{"id":"https://openalex.org/W2101552627","doi":"https://doi.org/10.1137/130916849","title":"Energy Norm A Posteriori Error Estimation for Parametric Operator Equations","display_name":"Energy Norm A Posteriori Error Estimation for Parametric Operator Equations","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2101552627","doi":"https://doi.org/10.1137/130916849","mag":"2101552627"},"language":"en","primary_location":{"id":"doi:10.1137/130916849","is_oa":true,"landing_page_url":"https://doi.org/10.1137/130916849","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1137/130916849","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081277992","display_name":"Alex Bespalov","orcid":"https://orcid.org/0000-0001-6181-6765"},"institutions":[{"id":"https://openalex.org/I79619799","display_name":"University of Birmingham","ror":"https://ror.org/03angcq70","country_code":"GB","type":"education","lineage":["https://openalex.org/I79619799"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Alex Bespalov","raw_affiliation_strings":["School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK","institution_ids":["https://openalex.org/I79619799"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073050362","display_name":"Catherine E. Powell","orcid":"https://orcid.org/0000-0003-3605-6797"},"institutions":[{"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":["US"],"is_corresponding":false,"raw_author_name":"Catherine E. Powell","raw_affiliation_strings":["Applied Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007116796","display_name":"David J. Silvester","orcid":"https://orcid.org/0000-0002-0652-5444"},"institutions":[{"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":["US"],"is_corresponding":false,"raw_author_name":"David Silvester","raw_affiliation_strings":["Applied Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["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":2.1537,"has_fulltext":true,"cited_by_count":46,"citation_normalized_percentile":{"value":0.89832023,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"36","issue":"2","first_page":"A339","last_page":"A363"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":1.0,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9980999827384949,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9958999752998352,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7576318979263306},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.6863523721694946},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5832751989364624},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5643081068992615},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.5474087595939636},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.5381184220314026},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.5232552289962769},{"id":"https://openalex.org/keywords/tensor-product","display_name":"Tensor product","score":0.5045512914657593},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5003108978271484},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4973757565021515},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.42017364501953125},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.41183003783226013},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3014927804470062}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7576318979263306},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.6863523721694946},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5832751989364624},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5643081068992615},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.5474087595939636},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.5381184220314026},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5232552289962769},{"id":"https://openalex.org/C51255310","wikidata":"https://www.wikidata.org/wiki/Q1163016","display_name":"Tensor product","level":2,"score":0.5045512914657593},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5003108978271484},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4973757565021515},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.42017364501953125},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.41183003783226013},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3014927804470062},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1137/130916849","is_oa":true,"landing_page_url":"https://doi.org/10.1137/130916849","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/9449bd78-313a-4b62-ae74-2eceeac36cae","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/9449bd78-313a-4b62-ae74-2eceeac36cae","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bespalov, A, Powell, C E & Silvester, D 2014, 'Energy norm a posteriori error estimation for parametric operator equations', SIAM Journal on Scientific Computing, vol. 36, no. 2, pp. A339-A363. https://doi.org/10.1137/130916849","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1967","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.658.8443","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.658.8443","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/1967/01/91684.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/a780d0a9-b012-4056-be69-1f39fc647500","is_oa":true,"landing_page_url":"https://research.birmingham.ac.uk/en/publications/a780d0a9-b012-4056-be69-1f39fc647500","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Bespalov, A, Powell, C E & Silvester, DJ 2014, 'Energy norm a posteriori error estimation for parametric operator equations', SIAM Journal on Scientific Computing, vol. 36, no. 2, pp. A339-A363. https://doi.org/10.1137/130916849","raw_type":"article"},{"id":"pmh:oai:pure.atira.dk:publications/9449bd78-313a-4b62-ae74-2eceeac36cae","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/energy-norm-a-posteriori-error-estimation-for-parametric-operator-equations(9449bd78-313a-4b62-ae74-2eceeac36cae).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:pure.atira.dk:publications/a780d0a9-b012-4056-be69-1f39fc647500","is_oa":true,"landing_page_url":"https://birmingham.elsevierpure.com/en/publications/a780d0a9-b012-4056-be69-1f39fc647500","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""}],"best_oa_location":{"id":"doi:10.1137/130916849","is_oa":true,"landing_page_url":"https://doi.org/10.1137/130916849","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.8199999928474426}],"awards":[{"id":"https://openalex.org/G2522733424","display_name":"Analysis of Numerical Methods for Partial Differential Equations with Random Data","funder_award_id":"EP/H021205/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G4574271719","display_name":null,"funder_award_id":"EP/H021205/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W73098682","https://openalex.org/W196710731","https://openalex.org/W1538934584","https://openalex.org/W1593505867","https://openalex.org/W1963837052","https://openalex.org/W1963971342","https://openalex.org/W1966827182","https://openalex.org/W1991777578","https://openalex.org/W1995404822","https://openalex.org/W1998021317","https://openalex.org/W2004814202","https://openalex.org/W2014750460","https://openalex.org/W2024300082","https://openalex.org/W2025769367","https://openalex.org/W2042128780","https://openalex.org/W2044390636","https://openalex.org/W2055460625","https://openalex.org/W2061604462","https://openalex.org/W2083054768","https://openalex.org/W2083754610","https://openalex.org/W2102031020","https://openalex.org/W2139798157","https://openalex.org/W2324833387","https://openalex.org/W2326704909"],"related_works":["https://openalex.org/W4388311650","https://openalex.org/W1974056099","https://openalex.org/W4245343541","https://openalex.org/W2386077341","https://openalex.org/W563589758","https://openalex.org/W5922282","https://openalex.org/W2954004777","https://openalex.org/W2949638731","https://openalex.org/W2951102138","https://openalex.org/W4389252581"],"abstract_inverted_index":{"Stochastic":[0],"Galerkin":[1],"approximation":[2,28,95,115],"is":[3,22,84,103],"an":[4,62],"increasingly":[5],"popular":[6],"approach":[7],"for":[8,112,125],"the":[9,30,41,80,89,94,114,122,126,135,142,146],"solution":[10],"of":[11,53,65,79,93,121,134],"elliptic":[12],"PDE":[13],"problems":[14],"with":[15,33],"correlated":[16],"random":[17],"data.":[18],"A":[19,69],"typical":[20],"strategy":[21],"to":[23],"combine":[24],"conventional":[25],"($h$-)finite":[26],"element":[27],"on":[29,36,141],"spatial":[31],"domain":[32],"spectral":[34],"($p$-)approximation":[35],"a":[37,49,76],"finite-dimensional":[38],"manifold":[39],"in":[40,48,58],"(stochastic)":[42],"parameter":[43],"domain.":[44],"The":[45],"issues":[46],"involved":[47],"posteriori":[50],"error":[51,72,101,123,136],"analysis":[52],"computed":[54],"solutions":[55],"are":[56],"outlined":[57],"this":[59],"paper":[60],"using":[61],"abstract":[63],"setting":[64],"parametric":[66],"operator":[67,83],"equations.":[68],"novel":[70],"energy":[71],"estimator":[73,102],"that":[74,99,138],"uses":[75],"parameter-free":[77],"part":[78],"underlying":[81],"differential":[82],"introduced":[85],"which":[86],"effectively":[87],"exploits":[88],"tensor":[90],"product":[91],"structure":[92],"space.":[96],"We":[97,107],"prove":[98,118],"our":[100],"reliable":[104],"and":[105,117,145],"efficient.":[106],"also":[108],"discuss":[109],"different":[110],"strategies":[111],"enriching":[113],"space":[116],"two-sided":[119],"estimates":[120,133],"reduction":[124,137],"corresponding":[127],"enhanced":[128],"approximations.":[129],"These":[130],"give":[131],"computable":[132],"depend":[139],"only":[140],"problem":[143],"data":[144],"original":[147],"approximation.":[148]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":12},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":2}],"updated_date":"2026-07-17T09:13:05.818461","created_date":"2025-10-10T00:00:00"}