{"id":"https://openalex.org/W1964200011","doi":"https://doi.org/10.1137/090756843","title":"Krylov Subspace Methods for Linear Systems with Tensor Product Structure","display_name":"Krylov Subspace Methods for Linear Systems with Tensor Product Structure","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W1964200011","doi":"https://doi.org/10.1137/090756843","mag":"1964200011"},"language":"en","primary_location":{"id":"doi:10.1137/090756843","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090756843","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":true,"oa_status":"green","oa_url":"http://infoscience.epfl.ch/record/165589","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084601962","display_name":"Daniel Kre\u00dfner","orcid":"https://orcid.org/0000-0003-3369-2958"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Daniel Kressner","raw_affiliation_strings":["kressner@math.ethz.ch and ctobler@math.ethz.ch#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"kressner@math.ethz.ch and ctobler@math.ethz.ch#TAB#","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034158272","display_name":"Christine Tobler","orcid":null},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Christine Tobler","raw_affiliation_strings":["[ETH, Zurich]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[ETH, Zurich]","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I35440088"],"apc_list":null,"apc_paid":null,"fwci":4.2523,"has_fulltext":false,"cited_by_count":176,"citation_normalized_percentile":{"value":0.93956044,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":"31","issue":"4","first_page":"1688","last_page":"1714"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9988999962806702,"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/T10305","display_name":"Power System Optimization and Stability","score":0.9868999719619751,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/krylov-subspace","display_name":"Krylov subspace","score":0.8421601057052612},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7360507249832153},{"id":"https://openalex.org/keywords/tensor-product","display_name":"Tensor product","score":0.7225000858306885},{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.6575064063072205},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.6335880756378174},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.5933246612548828},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5466450452804565},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.545153021812439},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.5161975026130676},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.4944179058074951},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.45380452275276184},{"id":"https://openalex.org/keywords/matrix-multiplication","display_name":"Matrix multiplication","score":0.4534674882888794},{"id":"https://openalex.org/keywords/tensor-product-of-hilbert-spaces","display_name":"Tensor product of Hilbert spaces","score":0.43795284628868103},{"id":"https://openalex.org/keywords/generalized-minimal-residual-method","display_name":"Generalized minimal residual method","score":0.42568567395210266},{"id":"https://openalex.org/keywords/cartesian-tensor","display_name":"Cartesian tensor","score":0.4117475748062134},{"id":"https://openalex.org/keywords/tensor-contraction","display_name":"Tensor contraction","score":0.41137850284576416},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.35359644889831543},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.32417190074920654},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26489973068237305},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.19503414630889893},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.18608322739601135},{"id":"https://openalex.org/keywords/tensor-density","display_name":"Tensor density","score":0.16566535830497742},{"id":"https://openalex.org/keywords/tensor-field","display_name":"Tensor field","score":0.1532973051071167}],"concepts":[{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.8421601057052612},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7360507249832153},{"id":"https://openalex.org/C51255310","wikidata":"https://www.wikidata.org/wiki/Q1163016","display_name":"Tensor product","level":2,"score":0.7225000858306885},{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.6575064063072205},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.6335880756378174},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.5933246612548828},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5466450452804565},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.545153021812439},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.5161975026130676},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.4944179058074951},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.45380452275276184},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.4534674882888794},{"id":"https://openalex.org/C93776189","wikidata":"https://www.wikidata.org/wiki/Q1404488","display_name":"Tensor product of Hilbert spaces","level":4,"score":0.43795284628868103},{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.42568567395210266},{"id":"https://openalex.org/C64835786","wikidata":"https://www.wikidata.org/wiki/Q17004583","display_name":"Cartesian tensor","level":5,"score":0.4117475748062134},{"id":"https://openalex.org/C124007464","wikidata":"https://www.wikidata.org/wiki/Q428091","display_name":"Tensor contraction","level":3,"score":0.41137850284576416},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.35359644889831543},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.32417190074920654},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26489973068237305},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.19503414630889893},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.18608322739601135},{"id":"https://openalex.org/C148125525","wikidata":"https://www.wikidata.org/wiki/Q904927","display_name":"Tensor density","level":4,"score":0.16566535830497742},{"id":"https://openalex.org/C166077713","wikidata":"https://www.wikidata.org/wiki/Q1758924","display_name":"Tensor field","level":3,"score":0.1532973051071167},{"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/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/090756843","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090756843","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:CiteSeerX.psu:10.1.1.297.963","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.963","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ethz.ch/~kressner/pub/tensorrevision.pdf","raw_type":"text"},{"id":"pmh:oai:infoscience.epfl.ch:165589","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/165589","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"pmh:oai:infoscience.epfl.ch:165589","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/165589","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W845130065","https://openalex.org/W1488435683","https://openalex.org/W1506342804","https://openalex.org/W1971226336","https://openalex.org/W1978115611","https://openalex.org/W1988395629","https://openalex.org/W1995356931","https://openalex.org/W1996297482","https://openalex.org/W2009253094","https://openalex.org/W2016075148","https://openalex.org/W2017710275","https://openalex.org/W2022615414","https://openalex.org/W2024165284","https://openalex.org/W2033029203","https://openalex.org/W2036191408","https://openalex.org/W2041118838","https://openalex.org/W2053693274","https://openalex.org/W2061025600","https://openalex.org/W2073469810","https://openalex.org/W2105812912","https://openalex.org/W2108138101","https://openalex.org/W2342316458","https://openalex.org/W2419656545"],"related_works":["https://openalex.org/W2186765725","https://openalex.org/W2403348579","https://openalex.org/W2947494864","https://openalex.org/W2948436065","https://openalex.org/W2981910625","https://openalex.org/W2312915193","https://openalex.org/W2765690628","https://openalex.org/W2360189018","https://openalex.org/W2053825454","https://openalex.org/W2053693274"],"abstract_inverted_index":{"The":[0,83],"numerical":[1],"solution":[2,94],"of":[3,23,71,104,115],"linear":[4,25,41,65],"systems":[5,32],"with":[6,33,81,134],"certain":[7],"tensor":[8,34,106,118],"product":[9,35],"structures":[10,14],"is":[11,89],"considered.":[12],"Such":[13],"arise,":[15],"for":[16,44,54],"example,":[17],"from":[18,68],"the":[19,69,87,93],"finite":[20],"element":[21],"discretization":[22],"a":[24,28,64,75,102,112,128],"PDE":[26],"on":[27],"d-dimensional":[29],"hypercube.":[30],"Linear":[31],"structure":[36],"can":[37,95],"be":[38,49,97],"regarded":[39],"as":[40],"matrix":[42],"equations":[43],"$d=2$":[45],"and":[46,73,110,126],"appear":[47],"to":[48,62,85,90],"their":[50],"most":[51],"natural":[52],"extension":[53],"$d>2$.":[55],"A":[56],"standard":[57],"Krylov":[58,119],"subspace":[59,120],"method":[60],"applied":[61],"such":[63],"system":[66],"suffers":[67],"curse":[70,88],"dimensionality":[72],"has":[74],"computational":[76,129],"cost":[77,130],"that":[78,92,131],"grows":[79,132],"exponentially":[80],"d.":[82,135],"key":[84],"breaking":[86],"note":[91],"often":[96],"very":[98],"well":[99],"approximated":[100],"by":[101],"vector":[103],"low":[105],"rank.":[107],"We":[108],"propose":[109],"analyze":[111],"new":[113],"class":[114],"methods,":[116,121],"so-called":[117],"which":[122],"exploit":[123],"this":[124],"fact":[125],"attain":[127],"linearly":[133]},"counts_by_year":[{"year":2025,"cited_by_count":11},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":13},{"year":2021,"cited_by_count":21},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":14},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":10},{"year":2015,"cited_by_count":16},{"year":2014,"cited_by_count":16},{"year":2013,"cited_by_count":9},{"year":2012,"cited_by_count":10}],"updated_date":"2026-07-31T08:31:51.225901","created_date":"2025-10-10T00:00:00"}
