{"id":"https://openalex.org/W2216935333","doi":"https://doi.org/10.1137/140974985","title":"A Multigrid Method for Adaptive Sparse Grids","display_name":"A Multigrid Method for Adaptive Sparse Grids","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2216935333","doi":"https://doi.org/10.1137/140974985","mag":"2216935333"},"language":"en","primary_location":{"id":"doi:10.1137/140974985","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140974985","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":true,"oa_status":"green","oa_url":"http://hdl.handle.net/1721.1/100938","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027402421","display_name":"Benjamin Peherstorfer","orcid":"https://orcid.org/0000-0002-1558-6775"},"institutions":[{"id":"https://openalex.org/I46020346","display_name":"American Institute of Aeronautics and Astronautics","ror":"https://ror.org/00a1rzv11","country_code":"US","type":"other","lineage":["https://openalex.org/I46020346"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Benjamin Peherstorfer","raw_affiliation_strings":["Department of Aeronautics and Astronautics, MIT, Cambridge, MA 02139"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Aeronautics and Astronautics, MIT, Cambridge, MA 02139","institution_ids":["https://openalex.org/I46020346"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005825055","display_name":"Stefan Zimmer","orcid":"https://orcid.org/0000-0001-6697-2548"},"institutions":[{"id":"https://openalex.org/I100066346","display_name":"University of Stuttgart","ror":"https://ror.org/04vnq7t77","country_code":"DE","type":"education","lineage":["https://openalex.org/I100066346"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Stefan Zimmer","raw_affiliation_strings":["Institute for Parallel and Distributed Systems, University of Stuttgart, 70569 Stuttgart, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Parallel and Distributed Systems, University of Stuttgart, 70569 Stuttgart, Germany","institution_ids":["https://openalex.org/I100066346"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108361959","display_name":"Christoph Zenger","orcid":null},"institutions":[{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Christoph Zenger","raw_affiliation_strings":["Department of Informatics, Technische Universit\u00e4t M\u00fcnchen, 85748 Garching, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Informatics, Technische Universit\u00e4t M\u00fcnchen, 85748 Garching, Germany","institution_ids":["https://openalex.org/I62916508"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5068403289","display_name":"Hans\u2010Joachim Bungartz","orcid":"https://orcid.org/0000-0002-0171-0712"},"institutions":[{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Hans-Joachim Bungartz","raw_affiliation_strings":["Department of Informatics, Technische Universit\u00e4t M\u00fcnchen, 85748 Garching, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Informatics, Technische Universit\u00e4t M\u00fcnchen, 85748 Garching, Germany","institution_ids":["https://openalex.org/I62916508"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.67,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.71457017,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"37","issue":"5","first_page":"S51","last_page":"S70"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"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":0.9998999834060669,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.998199999332428,"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9975000023841858,"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/multigrid-method","display_name":"Multigrid method","score":0.9294847249984741},{"id":"https://openalex.org/keywords/sparse-grid","display_name":"Sparse grid","score":0.9282700419425964},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.6495636701583862},{"id":"https://openalex.org/keywords/degrees-of-freedom","display_name":"Degrees of freedom (physics and chemistry)","score":0.6334512233734131},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6019582748413086},{"id":"https://openalex.org/keywords/sparse-approximation","display_name":"Sparse approximation","score":0.5986616611480713},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.542984664440155},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.46993502974510193},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.46478983759880066},{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.4477687180042267},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.44310665130615234},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4403541088104248},{"id":"https://openalex.org/keywords/sparse-matrix","display_name":"Sparse matrix","score":0.43961039185523987},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.4314730763435364},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.42653876543045044},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3703739643096924},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.2102208137512207},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.19504699110984802},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15811505913734436}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.9294847249984741},{"id":"https://openalex.org/C156439662","wikidata":"https://www.wikidata.org/wiki/Q7573793","display_name":"Sparse grid","level":2,"score":0.9282700419425964},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.6495636701583862},{"id":"https://openalex.org/C208081375","wikidata":"https://www.wikidata.org/wiki/Q274502","display_name":"Degrees of freedom (physics and chemistry)","level":2,"score":0.6334512233734131},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6019582748413086},{"id":"https://openalex.org/C124066611","wikidata":"https://www.wikidata.org/wiki/Q28684319","display_name":"Sparse approximation","level":2,"score":0.5986616611480713},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.542984664440155},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.46993502974510193},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.46478983759880066},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.4477687180042267},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.44310665130615234},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4403541088104248},{"id":"https://openalex.org/C56372850","wikidata":"https://www.wikidata.org/wiki/Q1050404","display_name":"Sparse matrix","level":3,"score":0.43961039185523987},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.4314730763435364},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.42653876543045044},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3703739643096924},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.2102208137512207},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.19504699110984802},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15811505913734436},{"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/140974985","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140974985","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"},{"id":"pmh:oai:dspace.mit.edu:1721.1/100938","is_oa":true,"landing_page_url":"http://hdl.handle.net/1721.1/100938","pdf_url":null,"source":{"id":"https://openalex.org/S4306400425","display_name":"DSpace@MIT (Massachusetts Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I63966007","host_organization_name":"Massachusetts Institute of Technology","host_organization_lineage":["https://openalex.org/I63966007"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Society for Industrial and Applied Mathematics","raw_type":"http://purl.org/eprint/type/JournalArticle"},{"id":"pmh:oai:informatik.uni-stuttgart.de:ART-2015-14","is_oa":false,"landing_page_url":"http://www2.informatik.uni-stuttgart.de/cgi-bin/NCSTRL/NCSTRL_view.pl?id=ART-2015-14&engl=1","pdf_url":null,"source":{"id":"https://openalex.org/S4306401306","display_name":"Fachbereich Informatik (University of Stuttgart)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I100066346","host_organization_name":"University of Stuttgart","host_organization_lineage":["https://openalex.org/I100066346"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"\\n            In: SIAM Journal on Scientific Computing. Vol. 37(5), pp. 51-70\\n          ","raw_type":"Text"},{"id":"pmh:oai:mediatum.ub.tum.de:node/1418876","is_oa":true,"landing_page_url":"http://mediatum.ub.tum.de/node?id=1418876","pdf_url":null,"source":{"id":"https://openalex.org/S4377196330","display_name":"mediaTUM  (Technical University of Munich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I62916508","host_organization_name":"Technical University of Munich","host_organization_lineage":["https://openalex.org/I62916508"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:dspace.mit.edu:1721.1/100938","is_oa":true,"landing_page_url":"http://hdl.handle.net/1721.1/100938","pdf_url":null,"source":{"id":"https://openalex.org/S4306400425","display_name":"DSpace@MIT (Massachusetts Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I63966007","host_organization_name":"Massachusetts Institute of Technology","host_organization_lineage":["https://openalex.org/I63966007"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Society for Industrial and Applied Mathematics","raw_type":"http://purl.org/eprint/type/JournalArticle"},"sustainable_development_goals":[{"score":0.8299999833106995,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W22984837","https://openalex.org/W79383899","https://openalex.org/W1601540052","https://openalex.org/W1717419277","https://openalex.org/W1800345484","https://openalex.org/W1974963628","https://openalex.org/W1992068214","https://openalex.org/W1999389926","https://openalex.org/W2002274796","https://openalex.org/W2010452605","https://openalex.org/W2015740317","https://openalex.org/W2029191491","https://openalex.org/W2031336791","https://openalex.org/W2034736441","https://openalex.org/W2035450252","https://openalex.org/W2039317494","https://openalex.org/W2054314432","https://openalex.org/W2057331906","https://openalex.org/W2065021724","https://openalex.org/W2132851968","https://openalex.org/W2146766088","https://openalex.org/W2151043893","https://openalex.org/W2169059480","https://openalex.org/W2394917699","https://openalex.org/W2403090514","https://openalex.org/W2749942571","https://openalex.org/W2753336384","https://openalex.org/W4248381180","https://openalex.org/W4255975175"],"related_works":["https://openalex.org/W2216935333","https://openalex.org/W2011091376","https://openalex.org/W2487261884","https://openalex.org/W2091883426","https://openalex.org/W2174948646","https://openalex.org/W3173235360","https://openalex.org/W2024017047","https://openalex.org/W4318256793","https://openalex.org/W2594370889","https://openalex.org/W2390720471"],"abstract_inverted_index":{"Sparse":[0],"grids":[1,158],"have":[2],"become":[3],"an":[4,35,72],"important":[5],"tool":[6],"to":[7,70,104],"reduce":[8],"the":[9,24,32,50,66,79,94,98,115,144,161,169,179,187],"number":[10,80,170],"of":[11,13,15,17,28,34,42,49,81,93,97,143,163,171,186],"degrees":[12,27],"freedom":[14,29],"discretizations":[16],"moderately":[18],"high-dimensional":[19],"partial":[20],"differential":[21],"equations;":[22],"however,":[23],"reduction":[25],"in":[26,107,121,136],"comes":[30],"at":[31],"cost":[33],"almost":[36],"dense":[37],"and":[38,154,175],"unconventionally":[39],"structured":[40],"system":[41],"linear":[43,55],"equations.":[44],"To":[45],"guarantee":[46],"overall":[47],"efficiency":[48],"sparse":[51,67,82,147,157,172],"grid":[52,68,83,100,148,173],"approach,":[53],"special":[54],"solvers":[56],"are":[57,183],"required.":[58],"We":[59],"present":[60],"a":[61,90,105,122],"multigrid":[62,132],"method":[63,133,165],"that":[64,75,102,160,178],"exploits":[65],"structure":[69],"achieve":[71],"optimal":[73],"runtime":[74,162],"scales":[76,166],"linearly":[77,167],"with":[78,168],"points.":[84],"Our":[85,130],"approach":[86],"is":[87],"based":[88],"on":[89,126,152],"novel":[91],"decomposition":[92],"right-hand":[95,116],"sides":[96,117],"coarse":[99],"equations":[101],"leads":[103],"reformulation":[106],"so-called":[108],"auxiliary":[109,113,138],"coefficients.":[110],"With":[111],"these":[112],"coefficients,":[114],"can":[118],"be":[119],"represented":[120],"nodal":[123],"point":[124],"basis":[125],"low-dimensional":[127],"full":[128],"grids.":[129],"proposed":[131],"directly":[134],"operates":[135],"this":[137],"coefficient":[139],"representation,":[140],"circumventing":[141],"most":[142],"computationally":[145],"cumbersome":[146],"structure.":[149],"Numerical":[150],"results":[151],"nonadaptive":[153],"spatially":[155],"adaptive":[156],"confirm":[159],"our":[164],"points":[174],"they":[176],"indicate":[177],"obtained":[180],"convergence":[181],"factors":[182],"bounded":[184],"independently":[185],"mesh":[188],"width.":[189]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}