{"id":"https://openalex.org/W2088411652","doi":"https://doi.org/10.1137/090746409","title":"On the Strong Convergence of Gradients in Stabilized Degenerate Convex Minimization Problems","display_name":"On the Strong Convergence of Gradients in Stabilized Degenerate Convex Minimization Problems","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2088411652","doi":"https://doi.org/10.1137/090746409","mag":"2088411652"},"language":"en","primary_location":{"id":"doi:10.1137/090746409","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090746409","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":["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/A5068216914","display_name":"Wolfgang Boiger","orcid":null},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Wolfgang Boiger","raw_affiliation_strings":["boiger@math.hu-berlin.de and cc@math.hu-berlin.de#TAB#"],"affiliations":[{"raw_affiliation_string":"boiger@math.hu-berlin.de and cc@math.hu-berlin.de#TAB#","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036808184","display_name":"Carsten Carstensen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Carsten Carstensen","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5068216914"],"corresponding_institution_ids":["https://openalex.org/I39343248"],"apc_list":null,"apc_paid":null,"fwci":0.3388,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.64685444,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"47","issue":"6","first_page":"4569","last_page":"4580"},"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":0.9994000196456909,"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":0.9994000196456909,"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/T11115","display_name":"Topology Optimization in Engineering","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/2205","display_name":"Civil and Structural Engineering"},"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/T11366","display_name":"Elasticity and Material Modeling","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"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/discretization","display_name":"Discretization","score":0.7634475827217102},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7433812618255615},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5941047072410583},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.5913494825363159},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.5886831283569336},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.5828866362571716},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5436276197433472},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4754050672054291},{"id":"https://openalex.org/keywords/energy-minimization","display_name":"Energy minimization","score":0.4702010750770569},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.41810840368270874},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.4116796553134918},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3586263656616211},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31392955780029297},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1901039183139801},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09081944823265076}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7634475827217102},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7433812618255615},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5941047072410583},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.5913494825363159},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.5886831283569336},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.5828866362571716},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5436276197433472},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4754050672054291},{"id":"https://openalex.org/C14961307","wikidata":"https://www.wikidata.org/wiki/Q5377176","display_name":"Energy minimization","level":2,"score":0.4702010750770569},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.41810840368270874},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.4116796553134918},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3586263656616211},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31392955780029297},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1901039183139801},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09081944823265076},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/090746409","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090746409","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"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","score":0.6800000071525574,"display_name":"Affordable and clean energy"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1492326914","https://openalex.org/W1968104256","https://openalex.org/W2041523977","https://openalex.org/W2054535420","https://openalex.org/W2092692435","https://openalex.org/W2159696646"],"related_works":["https://openalex.org/W2620834727","https://openalex.org/W205574387","https://openalex.org/W2064244791","https://openalex.org/W2351116219","https://openalex.org/W1483822002","https://openalex.org/W4298930346","https://openalex.org/W2952156591","https://openalex.org/W2052250766","https://openalex.org/W2985648438","https://openalex.org/W2050152410"],"abstract_inverted_index":{"Infimizing":[0],"sequences":[1],"in":[2,34,63],"nonconvex":[3,41],"variational":[4],"problems":[5],"typically":[6],"exhibit":[7],"enforced":[8],"finer":[9,11],"and":[10,127,148],"oscillations":[12,25],"called":[13],"microstructures":[14],"such":[15],"that":[16,66,93],"the":[17,40,53,64,86,109,119,151,157,166],"infimal":[18],"energy":[19],"is":[20,61,68],"not":[21,71],"attained.":[22],"Although":[23],"those":[24],"are":[26],"physically":[27],"meaningful,":[28],"finite":[29],"element":[30],"approximations":[31],"experience":[32],"difficulty":[33],"their":[35],"reconstruction.":[36],"The":[37,56],"relaxation":[38],"of":[39,98,108,153],"minimization":[42],"problem":[43,60],"by":[44,80],"(semi)":[45],"convexification":[46],"leads":[47,103],"to":[48,85,104,118,165],"a":[49,77,82,95],"macroscopic":[50,59,110],"model":[51],"for":[52,94,143,156],"effective":[54],"energy.":[55,88],"resulting":[57],"discrete":[58,87],"degenerate":[62],"sense":[65],"it":[67],"convex":[69],"but":[70],"strictly":[72],"convex.":[73],"This":[74],"paper":[75,141],"studies":[76],"modified":[78],"discretization":[79],"adding":[81],"stabilization":[83,101],"term":[84],"It":[89],"will":[90],"be":[91],"proven":[92],"wide":[96],"class":[97],"problems,":[99],"this":[100,140],"technique":[102],"strong":[105],"$H^1$":[106],"convergence":[107],"variables":[111],"even":[112],"on":[113,137],"unstructured":[114,145],"triangulations.":[115],"In":[116],"contrast":[117],"work":[120],"[C.":[121],"Carstensen,":[122],"P.":[123],"Plech\u00e1\u0109,":[124],"S.":[125],"Bartels,":[126],"A.":[128],"Prohl,":[129],"Interfaces":[130],"Free":[131],"Bound.,":[132],"6":[133],"(2004),":[134],"pp.":[135],"253\u2013269]":[136],"quasi-uniform":[138],"triangulations,":[139],"allows":[142],"general":[144],"shape-regular":[146],"triangulations":[147],"so":[149],"enables":[150],"use":[152],"adaptive":[154],"algorithms":[155],"stabilized":[158],"formulations.":[159],"(An":[160],"erratum":[161],"has":[162],"been":[163],"appended":[164],"originally":[167],"posted":[168],"pdf.)":[169]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}