{"id":"https://openalex.org/W2508747201","doi":"https://doi.org/10.1137/16m1095688","title":"Gauss-Quadrature Method for One-Dimensional Mean-Field SDEs","display_name":"Gauss-Quadrature Method for One-Dimensional Mean-Field SDEs","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2508747201","doi":"https://doi.org/10.1137/16m1095688","mag":"2508747201"},"language":"en","primary_location":{"id":"doi:10.1137/16m1095688","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1095688","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1608.06741","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107842711","display_name":"Peter E. Kloeden","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Peter Kloeden","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5062734513","display_name":"Tony Shardlow","orcid":"https://orcid.org/0000-0003-1154-855X"},"institutions":[{"id":"https://openalex.org/I1299528221","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842","country_code":"GB","type":"government","lineage":["https://openalex.org/I1299528221","https://openalex.org/I4210087105"]},{"id":"https://openalex.org/I4210131439","display_name":"Applied Mathematics (United States)","ror":"https://ror.org/03seew607","country_code":"US","type":"company","lineage":["https://openalex.org/I4210131439"]}],"countries":["GB","US"],"is_corresponding":false,"raw_author_name":"Tony Shardlow","raw_affiliation_strings":["EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)","institution_ids":["https://openalex.org/I1299528221","https://openalex.org/I4210131439"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01097071,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"39","issue":"6","first_page":"A2784","last_page":"A2807"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9993000030517578,"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":0.9993000030517578,"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9962999820709229,"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/T11012","display_name":"Gas Dynamics and Kinetic Theory","score":0.9873999953269958,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.7161635756492615},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6331998705863953},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.6273421049118042},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5978071689605713},{"id":"https://openalex.org/keywords/euler-method","display_name":"Euler method","score":0.5811431407928467},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.4782108962535858},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.476631760597229},{"id":"https://openalex.org/keywords/gaussian-quadrature","display_name":"Gaussian quadrature","score":0.47203126549720764},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.4589169919490814},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.4506044387817383},{"id":"https://openalex.org/keywords/gauss","display_name":"Gauss","score":0.43734776973724365},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.43488195538520813},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.4272230863571167},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4259372651576996},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4257645010948181},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4252851605415344},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.4174695312976837},{"id":"https://openalex.org/keywords/backward-euler-method","display_name":"Backward Euler method","score":0.4143965542316437},{"id":"https://openalex.org/keywords/nystr\u00f6m-method","display_name":"Nystr\u00f6m method","score":0.39271843433380127},{"id":"https://openalex.org/keywords/euler-equations","display_name":"Euler equations","score":0.3506178855895996},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.290863037109375},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19730666279792786},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.18681707978248596},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1176435649394989}],"concepts":[{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.7161635756492615},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6331998705863953},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.6273421049118042},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5978071689605713},{"id":"https://openalex.org/C75380026","wikidata":"https://www.wikidata.org/wiki/Q868454","display_name":"Euler method","level":3,"score":0.5811431407928467},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.4782108962535858},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.476631760597229},{"id":"https://openalex.org/C167196314","wikidata":"https://www.wikidata.org/wiki/Q767680","display_name":"Gaussian quadrature","level":4,"score":0.47203126549720764},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.4589169919490814},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.4506044387817383},{"id":"https://openalex.org/C161794534","wikidata":"https://www.wikidata.org/wiki/Q177493","display_name":"Gauss","level":2,"score":0.43734776973724365},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.43488195538520813},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.4272230863571167},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4259372651576996},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4257645010948181},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4252851605415344},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.4174695312976837},{"id":"https://openalex.org/C768646","wikidata":"https://www.wikidata.org/wiki/Q2736820","display_name":"Backward Euler method","level":3,"score":0.4143965542316437},{"id":"https://openalex.org/C48265008","wikidata":"https://www.wikidata.org/wiki/Q7071295","display_name":"Nystr\u00f6m method","level":3,"score":0.39271843433380127},{"id":"https://openalex.org/C38409319","wikidata":"https://www.wikidata.org/wiki/Q375175","display_name":"Euler equations","level":2,"score":0.3506178855895996},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.290863037109375},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19730666279792786},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.18681707978248596},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1176435649394989},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/16m1095688","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1095688","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:arXiv.org:1608.06741","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1608.06741","pdf_url":"https://arxiv.org/pdf/1608.06741","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":"mag:2508747201","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1608.06741.pdf","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.1608.06741","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1608.06741","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:1608.06741","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1608.06741","pdf_url":"https://arxiv.org/pdf/1608.06741","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":[],"funders":[{"id":"https://openalex.org/F4320321883","display_name":"Huazhong University of Science and Technology","ror":"https://ror.org/00p991c53"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2508747201.pdf"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W1190414567","https://openalex.org/W1593505867","https://openalex.org/W1967396173","https://openalex.org/W1970376871","https://openalex.org/W1971947347","https://openalex.org/W1990838746","https://openalex.org/W2003530131","https://openalex.org/W2020328455","https://openalex.org/W2028859398","https://openalex.org/W2048743591","https://openalex.org/W2065172718","https://openalex.org/W2086872802","https://openalex.org/W2104842206","https://openalex.org/W2119062437","https://openalex.org/W2129132283","https://openalex.org/W2219497451","https://openalex.org/W2964104047","https://openalex.org/W3013068944"],"related_works":["https://openalex.org/W2963148396","https://openalex.org/W2767225077","https://openalex.org/W2998835977","https://openalex.org/W3162348888","https://openalex.org/W3019868291","https://openalex.org/W2787713479","https://openalex.org/W3109274077","https://openalex.org/W1989197292","https://openalex.org/W2262160876","https://openalex.org/W3025554787","https://openalex.org/W3098653717","https://openalex.org/W2270941298","https://openalex.org/W1978255949","https://openalex.org/W2977633471","https://openalex.org/W2078735087","https://openalex.org/W2991285603","https://openalex.org/W1144018","https://openalex.org/W3193617592","https://openalex.org/W3122911651","https://openalex.org/W2788679035"],"abstract_inverted_index":{"Mean-field":[0],"SDEs,":[1],"also":[2,82,122],"known":[3],"as":[4,112,114,130],"McKean--Vlasov":[5],"equations,":[6],"are":[7,103,121],"stochastic":[8],"differential":[9],"equations":[10],"where":[11],"the":[12,18,24,35,41,44,59,84,95,98,107,136],"drift":[13],"and":[14],"diffusion":[15],"depend":[16],"on":[17],"current":[19,25],"distribution":[20,36],"in":[21,92,127],"addition":[22],"to":[23,43,54,87],"position.":[26],"We":[27,81],"describe":[28],"an":[29,63],"efficient":[30],"numerical":[31],"method":[32,111],"for":[33,47,77,124],"approximating":[34],"at":[37],"time":[38,55,79],"$t$":[39],"of":[40,94,97,109],"solution":[42],"initial-value":[45],"problem":[46],"one-dimensional":[48],"mean-field":[49],"SDEs.":[50],"The":[51,119],"idea":[52],"is":[53,75],"march":[56],"(e.g.,":[57],"using":[58],"Euler--Maruyama":[60,78],"time-stepping":[61,117],"method)":[62],"$m$-point":[64],"Gauss-quadrature":[65],"rule.":[66],"With":[67],"suitable":[68],"regularity":[69],"conditions,":[70],"convergence":[71],"with":[72,135],"first":[73],"order":[74],"proved":[76],"stepping.":[80],"estimate":[83],"work":[85],"needed":[86],"achieve":[88],"a":[89],"given":[90],"accuracy":[91],"terms":[93],"smoothness":[96],"underlying":[99],"problem.":[100],"Numerical":[101],"experiments":[102],"given,":[104],"which":[105],"show":[106],"effectiveness":[108],"this":[110],"well":[113],"two":[115],"second-order":[116],"methods.":[118],"methods":[120],"effective":[123],"ordinary":[125],"SDEs":[126],"one":[128],"dimension,":[129],"we":[131],"demonstrate":[132],"by":[133],"comparison":[134],"multilevel":[137],"Monte":[138],"Carlo":[139],"method.":[140]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}