{"id":"https://openalex.org/W1991061880","doi":"https://doi.org/10.1137/13092842x","title":"A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting","display_name":"A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W1991061880","doi":"https://doi.org/10.1137/13092842x","mag":"1991061880"},"language":"en","primary_location":{"id":"doi:10.1137/13092842x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/13092842x","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"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 Mathematical Analysis","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/A5066156899","display_name":"Jeff Calder","orcid":"https://orcid.org/0000-0002-9829-4128"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jeff Calder","raw_affiliation_strings":["(University of Michigan)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(University of Michigan)","institution_ids":["https://openalex.org/I27837315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070153856","display_name":"Selim Esedo\u1e21lu","orcid":null},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Selim Esedo\u1e21lu","raw_affiliation_strings":["(University of Michigan)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(University of Michigan)","institution_ids":["https://openalex.org/I27837315"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077692655","display_name":"Alfred O. Hero","orcid":"https://orcid.org/0000-0002-2531-9670"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alfred O. Hero","raw_affiliation_strings":["(University of Michigan)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(University of Michigan)","institution_ids":["https://openalex.org/I27837315"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I27837315"],"apc_list":null,"apc_paid":null,"fwci":4.766,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.95003068,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"46","issue":"1","first_page":"603","last_page":"638"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.996999979019165,"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"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.996999979019165,"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/T11106","display_name":"Data Management and Algorithms","score":0.9937999844551086,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T11830","display_name":"Point processes and geometric inequalities","score":0.9912999868392944,"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/mathematics","display_name":"Mathematics","score":0.8460192680358887},{"id":"https://openalex.org/keywords/antichain","display_name":"Antichain","score":0.8270485401153564},{"id":"https://openalex.org/keywords/independent-and-identically-distributed-random-variables","display_name":"Independent and identically distributed random variables","score":0.6322251558303833},{"id":"https://openalex.org/keywords/hamilton\u2013jacobi-equation","display_name":"Hamilton\u2013Jacobi equation","score":0.5745618939399719},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5732153654098511},{"id":"https://openalex.org/keywords/viscosity-solution","display_name":"Viscosity solution","score":0.5627484321594238},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.500049352645874},{"id":"https://openalex.org/keywords/sorting","display_name":"Sorting","score":0.48831436038017273},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.477954626083374},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.47461405396461487},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.43471038341522217},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4095173478126526},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.35342735052108765},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.08230471611022949},{"id":"https://openalex.org/keywords/partially-ordered-set","display_name":"Partially ordered set","score":0.07586997747421265}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8460192680358887},{"id":"https://openalex.org/C136134351","wikidata":"https://www.wikidata.org/wiki/Q431937","display_name":"Antichain","level":3,"score":0.8270485401153564},{"id":"https://openalex.org/C141513077","wikidata":"https://www.wikidata.org/wiki/Q378542","display_name":"Independent and identically distributed random variables","level":3,"score":0.6322251558303833},{"id":"https://openalex.org/C2778860007","wikidata":"https://www.wikidata.org/wiki/Q1060137","display_name":"Hamilton\u2013Jacobi equation","level":2,"score":0.5745618939399719},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5732153654098511},{"id":"https://openalex.org/C88731125","wikidata":"https://www.wikidata.org/wiki/Q7935676","display_name":"Viscosity solution","level":2,"score":0.5627484321594238},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.500049352645874},{"id":"https://openalex.org/C111696304","wikidata":"https://www.wikidata.org/wiki/Q2303697","display_name":"Sorting","level":2,"score":0.48831436038017273},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.477954626083374},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.47461405396461487},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.43471038341522217},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4095173478126526},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35342735052108765},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.08230471611022949},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.07586997747421265},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/13092842x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/13092842x","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"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 Mathematical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1855331886","display_name":null,"funder_award_id":"W911NF-09-1-0310","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G4286413650","display_name":"Collaborative Research: ATD (Algorithms for Threat Detection): Inverse Problems Methods in Chemical Threat Detection","funder_award_id":"0914567","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7452299184","display_name":null,"funder_award_id":"W911NF","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W583936781","https://openalex.org/W1487586009","https://openalex.org/W1549299680","https://openalex.org/W1554486237","https://openalex.org/W1559701175","https://openalex.org/W1575331340","https://openalex.org/W1595498733","https://openalex.org/W1669532169","https://openalex.org/W1965482388","https://openalex.org/W1969051672","https://openalex.org/W1970795056","https://openalex.org/W1977559136","https://openalex.org/W2002114058","https://openalex.org/W2032316144","https://openalex.org/W2034648643","https://openalex.org/W2035101851","https://openalex.org/W2035643168","https://openalex.org/W2037148846","https://openalex.org/W2044917762","https://openalex.org/W2051867489","https://openalex.org/W2059565438","https://openalex.org/W2075581822","https://openalex.org/W2097431626","https://openalex.org/W2106780336","https://openalex.org/W2113222527","https://openalex.org/W2114487471","https://openalex.org/W2116040950","https://openalex.org/W2116396741","https://openalex.org/W2116661285","https://openalex.org/W2121365620","https://openalex.org/W2121612399","https://openalex.org/W2126105956","https://openalex.org/W2143760297","https://openalex.org/W2147573606","https://openalex.org/W2152192852","https://openalex.org/W2158525989","https://openalex.org/W2166738685","https://openalex.org/W2170429144","https://openalex.org/W2259507532","https://openalex.org/W2261054240","https://openalex.org/W2332092548","https://openalex.org/W2536044253","https://openalex.org/W2952831829","https://openalex.org/W2993433457","https://openalex.org/W3021709410","https://openalex.org/W3103506376","https://openalex.org/W3125904670","https://openalex.org/W3140175389","https://openalex.org/W4292157029"],"related_works":["https://openalex.org/W1983345526","https://openalex.org/W2076944689","https://openalex.org/W2061585064","https://openalex.org/W1594944678","https://openalex.org/W1977330547","https://openalex.org/W1487586009","https://openalex.org/W3102569876","https://openalex.org/W2347733609","https://openalex.org/W2971079383","https://openalex.org/W1997500459"],"abstract_inverted_index":{"We":[0,85],"show":[1,72],"that":[2,22,73],"nondominated":[3,74],"sorting":[4,38,75],"of":[5,9,35,68,95],"a":[6,19,26,40,87],"sequence":[7],"$X_1,\\dots,X_n$":[8],"independent":[10],"and":[11,46,55,99],"identically":[12],"distributed":[13],"random":[14,81],"variables":[15],"in":[16,43,83],"$\\mathbb{R}^d$":[17],"has":[18],"continuum":[20],"limit":[21],"corresponds":[23],"to":[24,49,56],"solving":[25],"Hamilton--Jacobi":[27,97],"equation":[28,98],"involving":[29,58],"the":[30,51,59,92],"probability":[31],"density":[32,106],"function":[33],"$f$":[34],"$X_i$.":[36],"Nondominated":[37],"is":[39,47,76],"fundamental":[41],"problem":[42],"multiobjective":[44],"optimization":[45],"equivalent":[48],"finding":[50],"canonical":[52],"antichain":[53],"partition":[54],"problems":[57],"longest":[60],"chain":[61],"among":[62],"Euclidean":[63],"points.":[64],"As":[65],"an":[66],"application":[67],"this":[69,96],"result,":[70],"we":[71],"asymptotically":[77],"stable":[78],"under":[79],"bounded":[80],"perturbations":[82],"$X_1,\\dots,X_n$.":[84],"give":[86],"numerical":[88,102],"scheme":[89],"for":[90,104],"computing":[91],"viscosity":[93],"solution":[94],"present":[100],"some":[101],"simulations":[103],"various":[105],"functions.":[107]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":6}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}