{"id":"https://openalex.org/W2051456448","doi":"https://doi.org/10.1137/140976546","title":"Time Integration of Tensor Trains","display_name":"Time Integration of Tensor Trains","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2051456448","doi":"https://doi.org/10.1137/140976546","mag":"2051456448"},"language":"en","primary_location":{"id":"doi:10.1137/140976546","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140976546","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1407.2042","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Christian Lubich","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Christian Lubich","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Ivan V. Oseledets","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ivan V. Oseledets","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Bart Vandereycken","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bart Vandereycken","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.7121,"has_fulltext":false,"cited_by_count":171,"citation_normalized_percentile":{"value":0.90449438,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"53","issue":"2","first_page":"917","last_page":"941"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.7515000104904175,"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":0.7515000104904175,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.09910000115633011,"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/T11416","display_name":"Numerical methods for differential equations","score":0.050999999046325684,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/tensor-contraction","display_name":"Tensor contraction","score":0.7116000056266785},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.698199987411499},{"id":"https://openalex.org/keywords/integrator","display_name":"Integrator","score":0.6302000284194946},{"id":"https://openalex.org/keywords/tensor-product","display_name":"Tensor product","score":0.6011999845504761},{"id":"https://openalex.org/keywords/tensor-density","display_name":"Tensor density","score":0.5669000148773193},{"id":"https://openalex.org/keywords/tensor-product-of-hilbert-spaces","display_name":"Tensor product of Hilbert spaces","score":0.5480999946594238},{"id":"https://openalex.org/keywords/cartesian-tensor","display_name":"Cartesian tensor","score":0.5364000201225281},{"id":"https://openalex.org/keywords/symmetric-tensor","display_name":"Symmetric tensor","score":0.5088000297546387},{"id":"https://openalex.org/keywords/tensor-field","display_name":"Tensor field","score":0.4846999943256378},{"id":"https://openalex.org/keywords/projector","display_name":"Projector","score":0.453900009393692}],"concepts":[{"id":"https://openalex.org/C124007464","wikidata":"https://www.wikidata.org/wiki/Q428091","display_name":"Tensor contraction","level":3,"score":0.7116000056266785},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.698199987411499},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6665999889373779},{"id":"https://openalex.org/C79518650","wikidata":"https://www.wikidata.org/wiki/Q2081431","display_name":"Integrator","level":3,"score":0.6302000284194946},{"id":"https://openalex.org/C51255310","wikidata":"https://www.wikidata.org/wiki/Q1163016","display_name":"Tensor product","level":2,"score":0.6011999845504761},{"id":"https://openalex.org/C148125525","wikidata":"https://www.wikidata.org/wiki/Q904927","display_name":"Tensor density","level":4,"score":0.5669000148773193},{"id":"https://openalex.org/C93776189","wikidata":"https://www.wikidata.org/wiki/Q1404488","display_name":"Tensor product of Hilbert spaces","level":4,"score":0.5480999946594238},{"id":"https://openalex.org/C64835786","wikidata":"https://www.wikidata.org/wiki/Q17004583","display_name":"Cartesian tensor","level":5,"score":0.5364000201225281},{"id":"https://openalex.org/C20178491","wikidata":"https://www.wikidata.org/wiki/Q2204117","display_name":"Symmetric tensor","level":3,"score":0.5088000297546387},{"id":"https://openalex.org/C166077713","wikidata":"https://www.wikidata.org/wiki/Q1758924","display_name":"Tensor field","level":3,"score":0.4846999943256378},{"id":"https://openalex.org/C2776865275","wikidata":"https://www.wikidata.org/wiki/Q311666","display_name":"Projector","level":2,"score":0.453900009393692},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4426000118255615},{"id":"https://openalex.org/C2778351468","wikidata":"https://www.wikidata.org/wiki/Q17149634","display_name":"Matrix product state","level":4,"score":0.41940000653266907},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.4124999940395355},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.3653999865055084},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.3560999929904938},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.3522000014781952},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.33570000529289246},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.33219999074935913},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3125999867916107},{"id":"https://openalex.org/C47890412","wikidata":"https://www.wikidata.org/wiki/Q1179296","display_name":"Tangent vector","level":3,"score":0.3021000027656555},{"id":"https://openalex.org/C90952326","wikidata":"https://www.wikidata.org/wiki/Q3650511","display_name":"Tensor calculus","level":4,"score":0.29420000314712524},{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.29019999504089355},{"id":"https://openalex.org/C72434380","wikidata":"https://www.wikidata.org/wiki/Q230930","display_name":"State space","level":2,"score":0.2847000062465668},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C14440778","wikidata":"https://www.wikidata.org/wiki/Q550929","display_name":"Einstein tensor","level":4,"score":0.2669999897480011},{"id":"https://openalex.org/C126412783","wikidata":"https://www.wikidata.org/wiki/Q15104431","display_name":"Tensor operator","level":3,"score":0.2660999894142151},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C34683368","wikidata":"https://www.wikidata.org/wiki/Q6483803","display_name":"Lanczos tensor","level":5,"score":0.258899986743927},{"id":"https://openalex.org/C82927061","wikidata":"https://www.wikidata.org/wiki/Q178192","display_name":"Cross product","level":2,"score":0.25189998745918274},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.2513999938964844}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/140976546","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140976546","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1407.2042","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1407.2042","pdf_url":"https://arxiv.org/pdf/1407.2042","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1407.2042","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1407.2042","pdf_url":"https://arxiv.org/pdf/1407.2042","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1966702069","https://openalex.org/W1967859589","https://openalex.org/W1968119930","https://openalex.org/W1976837932","https://openalex.org/W1984347942","https://openalex.org/W1993482030","https://openalex.org/W1995406764","https://openalex.org/W1998126383","https://openalex.org/W2024165284","https://openalex.org/W2025267249","https://openalex.org/W2026528629","https://openalex.org/W2031216664","https://openalex.org/W2036133196","https://openalex.org/W2042308234","https://openalex.org/W2056404710","https://openalex.org/W2067067855","https://openalex.org/W2084995257","https://openalex.org/W2088522763","https://openalex.org/W2096434207","https://openalex.org/W2122816100","https://openalex.org/W2125410420","https://openalex.org/W2130727150","https://openalex.org/W2133512121","https://openalex.org/W2152279941","https://openalex.org/W4211077197"],"related_works":[],"abstract_inverted_index":{"A":[0],"robust":[1],"and":[2,55,73,82,91],"efficient":[3],"time":[4],"integrator":[5,79],"for":[6,42,56],"dynamical":[7],"tensor":[8,12,35,50,63,97],"approximation":[9],"in":[10,46,95],"the":[11,27,30,34,47,77,96],"train":[13,98],"or":[14],"matrix":[15],"product":[16,52],"state":[17,53],"format":[18,54,99],"is":[19,23],"presented.":[20],"The":[21,37,70],"method":[22],"based":[24],"on":[25],"splitting":[26],"projector":[28],"onto":[29],"tangent":[31],"space":[32],"of":[33,76],"manifold.":[36],"algorithm":[38],"can":[39],"be":[40],"used":[41],"updating":[43],"time-dependent":[44],"tensors":[45],"given":[48],"data-sparse":[49,68],"train/matrix":[51],"computing":[57],"an":[58],"approximate":[59],"solution":[60],"to":[61],"high-dimensional":[62],"differential":[64],"equations":[65],"within":[66],"this":[67],"format.":[69],"formulation,":[71],"implementation,":[72],"theoretical":[74],"properties":[75],"proposed":[78],"are":[80,100],"studied,":[81],"numerical":[83],"experiments":[84],"with":[85,92],"problems":[86],"from":[87],"quantum":[88],"molecular":[89],"dynamics":[90],"iterative":[93],"processes":[94],"included.":[101]},"counts_by_year":[{"year":2026,"cited_by_count":15},{"year":2025,"cited_by_count":24},{"year":2024,"cited_by_count":25},{"year":2023,"cited_by_count":19},{"year":2022,"cited_by_count":17},{"year":2021,"cited_by_count":23},{"year":2020,"cited_by_count":13},{"year":2019,"cited_by_count":12},{"year":2018,"cited_by_count":13},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":2}],"updated_date":"2026-04-18T07:56:08.524223","created_date":"2016-06-24T00:00:00"}
