{"id":"https://openalex.org/W1980600685","doi":"https://doi.org/10.1137/040607964","title":"Multilevel Hierarchical Matrices","display_name":"Multilevel Hierarchical Matrices","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W1980600685","doi":"https://doi.org/10.1137/040607964","mag":"1980600685"},"language":"en","primary_location":{"is_oa":false,"landing_page_url":"https://doi.org/10.1137/040607964","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":null,"is_accepted":false,"is_published":false},"type":"article","type_crossref":"journal-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/A5014285191","display_name":"Sabine Le Borne","orcid":"https://orcid.org/0000-0002-4399-4442"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Sabine Le Borne","raw_affiliation_strings":[],"affiliations":[]}],"institution_assertions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5014285191"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.442,"has_fulltext":true,"fulltext_origin":"ngrams","cited_by_count":8,"citation_normalized_percentile":{"value":0.674137,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":80,"max":81},"biblio":{"volume":"28","issue":"3","first_page":"871","last_page":"889"},"is_retracted":false,"is_paratext":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9995,"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":0.9995,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9994,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9991,"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/interpolation","display_name":"Interpolation","score":0.6360556},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5592122},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5140618}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.84096026},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.657503},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.6360556},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5592122},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.54257786},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5140618},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.51227707},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4057359},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.37572145},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37265146},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.24161643},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08040753},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"is_oa":false,"landing_page_url":"https://doi.org/10.1137/040607964","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":null,"is_accepted":false,"is_published":false}],"best_oa_location":null,"sustainable_development_goals":[],"grants":[],"datasets":[],"versions":[],"referenced_works_count":18,"referenced_works":["https://openalex.org/W1557742846","https://openalex.org/W1569304948","https://openalex.org/W1969720662","https://openalex.org/W1976016599","https://openalex.org/W2013144881","https://openalex.org/W2016121728","https://openalex.org/W2020062373","https://openalex.org/W2027764906","https://openalex.org/W2065205300","https://openalex.org/W2076467131","https://openalex.org/W2078645506","https://openalex.org/W2079059444","https://openalex.org/W2122137889","https://openalex.org/W2156164638","https://openalex.org/W2508429878","https://openalex.org/W3152461294","https://openalex.org/W425393852","https://openalex.org/W609598071"],"related_works":["https://openalex.org/W946352265","https://openalex.org/W3020787026","https://openalex.org/W2793786119","https://openalex.org/W2364741597","https://openalex.org/W2334479858","https://openalex.org/W2022062253","https://openalex.org/W2006251942","https://openalex.org/W1864774435","https://openalex.org/W1507702947","https://openalex.org/W1492103595"],"abstract_inverted_index":{"This":[0],"paper":[1],"deals":[2],"with":[3,120],"a":[4,20],"multilevel":[5],"construction":[6],"of":[7,14,22,76,87],"hierarchical":[8],"matrix":[9],"approximations":[10,104],"to":[11,48,96,105],"the":[12,35,56,63,74,83,106],"inverses":[13],"finite":[15,84],"element":[16,85],"stiffness":[17],"matrices.":[18],"Given":[19],"sequence":[21],"discretizations":[23],"$A_{\\ell}":[24],"x_{\\ell}":[25],"=":[26,32],"f_{\\ell}$,":[27],"$\\ell=0,\\ldots,n$,":[28],"where":[29,100],"$A_0":[30],"x_0":[31],"f_0$":[33],"denotes":[34],"coarse":[36],"grid":[37],"problem,":[38],"we":[39],"will":[40],"compute":[41],"$A^{-1}_0$":[42],"exactly":[43],"and":[44],"then":[45,92],"use":[46],"interpolation":[47,71],"obtain":[49],"an":[50,69],"$\\mathcal{H}$\u2010matrix":[51,58],"approximation":[52],"$A^{-\\cal":[53,60],"H}_{\\ell+1}$":[54],"from":[55],"approximate":[57],"inverse":[59,75,107],"H}_{\\ell}$":[61],"on":[62],"next":[64],"coarser":[65],"grid.":[66],"We":[67,91,116],"develop":[68],"exact":[70],"scheme":[72],"for":[73,124],"tridiagonal":[77],"matrices":[78],"as":[79,110],"they":[80],"appear":[81],"in":[82,112],"discretization":[86],"one\u2010dimensional":[88],"differential":[89],"equations.":[90],"generalize":[93],"this":[94,118],"approach":[95,119],"two":[97],"spatial":[98],"dimensions":[99],"these":[101],"efficiently":[102],"computed":[103],"may":[108],"serve":[109],"preconditioners":[111],"iterative":[113],"solution":[114],"methods.":[115],"illustrate":[117],"some":[121],"numerical":[122],"tests":[123],"convection\u2010dominated":[125],"convection\u2010diffusion":[126],"problems.":[127]},"cited_by_api_url":"https://api.openalex.org/works?filter=cites:W1980600685","counts_by_year":[{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2024-12-17T23:10:27.727914","created_date":"2016-06-24"}