{"id":"https://openalex.org/W1630018803","doi":"https://doi.org/10.1137/130943984","title":"Higher Order QMC Petrov--Galerkin Discretization for Affine Parametric Operator Equations with Random Field Inputs","display_name":"Higher Order QMC Petrov--Galerkin Discretization for Affine Parametric Operator Equations with Random Field Inputs","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W1630018803","doi":"https://doi.org/10.1137/130943984","mag":"1630018803"},"language":"en","primary_location":{"id":"doi:10.1137/130943984","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130943984","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1309.4624","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Josef Dick","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Josef Dick","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Frances Y. Kuo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Frances Y. Kuo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Quoc T. Le Gia","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Quoc T. Le Gia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Dirk Nuyens","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dirk Nuyens","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Christoph Schwab","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Christoph Schwab","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":11.2436,"has_fulltext":true,"cited_by_count":87,"citation_normalized_percentile":{"value":0.9914208,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"52","issue":"6","first_page":"2676","last_page":"2702"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.7423999905586243,"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"}},"topics":[{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.7423999905586243,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.15080000460147858,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.025200000032782555,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/rate-of-convergence","display_name":"Rate of convergence","score":0.5964999794960022},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5564000010490417},{"id":"https://openalex.org/keywords/parametric-statistics","display_name":"Parametric statistics","score":0.5210000276565552},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4537999927997589},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.45260000228881836},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.43700000643730164},{"id":"https://openalex.org/keywords/random-field","display_name":"Random field","score":0.41190001368522644},{"id":"https://openalex.org/keywords/product-topology","display_name":"Product topology","score":0.4016000032424927},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.398499995470047}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8848999738693237},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5964999794960022},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5564000010490417},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5257999897003174},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.5210000276565552},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4537999927997589},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.45260000228881836},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.43700000643730164},{"id":"https://openalex.org/C130402806","wikidata":"https://www.wikidata.org/wiki/Q5361768","display_name":"Random field","level":2,"score":0.41190001368522644},{"id":"https://openalex.org/C123222754","wikidata":"https://www.wikidata.org/wiki/Q1070472","display_name":"Product topology","level":2,"score":0.4016000032424927},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.398499995470047},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38850000500679016},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3869999945163727},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.3653999865055084},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3483999967575073},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3370000123977661},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.33649998903274536},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.32919999957084656},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.3091000020503998},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.29739999771118164},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.2897999882698059},{"id":"https://openalex.org/C197656079","wikidata":"https://www.wikidata.org/wiki/Q17147719","display_name":"Polynomial chaos","level":3,"score":0.28450000286102295},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.28360000252723694},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.2648000121116638},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.251800000667572},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.25099998712539673},{"id":"https://openalex.org/C13982400","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Linear differential equation","level":3,"score":0.250900000333786}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/130943984","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130943984","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1309.4624","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1309.4624","pdf_url":"https://arxiv.org/pdf/1309.4624","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"text"},{"id":"pmh:oai:unsworks.unsw.edu.au:1959.4/unsworks_48521","is_oa":true,"landing_page_url":"http://handle.unsw.edu.au/1959.4/unsworks_48521","pdf_url":null,"source":{"id":"https://openalex.org/S4377196481","display_name":"UNSWorks (UNSW Sydney)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I31746571","host_organization_name":"UNSW Sydney","host_organization_lineage":["https://openalex.org/I31746571"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/523282","is_oa":true,"landing_page_url":"http://epubs.siam.org/doi/abs/10.1137/130943984","pdf_url":null,"source":{"id":"https://openalex.org/S4306401954","display_name":"Lirias (KU Leuven)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I99464096","host_organization_name":"KU Leuven","host_organization_lineage":["https://openalex.org/I99464096"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Numerical Analysis, vol. 52 (6), Art.No. 6, (2676-2702)","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:unsworks.library.unsw.edu.au:1959.4/unsworks_48521","is_oa":true,"landing_page_url":"http://hdl.handle.net/1959.4/unsworks_48521","pdf_url":null,"source":{"id":"https://openalex.org/S4306401737","display_name":"UNSWorks (University of New South Wales, Sydney, Australia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40053085","host_organization_name":"Australian Defence Force Academy","host_organization_lineage":["https://openalex.org/I40053085"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Numerical Analysis, 52, 6, 2676-2702","raw_type":"http://purl.org/coar/resource_type/c_6501"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1309.4624","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1309.4624","pdf_url":"https://arxiv.org/pdf/1309.4624","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321730","display_name":"Fonds Wetenschappelijk Onderzoek","ror":"https://ror.org/03qtxy027"},{"id":"https://openalex.org/F4320327336","display_name":"Vlaamse regering","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1980498334","https://openalex.org/W1981743146","https://openalex.org/W2000408025","https://openalex.org/W2001012793","https://openalex.org/W2010177608","https://openalex.org/W2014551121","https://openalex.org/W2014789512","https://openalex.org/W2030404880","https://openalex.org/W2030470940","https://openalex.org/W2042083287","https://openalex.org/W2049222546","https://openalex.org/W2070980742","https://openalex.org/W2083754610","https://openalex.org/W2110065034","https://openalex.org/W2111894258","https://openalex.org/W2145169372","https://openalex.org/W2146736312","https://openalex.org/W2152162686","https://openalex.org/W2685500084"],"related_works":[],"abstract_inverted_index":{"We":[0,45],"construct":[1],"quasi--Monte":[2],"Carlo":[3],"methods":[4],"to":[5,54,79,125,145],"approximate":[6],"the":[7,33,47,50,55,60,65,68,75,80,133,146],"expected":[8],"values":[9],"of":[10,13,16,27,36,49,59,62,64,67,85,96,130,137],"linear":[11],"functionals":[12],"Petrov--Galerkin":[14],"discretizations":[15],"parametric":[17],"operator":[18],"equations":[19,40],"which":[20,153,184],"depend":[21],"on":[22],"a":[23,113,127,170,187],"possibly":[24],"infinite":[25],"sequence":[26],"parameters.":[28],"Such":[29],"problems":[30],"arise":[31],"in":[32,57,82,103,117],"numerical":[34],"solution":[35],"differential":[37],"and":[38,175,179],"integral":[39],"with":[41,52,106,132],"random":[42],"field":[43],"inputs.":[44],"analyze":[46],"regularity":[48],"solutions":[51],"respect":[53],"parameters":[56],"terms":[58],"rate":[61,129,142,147],"decay":[63],"fluctuations":[66,81],"input":[69],"field.":[70],"If":[71],"$p\\in":[72],"(0,1]$":[73],"denotes":[74],"\u201csummability":[76],"exponent\u201d":[77],"corresponding":[78],"affine-parametric":[83],"families":[84],"operators,":[86],"then":[87],"we":[88,168,185],"prove":[89],"that":[90],"deterministic":[91],"\u201cinterlaced":[92],"polynomial":[93],"lattice":[94,160],"rules\u201d":[95],"order":[97,180],"$\\alpha":[98],"=":[99],"\\lfloor":[100],"1/p":[101],"\\rfloor+1$":[102],"$s$":[104],"dimensions":[105],"$N$":[107],"points":[108],"can":[109],"be":[110],"constructed":[111],"using":[112],"fast":[114,189],"component-by-component":[115,190],"algorithm,":[116],"$\\mathcal{O}(\\alpha\\,s\\,":[118],"N\\log":[119],"N":[120],"+":[121],"\\alpha^2\\,s^2":[122],"N)$":[123],"operations,":[124],"achieve":[126],"convergence":[128,141],"$\\mathcal{O}(N^{-1/p})$,":[131],"implied":[134],"constant":[135],"independent":[136],"$s$.":[138],"This":[139],"dimension-independent":[140],"is":[143],"superior":[144],"$\\mathcal{O}(N^{-1/p+1/2})$":[148],"for":[149,157,183],"$2/3\\leq":[150],"p\\leq":[151],"1$,":[152],"was":[154],"recently":[155],"established":[156],"randomly":[158],"shifted":[159],"rules":[161],"under":[162],"comparable":[163],"assumptions.":[164],"In":[165],"our":[166],"analysis":[167],"use":[169],"nonstandard":[171],"Banach":[172],"space":[173],"setting":[174],"introduce":[176],"\u201csmoothness-driven":[177],"product":[178],"dependent\u201d":[181],"weights":[182],"develop":[186],"new":[188],"construction.":[191]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":10},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2016-06-24T00:00:00"}
