{"id":"https://openalex.org/W4394684789","doi":"https://doi.org/10.1007/s10444-024-10123-w","title":"Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization","display_name":"Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization","publication_year":2024,"publication_date":"2024-04-01","ids":{"openalex":"https://openalex.org/W4394684789","doi":"https://doi.org/10.1007/s10444-024-10123-w"},"language":"en","primary_location":{"id":"doi:10.1007/s10444-024-10123-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-024-10123-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-024-10123-w.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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":"Advances in Computational Mathematics","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/s10444-024-10123-w.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058613122","display_name":"Ulrich Langer","orcid":"https://orcid.org/0000-0003-3797-7475"},"institutions":[{"id":"https://openalex.org/I121883995","display_name":"Johannes Kepler University of Linz","ror":"https://ror.org/052r2xn60","country_code":"AT","type":"education","lineage":["https://openalex.org/I121883995"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Ulrich Langer","raw_affiliation_strings":["Institute of Numerical Mathematics, Johannes Kepler University Linz, Altenberger Stra\u00dfe 69, 4040, Linz, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Numerical Mathematics, Johannes Kepler University Linz, Altenberger Stra\u00dfe 69, 4040, Linz, Austria","institution_ids":["https://openalex.org/I121883995"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037762950","display_name":"Olaf Steinbach","orcid":"https://orcid.org/0000-0002-2552-3022"},"institutions":[{"id":"https://openalex.org/I4092182","display_name":"Graz University of Technology","ror":"https://ror.org/00d7xrm67","country_code":"AT","type":"education","lineage":["https://openalex.org/I4092182"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Olaf Steinbach","raw_affiliation_strings":["Institut f\u00fcr Angewandte Mathematik, Technische Universit\u00e4t Graz, Steyrergasse 30, 8010, Graz, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Angewandte Mathematik, Technische Universit\u00e4t Graz, Steyrergasse 30, 8010, Graz, Austria","institution_ids":["https://openalex.org/I4092182"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014561752","display_name":"Huidong Yang","orcid":"https://orcid.org/0000-0001-8639-2030"},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Huidong Yang","raw_affiliation_strings":["Computational Science Center, Universit\u00e4t Wien, Oskar\u2013Morgenstern\u2013Platz 1, 1090, Wien, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computational Science Center, Universit\u00e4t Wien, Oskar\u2013Morgenstern\u2013Platz 1, 1090, Wien, Austria","institution_ids":["https://openalex.org/I129774422"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5037762950"],"corresponding_institution_ids":["https://openalex.org/I4092182"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.6364,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.79540594,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"50","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T13862","display_name":"Soil, Finite Element Methods","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.7523735761642456},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.7257991433143616},{"id":"https://openalex.org/keywords/computational-science-and-engineering","display_name":"Computational Science and Engineering","score":0.6999138593673706},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6212635636329651},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.47734335064888},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4545449912548065},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4457917809486389},{"id":"https://openalex.org/keywords/spacetime","display_name":"Spacetime","score":0.42716965079307556},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.42453524470329285},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.39193177223205566},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.33224356174468994},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.30986741185188293},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10590755939483643},{"id":"https://openalex.org/keywords/structural-engineering","display_name":"Structural engineering","score":0.08017930388450623},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0793144702911377},{"id":"https://openalex.org/keywords/orthodontics","display_name":"Orthodontics","score":0.07416611909866333},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.07361751794815063},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.06972664594650269}],"concepts":[{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.7523735761642456},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.7257991433143616},{"id":"https://openalex.org/C68597687","wikidata":"https://www.wikidata.org/wiki/Q362601","display_name":"Computational Science and Engineering","level":2,"score":0.6999138593673706},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6212635636329651},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.47734335064888},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4545449912548065},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4457917809486389},{"id":"https://openalex.org/C130187892","wikidata":"https://www.wikidata.org/wiki/Q133327","display_name":"Spacetime","level":2,"score":0.42716965079307556},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.42453524470329285},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39193177223205566},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.33224356174468994},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.30986741185188293},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10590755939483643},{"id":"https://openalex.org/C66938386","wikidata":"https://www.wikidata.org/wiki/Q633538","display_name":"Structural engineering","level":1,"score":0.08017930388450623},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0793144702911377},{"id":"https://openalex.org/C29694066","wikidata":"https://www.wikidata.org/wiki/Q118301","display_name":"Orthodontics","level":1,"score":0.07416611909866333},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.07361751794815063},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.06972664594650269},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","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":1,"locations":[{"id":"doi:10.1007/s10444-024-10123-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-024-10123-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-024-10123-w.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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":"Advances in Computational Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10444-024-10123-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-024-10123-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-024-10123-w.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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":"Advances in Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320310430","display_name":"TU Graz, Internationale Beziehungen und Mobilit\u00e4tsprogramme","ror":"https://ror.org/00d7xrm67"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4394684789.pdf"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W93748562","https://openalex.org/W433418221","https://openalex.org/W1547403762","https://openalex.org/W1572617741","https://openalex.org/W1601728065","https://openalex.org/W1966886095","https://openalex.org/W1980053021","https://openalex.org/W1993072133","https://openalex.org/W2030036911","https://openalex.org/W2044959876","https://openalex.org/W2059880466","https://openalex.org/W2076257978","https://openalex.org/W2078843646","https://openalex.org/W2097418791","https://openalex.org/W2141870784","https://openalex.org/W2148733093","https://openalex.org/W2155750845","https://openalex.org/W2402468387","https://openalex.org/W2498157954","https://openalex.org/W2506719423","https://openalex.org/W2545270568","https://openalex.org/W2602742384","https://openalex.org/W3014898465","https://openalex.org/W3107501872","https://openalex.org/W3135041483","https://openalex.org/W3135337288","https://openalex.org/W3204708751","https://openalex.org/W3206753123","https://openalex.org/W3213458207","https://openalex.org/W4302819252","https://openalex.org/W4324356881","https://openalex.org/W4385235890","https://openalex.org/W4391765562","https://openalex.org/W4395050813"],"related_works":["https://openalex.org/W4393232657","https://openalex.org/W4390638272","https://openalex.org/W2607684300","https://openalex.org/W2472237121","https://openalex.org/W4323316863","https://openalex.org/W1985111449","https://openalex.org/W4304789336","https://openalex.org/W4365152763","https://openalex.org/W2340692695","https://openalex.org/W4306175439"],"abstract_inverted_index":{"Abstract":[0],"As":[1],"in":[2,27,77,116,206],"our":[3],"previous":[4],"work":[5],"(":[6],"SINUM":[7],"59(2):660\u2013674,":[8],"2021)":[9],"we":[10],"consider":[11],"space-time":[12,29,128,175],"tracking":[13],"optimal":[14,154],"control":[15],"problems":[16,23],"for":[17,90,191,217],"linear":[18,181],"parabolic":[19],"initial":[20],"boundary":[21],"value":[22],"that":[24,40],"are":[25,41,197],"given":[26],"the":[28,44,52,66,74,78,91,102,110,119,127,138,141,153],"cylinder":[30],"$$Q":[31],"=":[32,157],"\\Omega":[33],"\\times":[34],"(0,T)$$":[35],"<mml:math":[36,49,57,69,81,99,107,114,124,145,159,167],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mi>\u03a9</mml:mi><mml:mo>\u00d7</mml:mo><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>":[37],",":[38,134],"and":[39,109,126,135,182,208],"controlled":[42],"by":[43,73,171,200],"right-hand":[45],"side":[46],"$$z_\\varrho":[47],"$$":[48,123],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>z</mml:mi><mml:mi>\u03f1</mml:mi></mml:msub></mml:math>":[50],"from":[51],"Bochner":[53],"space":[54,210],"$$L^2(0,T;H^{-1}(\\Omega":[55,79],"))$$":[56,80],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>\u037e</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>\u03a9</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>":[58,82],".":[59,147,161,193],"So":[60],"it":[61],"is":[62,169],"natural":[63],"to":[64,152],"replace":[65],"usual":[67],"$$L^2(Q)$$":[68],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>":[70],"norm":[71],"regularization":[72,76,120],"energy":[75],"norm.":[83],"We":[84,212],"derive":[85],"new":[86,149],"a":[87,174,201],"priori":[88],"estimates":[89,150],"error":[92],"$$\\Vert":[93],"\\widetilde{u}_{\\varrho":[94],"h}":[95],"-":[96],"\\overline{u}\\Vert":[97],"_{L^2(Q)}$$":[98],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mrow><mml:mo>\u2016</mml:mo></mml:mrow><mml:msub><mml:mover><mml:mi>u</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mrow><mml:mi>\u03f1</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mover><mml:mi>u</mml:mi><mml:mo>\u00af</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mo>\u2016</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:math>":[100],"between":[101],"computed":[103,170],"state":[104,112,143,164],"$$\\widetilde{u}_{\\varrho":[105,165],"h}$$":[106,166],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mover><mml:mi>u</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mrow><mml:mi>\u03f1</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:math>":[108,168],"desired":[111,142],"$$\\overline{u}$$":[113,144],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mi>u</mml:mi><mml:mo>\u00af</mml:mo></mml:mover></mml:math>":[115,146],"terms":[117],"of":[118,140,173,203],"parameter":[121],"$$\\varrho":[122,156],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>\u03f1</mml:mi></mml:math>":[125],"finite":[129,176],"element":[130,177],"mesh":[131],"size":[132],"h":[133],"depending":[136],"on":[137,186],"regularity":[139],"These":[148],"lead":[151],"choice":[155],"h^2$$":[158],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mi>\u03f1</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math>":[160],"The":[162,194],"approximate":[163],"means":[172],"method":[178],"using":[179],"piecewise":[180],"continuous":[183],"basis":[184],"functions":[185],"completely":[187],"unstructured":[188],"simplicial":[189],"meshes":[190],"Q":[192],"theoretical":[195],"results":[196],"quantitatively":[198],"illustrated":[199],"series":[202],"numerical":[204],"examples":[205],"two":[207],"three":[209],"dimensions.":[211],"also":[213],"provide":[214],"performance":[215],"studies":[216],"different":[218],"solvers.":[219]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2}],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2025-10-10T00:00:00"}
