{"id":"https://openalex.org/W2625734760","doi":"https://doi.org/10.1137/15m1049269","title":"Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems","display_name":"Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2625734760","doi":"https://doi.org/10.1137/15m1049269","mag":"2625734760"},"language":"en","primary_location":{"id":"doi:10.1137/15m1049269","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1049269","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":null,"license_id":null,"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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012210042","display_name":"T. Croft","orcid":null},"institutions":[{"id":"https://openalex.org/I79510175","display_name":"Cardiff University","ror":"https://ror.org/03kk7td41","country_code":"GB","type":"education","lineage":["https://openalex.org/I79510175"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Thomas L. D. Croft","raw_affiliation_strings":["School of Mathematics, Cardiff University, Cardiff, CF24 4AG, Wales, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Cardiff University, Cardiff, CF24 4AG, Wales, UK","institution_ids":["https://openalex.org/I79510175"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072312564","display_name":"Timothy N. Phillips","orcid":"https://orcid.org/0000-0001-6455-1205"},"institutions":[{"id":"https://openalex.org/I79510175","display_name":"Cardiff University","ror":"https://ror.org/03kk7td41","country_code":"GB","type":"education","lineage":["https://openalex.org/I79510175"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Timothy N. Phillips","raw_affiliation_strings":["School of Mathematics, Cardiff University, Cardiff, CF24 4AG, Wales, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Cardiff University, Cardiff, CF24 4AG, Wales, UK","institution_ids":["https://openalex.org/I79510175"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.521,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.63763987,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"39","issue":"4","first_page":"A1366","last_page":"A1388"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9980000257492065,"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"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9977999925613403,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9927999973297119,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8404484391212463},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.5900810956954956},{"id":"https://openalex.org/keywords/coercivity","display_name":"Coercivity","score":0.5800649523735046},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5464205145835876},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5423423051834106},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.5004780292510986},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.4891984462738037},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.48532360792160034},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.440616637468338},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4258113503456116},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.21070095896720886},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.12363600730895996},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10782989859580994},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0633612871170044}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8404484391212463},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.5900810956954956},{"id":"https://openalex.org/C126530901","wikidata":"https://www.wikidata.org/wiki/Q432635","display_name":"Coercivity","level":2,"score":0.5800649523735046},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5464205145835876},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5423423051834106},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5004780292510986},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.4891984462738037},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.48532360792160034},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.440616637468338},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4258113503456116},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.21070095896720886},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.12363600730895996},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10782989859580994},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0633612871170044},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"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/C26873012","wikidata":"https://www.wikidata.org/wiki/Q214781","display_name":"Condensed matter physics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/15m1049269","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1049269","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:https://orca.cardiff.ac.uk:67639","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306401195","display_name":"ORCA Online Research @Cardiff (Cardiff University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79510175","host_organization_name":"Cardiff University","host_organization_lineage":["https://openalex.org/I79510175"],"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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1689917686","display_name":"DTA - Cardiff University","funder_award_id":"EP/J500197/1","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":22,"referenced_works":["https://openalex.org/W30858885","https://openalex.org/W1965727018","https://openalex.org/W1973117158","https://openalex.org/W1978376511","https://openalex.org/W1980428773","https://openalex.org/W1989511024","https://openalex.org/W2004026774","https://openalex.org/W2015148405","https://openalex.org/W2057577185","https://openalex.org/W2059537627","https://openalex.org/W2071942565","https://openalex.org/W2073370369","https://openalex.org/W2074330728","https://openalex.org/W2079775126","https://openalex.org/W2080131969","https://openalex.org/W2132292023","https://openalex.org/W2149384201","https://openalex.org/W2963858059","https://openalex.org/W3099723359","https://openalex.org/W4243738351","https://openalex.org/W4250155727","https://openalex.org/W4255975175"],"related_works":["https://openalex.org/W2044642251","https://openalex.org/W1979864087","https://openalex.org/W3093027776","https://openalex.org/W1986796332","https://openalex.org/W4283367861","https://openalex.org/W2909854803","https://openalex.org/W2609179044","https://openalex.org/W3015996521","https://openalex.org/W2022128204","https://openalex.org/W850259343"],"abstract_inverted_index":{"Proper":[0],"generalized":[1],"decompositions":[2],"(PGDs)":[3],"are":[4,40,53,66],"a":[5,18,27,84,128,132,163],"family":[6],"of":[7,24,30,50,88,99,131,153,176,178],"methods":[8],"for":[9,37,149],"efficiently":[10],"solving":[11],"high-dimensional":[12],"PDEs,":[13],"which":[14,39,52,106],"seek":[15],"to":[16,21,69,109],"find":[17],"low-rank":[19,74],"approximation":[20,87],"the":[22,25,47,59,73,97,124,154,179],"solution":[23],"PDE":[26],"priori.":[28],"Convergence":[29],"PGD":[31,75,86,100,142],"algorithms":[32,101,143],"can":[33,78],"only":[34,54],"be":[35,70],"proven":[36],"problems":[38,51,114],"continuous,":[41],"symmetric,":[42],"and":[43,111,115,119,138,172],"strongly":[44,112],"coercive.":[45],"In":[46,92],"particular":[48],"case":[49],"weakly":[55,89,133],"coercive":[56,90,113,134],"we":[57,95,136,173],"have":[58],"additional":[60],"issue":[61],"that":[62,158],"weak":[63],"coercivity":[64],"estimates":[65,148],"not":[67],"guaranteed":[68],"inherited":[71],"by":[72],"approximation.":[76],"This":[77],"cause":[79],"stability":[80],"issues":[81],"when":[82],"employing":[83],"Galerkin":[85,170],"problems.":[91],"this":[93],"paper":[94],"propose":[96],"use":[98],"based":[102,144],"on":[103,145],"least-squares":[104,141,147,160],"formulations":[105],"always":[107],"lead":[108],"symmetric":[110],"hence":[116],"provide":[117,162,174],"stable":[118],"provably":[120],"convergent":[121],"algorithms.":[122,180],"Taking":[123],"Stokes":[125],"problem":[126],"as":[127],"prototypical":[129],"example":[130],"problem,":[135],"develop":[137],"compare":[139],"rigorous":[140],"continuous":[146],"two":[150],"different":[151],"reformulations":[152],"problem.":[155],"We":[156],"show":[157],"these":[159],"PGDs":[161],"much":[164],"stabler":[165],"algorithm":[166],"than":[167],"an":[168],"equivalent":[169],"PGDs,":[171],"proofs":[175],"convergence":[177]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
