{"id":"https://openalex.org/W4318700203","doi":"https://doi.org/10.1137/22m1485759","title":"Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems","display_name":"Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems","publication_year":2023,"publication_date":"2023-01-23","ids":{"openalex":"https://openalex.org/W4318700203","doi":"https://doi.org/10.1137/22m1485759"},"language":"en","primary_location":{"id":"doi:10.1137/22m1485759","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1485759","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","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/A5082790650","display_name":"Yunhui He","orcid":"https://orcid.org/0000-0003-3476-2487"},"institutions":[{"id":"https://openalex.org/I141945490","display_name":"University of British Columbia","ror":"https://ror.org/03rmrcq20","country_code":"CA","type":"education","lineage":["https://openalex.org/I141945490"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Yunhui He","raw_affiliation_strings":["Department of Computer Science, The University of British Columbia, Vancouver V6T 1Z4, BC, Canada"],"raw_orcid":"https://orcid.org/0000-0003-3476-2487","affiliations":[{"raw_affiliation_string":"Department of Computer Science, The University of British Columbia, Vancouver V6T 1Z4, BC, Canada","institution_ids":["https://openalex.org/I141945490"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100362043","display_name":"Jun Liu","orcid":"https://orcid.org/0009-0004-8026-3704"},"institutions":[{"id":"https://openalex.org/I87547150","display_name":"Southern Illinois University Edwardsville","ror":"https://ror.org/04cqs5j56","country_code":"US","type":"education","lineage":["https://openalex.org/I2801502357","https://openalex.org/I87547150"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jun Liu","raw_affiliation_strings":["Corresponding author. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026 USA"],"raw_orcid":"https://orcid.org/0000-0003-1904-2350","affiliations":[{"raw_affiliation_string":"Corresponding author. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026 USA","institution_ids":["https://openalex.org/I87547150"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5082790650"],"corresponding_institution_ids":["https://openalex.org/I141945490"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01033621,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"44","issue":"1","first_page":"1","last_page":"26"},"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":0.9998000264167786,"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":0.9998000264167786,"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":0.9975000023841858,"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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9965000152587891,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.8275530338287354},{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.6923172473907471},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.5910032987594604},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.5795665383338928},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.5771334767341614},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5573493838310242},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.53925621509552},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.5055253505706787},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.44033506512641907},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.437171995639801},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.43270331621170044},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3147422671318054},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.21871009469032288},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16759136319160461},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.09061846137046814}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.8275530338287354},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.6923172473907471},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.5910032987594604},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.5795665383338928},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.5771334767341614},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5573493838310242},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.53925621509552},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.5055253505706787},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.44033506512641907},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.437171995639801},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.43270331621170044},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3147422671318054},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.21871009469032288},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16759136319160461},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.09061846137046814},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/22m1485759","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1485759","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":53,"referenced_works":["https://openalex.org/W433418221","https://openalex.org/W581038412","https://openalex.org/W634415007","https://openalex.org/W1579781026","https://openalex.org/W1600371943","https://openalex.org/W1820125862","https://openalex.org/W1969173072","https://openalex.org/W1974791647","https://openalex.org/W1976636768","https://openalex.org/W1977135073","https://openalex.org/W1978837662","https://openalex.org/W1989116194","https://openalex.org/W1992056335","https://openalex.org/W1998163691","https://openalex.org/W2002177035","https://openalex.org/W2007176686","https://openalex.org/W2011275179","https://openalex.org/W2012305345","https://openalex.org/W2026325708","https://openalex.org/W2026365342","https://openalex.org/W2026993378","https://openalex.org/W2053514102","https://openalex.org/W2058940783","https://openalex.org/W2062226392","https://openalex.org/W2071580657","https://openalex.org/W2078611566","https://openalex.org/W2079482881","https://openalex.org/W2087003257","https://openalex.org/W2088055586","https://openalex.org/W2115178538","https://openalex.org/W2126237310","https://openalex.org/W2129121555","https://openalex.org/W2155750845","https://openalex.org/W2160573428","https://openalex.org/W2162221417","https://openalex.org/W2326758409","https://openalex.org/W2397487207","https://openalex.org/W2556507607","https://openalex.org/W2767660003","https://openalex.org/W2805917124","https://openalex.org/W2912221298","https://openalex.org/W2952206045","https://openalex.org/W2963572835","https://openalex.org/W2963740496","https://openalex.org/W2963938374","https://openalex.org/W2964204683","https://openalex.org/W2995987820","https://openalex.org/W2996447514","https://openalex.org/W3108568571","https://openalex.org/W3144001861","https://openalex.org/W3216461233","https://openalex.org/W4298228534","https://openalex.org/W4308904453"],"related_works":["https://openalex.org/W2155025153","https://openalex.org/W4253880741","https://openalex.org/W2361205607","https://openalex.org/W1524826225","https://openalex.org/W2804034492","https://openalex.org/W2040165403","https://openalex.org/W4235544854","https://openalex.org/W1965116625","https://openalex.org/W993433986","https://openalex.org/W1990747776"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,34,93],"study":[4],"and":[5,16,28,30,81,95],"compare":[6],"two":[7,181],"multigrid":[8,146],"relaxation":[9,47,72,101],"schemes":[10,124],"with":[11,25,54,155,159],"coarsening":[12,150,162],"by":[13,143,151,163],"two,":[14],"three,":[15],"four":[17,154],"for":[18,50,117],"solving":[19],"elliptic":[20],"sparse":[21],"optimal":[22,60,71],"control":[23,26],"problems":[24],"constraints":[27],"combined":[29],"cost":[31,56],"functional.":[32],"First,":[33],"perform":[35],"a":[36,43,97,118,196],"detailed":[37],"local":[38],"Fourier":[39],"analysis":[40,67],"(LFA)":[41],"of":[42],"well-known":[44],"collective":[45],"Jacobi":[46],"(CJR)":[48],"scheme":[49,115,203],"the":[51,59,70,77,82,113,132,137,144,160,191,201,205],"unconstrained":[52],"case":[53],"only":[55],"functional,":[57],"where":[58,136,180],"smoothing":[61,110],"factors":[62,111],"are":[63,125,141,167,187],"derived.":[64],"This":[65],"insightful":[66],"reveals":[68],"that":[69],"parameters":[73],"depend":[74],"on":[75],"both":[76],"mesh":[78],"step":[79],"size":[80],"regularization":[83],"parameter":[84],",":[85],"which":[86,104],"was":[87],"not":[88],"investigated":[89],"in":[90,204],"literature.":[91],"Second,":[92],"propose":[94],"analyze":[96],"new":[98],"mass-based":[99],"Braess--Sarazin":[100],"(BSR)":[102],"scheme,":[103,179],"is":[105,157],"proven":[106],"to":[107,128,169,189],"provide":[108],"smaller":[109],"than":[112,200],"CJR":[114,202],"when":[116],"small":[119],"constant":[120],".":[121],"Finally,":[122],"these":[123],"successfully":[126],"extended":[127],"control-constrained":[129],"cases":[130],"through":[131],"semismooth":[133],"Newton":[134],"method,":[135],"corresponding":[138],"Jacobian":[139],"systems":[140],"treated":[142],"proposed":[145,175],"schemes.":[147],"The":[148,174],"nonstandard":[149],"three":[152],"or":[153],"BSR":[156,177],"competitive":[158],"standard":[161],"two.":[164],"Numerical":[165],"examples":[166],"presented":[168],"validate":[170],"our":[171],"theoretical":[172],"outcomes.":[173],"inexact":[176],"(IBSR)":[178],"preconditioned":[182],"conjugate":[183],"gradients":[184],"(PCG)":[185],"iterations":[186],"applied":[188],"solve":[190],"Schur":[192],"complement":[193],"system,":[194],"yields":[195],"better":[197],"computational":[198],"efficiency":[199],"conducted":[206],"numerical":[207],"comparison.":[208]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}