682889938354},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.548740804195404},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5272281169891357},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.4945964515209198},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.455385684967041},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.44146138429641724},{"id":"https://openalex.org/keywords/time-horizon","display_name":"Time horizon","score":0.4347497820854187},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.42788293957710266},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.4242624342441559},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41203680634498596},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23252913355827332},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.10101443529129028}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8119585514068604},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6890888810157776},{"id":"https://openalex.org/C14646407","wikidata":"https://www.wikidata.org/wiki/Q1430750","display_name":"Bellman equation","level":2,"score":0.5588682889938354},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.548740804195404},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5272281169891357},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.4945964515209198},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.455385684967041},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.44146138429641724},{"id":"https://openalex.org/C28761237","wikidata":"https://www.wikidata.org/wiki/Q7805321","display_name":"Time horizon","level":2,"score":0.4347497820854187},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.42788293957710266},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.4242624342441559},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41203680634498596},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23252913355827332},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.10101443529129028},{"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},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/100790069","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100790069","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"},{"id":"pmh:oai:iris.uniroma1.it:11573/454331","is_oa":false,"landing_page_url":"http://hdl.handle.net/11573/454331","pdf_url":null,"source":{"id":"https://openalex.org/S4377196107","display_name":"IRIS Research product catalog (Sapienza University of Rome)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6000000238418579,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W158030646","https://openalex.org/W1487586009","https://openalex.org/W1558902221","https://openalex.org/W1565627164","https://openalex.org/W1585160083","https://openalex.org/W1808956822","https://openalex.org/W1985506636","https://openalex.org/W2023042366","https://openalex.org/W2028316825","https://openalex.org/W2035299975","https://openalex.org/W2038686546","https://openalex.org/W2057470999","https://openalex.org/W2098770975","https://openalex.org/W4210364010"],"related_works":["https://openalex.org/W2050111419","https://openalex.org/W2898988421","https://openalex.org/W1859185863","https://openalex.org/W2964126589","https://openalex.org/W4285537323","https://openalex.org/W1932159282","https://openalex.org/W4307691181","https://openalex.org/W3016225735","https://openalex.org/W2017412075","https://openalex.org/W1518875602"],"abstract_inverted_index":{"Mean":[0],"field":[1],"games":[2,9],"describe":[3,121],"the":[4,12,23,28,32,61,73,91,102,107],"asymptotic":[5],"behavior":[6],"of":[7,14,34,39,90,101,110],"differential":[8],"in":[10,30],"which":[11,31,77],"number":[13,38],"players":[15],"tends":[16],"to":[17,55,86],"+infinity.":[18],"Here":[19],"we":[20,112],"focus":[21],"on":[22,117],"optimal":[24,74,80],"planning":[25,75],"problem,":[26,76],"i.e.,":[27],"problem":[29],"positions":[33],"a":[35,44,50,56,67,98,114],"very":[36],"large":[37],"identical":[40],"rational":[41],"agents,":[42],"with":[43],"common":[45],"value":[46],"function,":[47],"evolve":[48],"from":[49],"given":[51],"initial":[52],"spatial":[53],"density":[54,59],"desired":[57],"target":[58],"at":[60],"final":[62],"horizon":[63],"time.":[64],"We":[65,95,120],"propose":[66,113],"finite":[68],"difference":[69],"semi-implicit":[70,103],"scheme":[71],"for":[72],"has":[78],"an":[79],"control":[81,93],"formulation.":[82],"The":[83],"latter":[84],"leads":[85],"existence":[87],"and":[88],"uniqueness":[89],"discrete":[92],"problem.":[94],"also":[96],"study":[97],"penalized":[99],"version":[100],"scheme.":[104],"For":[105],"solving":[106],"resulting":[108],"system":[109],"equations,":[111],"strategy":[115],"based":[116],"Newton":[118],"iterations.":[119],"some":[122],"numerical":[123],"experiments.":[124]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":12},{"year":2024,"cited_by_count":14},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":14},{"year":2021,"cited_by_count":22},{"year":2020,"cited_by_count":20},{"year":2019,"cited_by_count":12},{"year":2018,"cited_by_count":23},{"year":2017,"cited_by_count":13},{"year":2016,"cited_by_count":15},{"year":2015,"cited_by_count":11},{"year":2014,"cited_by_count":8},{"year":2013,"cited_by_count":12},{"year":2012,"cited_by_count":10}],"updated_date":"2026-06-18T10:00:31.954636","created_date":"2025-10-10T00:00:00"}