{"id":"https://openalex.org/W2008377138","doi":"https://doi.org/10.1007/s00211-013-0575-z","title":"On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems","display_name":"On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems","publication_year":2013,"publication_date":"2013-07-25","ids":{"openalex":"https://openalex.org/W2008377138","doi":"https://doi.org/10.1007/s00211-013-0575-z","mag":"2008377138"},"language":"en","primary_location":{"id":"doi:10.1007/s00211-013-0575-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00211-013-0575-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00211-013-0575-z.pdf","source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00211-013-0575-z.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080347173","display_name":"Takehiko Kinoshita","orcid":"https://orcid.org/0000-0001-9756-4571"},"institutions":[{"id":"https://openalex.org/I22299242","display_name":"Kyoto University","ror":"https://ror.org/02kpeqv85","country_code":"JP","type":"education","lineage":["https://openalex.org/I22299242"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Takehiko Kinoshita","raw_affiliation_strings":["Research Institute for Mathematical Sciences, Kyoto University, 606-8502 , Kyoto, Japan","Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan 606-8502#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Research Institute for Mathematical Sciences, Kyoto University, 606-8502 , Kyoto, Japan","institution_ids":["https://openalex.org/I22299242"]},{"raw_affiliation_string":"Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan 606-8502#TAB#","institution_ids":["https://openalex.org/I22299242"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035428319","display_name":"Takuma Kimura","orcid":"https://orcid.org/0000-0003-4147-1864"},"institutions":[{"id":"https://openalex.org/I150744194","display_name":"Waseda University","ror":"https://ror.org/00ntfnx83","country_code":"JP","type":"education","lineage":["https://openalex.org/I150744194"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Takuma Kimura","raw_affiliation_strings":["Faculty of Science and Engineering, JST CREST, Waseda University, 169-8555 , Tokyo, Japan","Faculty of Science and Engineering, JST CREST, Waseda University, Tokyo, Japan 169-8555#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Science and Engineering, JST CREST, Waseda University, 169-8555 , Tokyo, Japan","institution_ids":["https://openalex.org/I150744194"]},{"raw_affiliation_string":"Faculty of Science and Engineering, JST CREST, Waseda University, Tokyo, Japan 169-8555#TAB#","institution_ids":["https://openalex.org/I150744194"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103127770","display_name":"Mitsuhiro T. Nakao","orcid":"https://orcid.org/0000-0001-5228-0591"},"institutions":[{"id":"https://openalex.org/I4210117716","display_name":"National Institute of Technology, Sasebo College","ror":"https://ror.org/02vvp2k82","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210117716","https://openalex.org/I4210120810"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Mitsuhiro T. Nakao","raw_affiliation_strings":["Sasebo National College of Technology, 857-1193 , Nagasaki, Japan","Sasebo National College of Technology, Nagasaki, Japan 857-1193#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Sasebo National College of Technology, 857-1193 , Nagasaki, Japan","institution_ids":["https://openalex.org/I4210117716"]},{"raw_affiliation_string":"Sasebo National College of Technology, Nagasaki, Japan 857-1193#TAB#","institution_ids":["https://openalex.org/I4210117716"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5080347173"],"corresponding_institution_ids":["https://openalex.org/I22299242"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":2.6306,"has_fulltext":true,"cited_by_count":21,"citation_normalized_percentile":{"value":0.87991249,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"126","issue":"4","first_page":"679","last_page":"701"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9990000128746033,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9990000128746033,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.991100013256073,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9901000261306763,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8650138974189758},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6808918714523315},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.5654381513595581},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.5230263471603394},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5200483798980713},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.5197793841362},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.5106871724128723},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5008177757263184},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.4895073175430298},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4872671365737915},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.46599140763282776},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.42249929904937744},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4198249578475952},{"id":"https://openalex.org/keywords/a-priori-estimate","display_name":"A priori estimate","score":0.41277146339416504},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.41013988852500916},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.22164389491081238}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8650138974189758},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6808918714523315},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.5654381513595581},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5230263471603394},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5200483798980713},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.5197793841362},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.5106871724128723},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5008177757263184},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4895073175430298},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4872671365737915},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.46599140763282776},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.42249929904937744},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4198249578475952},{"id":"https://openalex.org/C2775918384","wikidata":"https://www.wikidata.org/wiki/Q4660961","display_name":"A priori estimate","level":2,"score":0.41277146339416504},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.41013988852500916},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.22164389491081238},{"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"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/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},{"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00211-013-0575-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00211-013-0575-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00211-013-0575-z.pdf","source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},{"id":"pmh:oai:irdb.nii.ac.jp:01221:0000147109","is_oa":true,"landing_page_url":"http://hdl.handle.net/2433/187149","pdf_url":null,"source":{"id":"https://openalex.org/S7407056385","display_name":"Institutional Repositories DataBase (IRDB)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I184597095","host_organization_name":"National Institute of Informatics","host_organization_lineage":["https://openalex.org/I184597095"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal article"}],"best_oa_location":{"id":"doi:10.1007/s00211-013-0575-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00211-013-0575-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00211-013-0575-z.pdf","source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2008377138.pdf","grobid_xml":"https://content.openalex.org/works/W2008377138.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W83184700","https://openalex.org/W178772886","https://openalex.org/W268215538","https://openalex.org/W1518139620","https://openalex.org/W1547403762","https://openalex.org/W1963952659","https://openalex.org/W1985568784","https://openalex.org/W1992702716","https://openalex.org/W2015466611","https://openalex.org/W2054838492","https://openalex.org/W2055723554","https://openalex.org/W2065972561","https://openalex.org/W2072685235","https://openalex.org/W2092068986","https://openalex.org/W2105823053","https://openalex.org/W2114969977","https://openalex.org/W2582171088"],"related_works":["https://openalex.org/W2951949309","https://openalex.org/W4300232707","https://openalex.org/W2119544289","https://openalex.org/W2992612216","https://openalex.org/W2036364217","https://openalex.org/W61959755","https://openalex.org/W2394496373","https://openalex.org/W2050981166","https://openalex.org/W2020718081","https://openalex.org/W2089181747"],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,8,28,38,48,61,75,84,94,97,106,114,137],"guaranteed":[3],"a":[4,20,50,55,80,90],"posteriori":[5,91],"estimates":[6,53,70],"for":[7,41,54,60,74,96,126],"inverse":[9,77],"parabolic":[10,133],"operators":[11],"with":[12,31,71],"homogeneous":[13],"initial-boundary":[14],"conditions.":[15],"Our":[16],"estimation":[17],"technique":[18],"uses":[19],"full-discrete":[21],"numerical":[22,101,127],"scheme,":[23],"which":[24,135],"is":[25],"based":[26],"on":[27],"Galerkin":[29],"method":[30,108],"an":[32,65,72,89,119],"interpolation":[33],"in":[34,43],"time":[35],"by":[36],"using":[37],"fundamental":[39],"solution":[40],"semidiscretization":[42],"space.":[44],"In":[45,100],"our":[46,141],"technique,":[47],"constructive":[49],"priori":[51],"error":[52],"full":[56],"discretization":[57],"of":[58,83,93,129,131,140],"solutions":[59,130],"heat":[62],"equation":[63],"play":[64],"essential":[66],"role.":[67],"Combining":[68],"these":[69],"argument":[73],"discretized":[76],"operator":[78],"and":[79],"contraction":[81],"property":[82],"Newton-type":[85],"formulation,":[86],"we":[87,103,121],"derive":[88],"estimate":[92],"norm":[95],"infinite-dimensional":[98],"operator.":[99],"examples,":[102],"show":[104],"that":[105],"proposed":[107],"should":[109],"be":[110],"more":[111],"efficient":[112],"than":[113],"existing":[115],"method.":[116],"Moreover,":[117],"as":[118],"application,":[120],"give":[122],"some":[123],"prototype":[124],"results":[125],"verification":[128],"nonlinear":[132],"problems,":[134],"confirm":[136],"actual":[138],"usefulness":[139],"technique.":[142]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-06-19T15:47:20.252518","created_date":"2025-10-10T00:00:00"}