{"id":"https://openalex.org/W2085409527","doi":"https://doi.org/10.1137/050629252","title":"Space-Time Approximation with Sparse Grids","display_name":"Space-Time Approximation with Sparse Grids","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W2085409527","doi":"https://doi.org/10.1137/050629252","mag":"2085409527"},"language":"en","primary_location":{"id":"doi:10.1137/050629252","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050629252","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/A5028074195","display_name":"Michael Griebel","orcid":"https://orcid.org/0000-0002-3147-6680"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michael Griebel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030030665","display_name":"Daniel Oeltz","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Daniel Oeltz","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5027636165","display_name":"Panayot S. Vassilevski","orcid":"https://orcid.org/0000-0002-5294-7242"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Panayot Vassilevski","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.6637,"has_fulltext":false,"cited_by_count":25,"citation_normalized_percentile":{"value":0.84049163,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"28","issue":"2","first_page":"701","last_page":"727"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9998000264167786,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9998000264167786,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9994999766349792,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9922999739646912,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7451223134994507},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6885825395584106},{"id":"https://openalex.org/keywords/sparse-grid","display_name":"Sparse grid","score":0.6772566437721252},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.644575834274292},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5356352925300598},{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.5054590702056885},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.4900764226913452},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.4640549421310425},{"id":"https://openalex.org/keywords/degrees-of-freedom","display_name":"Degrees of freedom (physics and chemistry)","score":0.4461916387081146},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.44450968503952026},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42976075410842896},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.415374219417572},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3597054183483124},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.21344873309135437},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18131119012832642},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.16933465003967285},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.13347142934799194}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7451223134994507},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6885825395584106},{"id":"https://openalex.org/C156439662","wikidata":"https://www.wikidata.org/wiki/Q7573793","display_name":"Sparse grid","level":2,"score":0.6772566437721252},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.644575834274292},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5356352925300598},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.5054590702056885},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.4900764226913452},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.4640549421310425},{"id":"https://openalex.org/C208081375","wikidata":"https://www.wikidata.org/wiki/Q274502","display_name":"Degrees of freedom (physics and chemistry)","level":2,"score":0.4461916387081146},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.44450968503952026},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42976075410842896},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.415374219417572},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3597054183483124},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.21344873309135437},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18131119012832642},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.16933465003967285},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.13347142934799194},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C502989409","wikidata":"https://www.wikidata.org/wiki/Q11425","display_name":"Animation","level":2,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/050629252","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050629252","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"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.7400000095367432}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":43,"referenced_works":["https://openalex.org/W20389081","https://openalex.org/W207899519","https://openalex.org/W600139552","https://openalex.org/W1524015806","https://openalex.org/W1528500042","https://openalex.org/W1538588718","https://openalex.org/W1585070348","https://openalex.org/W1646309082","https://openalex.org/W1670124440","https://openalex.org/W1966348919","https://openalex.org/W1970350533","https://openalex.org/W1981476348","https://openalex.org/W1989594010","https://openalex.org/W1989951672","https://openalex.org/W1995214178","https://openalex.org/W2004261587","https://openalex.org/W2012909180","https://openalex.org/W2015740317","https://openalex.org/W2020527870","https://openalex.org/W2020919250","https://openalex.org/W2021382991","https://openalex.org/W2034167241","https://openalex.org/W2041523977","https://openalex.org/W2042587009","https://openalex.org/W2043997216","https://openalex.org/W2046680696","https://openalex.org/W2052806039","https://openalex.org/W2057331906","https://openalex.org/W2075536798","https://openalex.org/W2091465347","https://openalex.org/W2098914003","https://openalex.org/W2106088594","https://openalex.org/W2115566547","https://openalex.org/W2142304406","https://openalex.org/W2247100393","https://openalex.org/W2369386221","https://openalex.org/W2485790164","https://openalex.org/W2934526567","https://openalex.org/W3143214667","https://openalex.org/W3147515309","https://openalex.org/W3161027129","https://openalex.org/W4205634890","https://openalex.org/W4255975175"],"related_works":["https://openalex.org/W2077078424","https://openalex.org/W2182328658","https://openalex.org/W1519938374","https://openalex.org/W84272617","https://openalex.org/W613282331","https://openalex.org/W2216935333","https://openalex.org/W843765170","https://openalex.org/W2161759000","https://openalex.org/W2809041471","https://openalex.org/W2361920089"],"abstract_inverted_index":{"In":[0],"this":[1],"article":[2],"we":[3,160,175,194,214,225],"introduce":[4],"approximation":[5,11,89,106,120],"spaces,":[6],"especially":[7],"suited":[8,109],"for":[9,66,76,110,131,220,249],"the":[10,21,52,77,80,87,119,123,162,182,185,189,200,221,239,243],"of":[12,14,25,51,55,61,74,79,101,125,188,242,246,263],"solutions":[13],"parabolic":[15,112],"problems,":[16],"which":[17,97,224],"are":[18,266],"based":[19],"on":[20,238],"tensor":[22],"product":[23],"construction":[24],"a":[26,32,48,146,166,208],"multiscale":[27,33,134,143],"basis":[28,34,164,171,219],"in":[29,35,228],"space":[30,54],"and":[31,113,122,203,261],"time.":[36],"Proper":[37],"truncation":[38],"then":[39,137,226],"leads":[40],"to":[41,140,150,153,268],"so-called":[42],"space-time":[43,82,93,129,230,251],"sparse":[44,127,231],"grid":[45,128,178,205,232],"spaces.":[46],"For":[47],"uniform":[49],"discretization":[50,78],"spatial":[53,157,168,222,255],"dimension":[56,124],"d":[57,67],"with":[58,253],"O(Nd)":[59,72],"degrees":[60,73,100],"freedom,":[62],"these":[63,105,126,247],"spaces":[64,96,107,130,248],"involve":[65],">":[68],"1":[69],"also":[70,155,265],"only":[71],"freedom":[75],"whole":[81],"problem.":[83],"But":[84],"they":[85],"provide":[86],"same":[88],"rate":[90,241],"as":[91,172],"classical":[92],"finite":[94,169],"element":[95,170],"need":[98],"O(Nd+1)":[99],"freedoms.":[102],"This":[103,212],"makes":[104],"well":[108],"conventional":[111],"time-dependent":[114],"optimization":[115],"problems.":[116],"We":[117,136,234],"analyze":[118],"properties":[121],"general":[132],"stable":[133],"bases.":[135],"restrict":[138],"ourselves":[139],"an":[141,216],"interpolatory":[142,196],"basis,":[144],"i.e.,":[145],"hierarchical":[147,163,218],"basis.":[148],"Here,":[149],"be":[151],"able":[152],"handle":[154],"complicated":[156],"domains":[158],"Omega,":[159],"construct":[161],"from":[165],"given":[167],"follows:":[173],"First":[174],"determine":[176],"coarse":[177,202],"points":[179,206],"recursively":[180],"over":[181],"levels":[183],"by":[184,207],"coarsening":[186],"step":[187],"algebraic":[190,217],"multigrid":[191],"method.":[192],"Then,":[193],"derive":[195],"prolongation":[197],"operators":[198],"between":[199],"respective":[201],"fine":[204],"least":[209],"squares":[210],"approach.":[211,233],"way":[213],"obtain":[215],"domain":[223],"use":[227],"our":[229],"give":[235],"numerical":[236],"results":[237],"convergence":[240],"interpolation":[244],"error":[245],"various":[250],"problems":[252],"two":[254],"dimensions.":[256],"Implementational":[257],"issues,":[258],"data":[259],"structures,":[260],"questions":[262],"adaptivity":[264],"addressed":[267],"some":[269],"extent.":[270]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
