{"id":"https://openalex.org/W2804300145","doi":"https://doi.org/10.1137/18m1189373","title":"On the Best Approximation of the Hierarchical Matrix Product","display_name":"On the Best Approximation of the Hierarchical Matrix Product","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2804300145","doi":"https://doi.org/10.1137/18m1189373","mag":"2804300145"},"language":"en","primary_location":{"id":"doi:10.1137/18m1189373","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1189373","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","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012055655","display_name":"J\u00fcrgen D\u00f6lz","orcid":"https://orcid.org/0000-0003-3322-1187"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"J\u00fcrgen D\u00f6lz","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005562022","display_name":"Helmut Harbrecht","orcid":"https://orcid.org/0000-0003-0093-5706"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Helmut Harbrecht","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5069886499","display_name":"Michael Multerer","orcid":"https://orcid.org/0000-0003-0170-0239"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michael D. Multerer","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6065,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.68803672,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"40","issue":"1","first_page":"147","last_page":"174"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9993000030517578,"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":0.9983000159263611,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/matrix-multiplication","display_name":"Matrix multiplication","score":0.845051646232605},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.7372259497642517},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6090694069862366},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5747737884521484},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5699400901794434},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5658695101737976},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"Low-rank approximation","score":0.4607556462287903},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.45921561121940613},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.4421662390232086},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4305148720741272},{"id":"https://openalex.org/keywords/numerical-linear-algebra","display_name":"Numerical linear algebra","score":0.41868042945861816},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3367235064506531},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.3188926875591278},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.1139984130859375},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.09374204277992249}],"concepts":[{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.845051646232605},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.7372259497642517},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6090694069862366},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5747737884521484},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5699400901794434},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5658695101737976},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.4607556462287903},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.45921561121940613},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4421662390232086},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4305148720741272},{"id":"https://openalex.org/C163834973","wikidata":"https://www.wikidata.org/wiki/Q2004891","display_name":"Numerical linear algebra","level":3,"score":0.41868042945861816},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3367235064506531},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.3188926875591278},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.1139984130859375},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.09374204277992249},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"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/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/18m1189373","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1189373","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"},{"id":"pmh:oai:edoc.unibas.ch:69165","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306401836","display_name":"edoc (University of Basel)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1850255","host_organization_name":"University of Basel","host_organization_lineage":["https://openalex.org/I1850255"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:tubiblio.ulb.tu-darmstadt.de:111267","is_oa":false,"landing_page_url":"http://tubiblio.ulb.tu-darmstadt.de/111267/","pdf_url":null,"source":{"id":"https://openalex.org/S4377196390","display_name":"TUbilio (Technical University of Darmstadt)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I31512782","host_organization_name":"Technische Universit\u00e4t Darmstadt","host_organization_lineage":["https://openalex.org/I31512782"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Artikel"},{"id":"doi:10.5451/unibas-ep69165","is_oa":true,"landing_page_url":"https://doi.org/10.5451/unibas-ep69165","pdf_url":null,"source":{"id":"https://openalex.org/S4306401836","display_name":"edoc (University of Basel)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1850255","host_organization_name":"University of Basel","host_organization_lineage":["https://openalex.org/I1850255"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:edoc.unibas.ch:69165","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306401836","display_name":"edoc (University of Basel)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1850255","host_organization_name":"University of Basel","host_organization_lineage":["https://openalex.org/I1850255"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5253088208","display_name":"H-Matrix Techniques and Uncertainty Quantification in Electromagnetism","funder_award_id":"174987","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"},{"id":"https://openalex.org/F4320323384","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1569304948","https://openalex.org/W1599063302","https://openalex.org/W1712535590","https://openalex.org/W1968376936","https://openalex.org/W1973786815","https://openalex.org/W1983103429","https://openalex.org/W2005423095","https://openalex.org/W2016075148","https://openalex.org/W2018419001","https://openalex.org/W2027832430","https://openalex.org/W2030565709","https://openalex.org/W2062583795","https://openalex.org/W2078465266","https://openalex.org/W2089958289","https://openalex.org/W2091078728","https://openalex.org/W2117756735","https://openalex.org/W2122137889","https://openalex.org/W2156164638","https://openalex.org/W2188296733","https://openalex.org/W2262775741","https://openalex.org/W2281964624","https://openalex.org/W2963863488"],"related_works":["https://openalex.org/W3099313426","https://openalex.org/W4287593139","https://openalex.org/W752783541","https://openalex.org/W1506547947","https://openalex.org/W4206811032","https://openalex.org/W2995605830","https://openalex.org/W4239424132","https://openalex.org/W2596457687","https://openalex.org/W3212757063","https://openalex.org/W2086123442"],"abstract_inverted_index":{"The":[0],"multiplication":[1,25,68,80,174,186],"of":[2,15,26,41,69,81,88,90,113,132,143,158,161],"matrices":[3,83,163],"is":[4,44,135],"an":[5,33],"important":[6],"arithmetic":[7],"operation":[8,19],"in":[9,32,179],"computational":[10,39],"mathematics.":[11],"In":[12,57],"the":[13,24,38,42,52,67,79,110,126,130,133,141,144,150,155,159,172,184],"context":[14],"hierarchical":[16,70,82,114,162],"matrices,":[17,30],"this":[18,58,100,177],"can":[20,175],"be":[21],"realized":[22],"by":[23,77],"structured":[27],"blockwise":[28],"low-rank":[29,86,127],"resulting":[31],"almost":[34,136],"linear":[35,137],"cost.":[36],"However,":[37],"efficiency":[40],"algorithm":[43,187],"based":[45],"on":[46,125],"a":[47,62,103,165],"recursive":[48],"scheme":[49],"which":[50],"makes":[51],"error":[53,189],"analysis":[54,119],"quite":[55],"involved.":[56],"article,":[59],"we":[60,105],"propose":[61,94],"new":[63,151,173],"algorithmic":[64],"framework":[65,134],"for":[66,164],"matrices.":[71],"It":[72],"improves":[73],"currently":[74],"known":[75],"implementations":[76],"reducing":[78],"to":[84,98,108,140],"suitable":[85],"approximations":[87],"sums":[89],"matrix":[91,115],"products.":[92,116],"We":[93],"several":[95],"compression":[96],"schemes":[97],"address":[99],"task.":[101],"As":[102],"consequence,":[104],"are":[106],"able":[107],"compute":[109],"best":[111,156],"approximation":[112,128,157],"A":[117],"cost":[118,131],"shows":[120],"that,":[121],"under":[122],"reasonable":[123],"assumptions":[124],"method,":[129],"with":[138],"respect":[139],"size":[142],"matrix.":[145],"Numerical":[146],"experiments":[147],"show":[148,170],"that":[149,171],"approach":[152],"produces":[153],"indeed":[154],"product":[160],"given":[166],"tolerance.":[167],"They":[168],"also":[169],"accomplish":[176],"task":[178],"less":[180],"computation":[181],"time":[182],"than":[183],"established":[185],"without":[188],"control.":[190]},"counts_by_year":[{"year":2024,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2025-10-10T00:00:00"}