{"id":"https://openalex.org/W2767138068","doi":"https://doi.org/10.1137/16m1082330","title":"Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study","display_name":"Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2767138068","doi":"https://doi.org/10.1137/16m1082330","mag":"2767138068"},"language":"en","primary_location":{"id":"doi:10.1137/16m1082330","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1082330","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":"https://www.osti.gov/biblio/1808764","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057085539","display_name":"Robert D. Falgout","orcid":"https://orcid.org/0000-0003-4884-0087"},"institutions":[{"id":"https://openalex.org/I1282311441","display_name":"Lawrence Livermore National Laboratory","ror":"https://ror.org/041nk4h53","country_code":"US","type":"facility","lineage":["https://openalex.org/I1282311441","https://openalex.org/I1330989302","https://openalex.org/I198811213","https://openalex.org/I4210138311"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"R. D. Falgout","raw_affiliation_strings":["Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)"],"affiliations":[{"raw_affiliation_string":"Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)","institution_ids":["https://openalex.org/I1282311441"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008694561","display_name":"Thomas A. Manteuffel","orcid":"https://orcid.org/0000-0002-8897-4146"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"T. A. Manteuffel","raw_affiliation_strings":["Univ. of Colorado, Boulder, CO (United States)"],"affiliations":[{"raw_affiliation_string":"Univ. of Colorado, Boulder, CO (United States)","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044543218","display_name":"B. O'Neill","orcid":"https://orcid.org/0000-0002-8165-5839"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"B. O'Neill","raw_affiliation_strings":["Univ. of Colorado, Boulder, CO (United States)"],"affiliations":[{"raw_affiliation_string":"Univ. of Colorado, Boulder, CO (United States)","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5038235793","display_name":"Jacob B. Schroder","orcid":"https://orcid.org/0000-0002-1076-9206"},"institutions":[{"id":"https://openalex.org/I1282311441","display_name":"Lawrence Livermore National Laboratory","ror":"https://ror.org/041nk4h53","country_code":"US","type":"facility","lineage":["https://openalex.org/I1282311441","https://openalex.org/I1330989302","https://openalex.org/I198811213","https://openalex.org/I4210138311"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J. B. Schroder","raw_affiliation_strings":["Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)"],"affiliations":[{"raw_affiliation_string":"Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)","institution_ids":["https://openalex.org/I1282311441"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5057085539"],"corresponding_institution_ids":["https://openalex.org/I1282311441"],"apc_list":null,"apc_paid":null,"fwci":5.9262,"has_fulltext":false,"cited_by_count":52,"citation_normalized_percentile":{"value":0.97097242,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"39","issue":"5","first_page":"S298","last_page":"S322"},"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":0.9997000098228455,"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":0.9997000098228455,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9959999918937683,"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"}},{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9948999881744385,"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/multigrid-method","display_name":"Multigrid method","score":0.9043901562690735},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.7325970530509949},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6177799701690674},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.5324172973632812},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4958537518978119},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.46474382281303406},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.45202872157096863},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.44198840856552124},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.4370556175708771},{"id":"https://openalex.org/keywords/speedup","display_name":"Speedup","score":0.41311365365982056},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3962269425392151},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.31726741790771484},{"id":"https://openalex.org/keywords/parallel-computing","display_name":"Parallel computing","score":0.2834165692329407},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11803299188613892},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0951131284236908}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.9043901562690735},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.7325970530509949},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6177799701690674},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.5324172973632812},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4958537518978119},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46474382281303406},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.45202872157096863},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.44198840856552124},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.4370556175708771},{"id":"https://openalex.org/C68339613","wikidata":"https://www.wikidata.org/wiki/Q1549489","display_name":"Speedup","level":2,"score":0.41311365365982056},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3962269425392151},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.31726741790771484},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.2834165692329407},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11803299188613892},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0951131284236908},{"id":"https://openalex.org/C205649164","wikidata":"https://www.wikidata.org/wiki/Q1071","display_name":"Geography","level":0,"score":0.0},{"id":"https://openalex.org/C13280743","wikidata":"https://www.wikidata.org/wiki/Q131089","display_name":"Geodesy","level":1,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/16m1082330","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1082330","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:osti.gov:1808764","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/1808764","pdf_url":null,"source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"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":null}],"best_oa_location":{"id":"pmh:oai:osti.gov:1808764","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/1808764","pdf_url":null,"source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"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":null},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3629768340","display_name":null,"funder_award_id":"DE-NA0002376","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G5400081543","display_name":null,"funder_award_id":"LLNL-JRNL-692258","funder_id":"https://openalex.org/F4320338286","funder_display_name":"Lawrence Livermore National Laboratory"},{"id":"https://openalex.org/G797722205","display_name":null,"funder_award_id":"B600360","funder_id":"https://openalex.org/F4320338286","funder_display_name":"Lawrence Livermore National Laboratory"},{"id":"https://openalex.org/G8845741795","display_name":null,"funder_award_id":"DE-FC02-03ER25574","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"}],"funders":[{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320338286","display_name":"Lawrence Livermore National Laboratory","ror":"https://ror.org/041nk4h53"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W190449063","https://openalex.org/W1501940787","https://openalex.org/W1510003436","https://openalex.org/W1820706094","https://openalex.org/W1826824261","https://openalex.org/W1977424646","https://openalex.org/W1987455271","https://openalex.org/W1989707864","https://openalex.org/W1991531168","https://openalex.org/W1995664184","https://openalex.org/W2006910694","https://openalex.org/W2012630580","https://openalex.org/W2013561596","https://openalex.org/W2018943560","https://openalex.org/W2022396161","https://openalex.org/W2027491117","https://openalex.org/W2047496230","https://openalex.org/W2065884217","https://openalex.org/W2070139122","https://openalex.org/W2072572259","https://openalex.org/W2082146198","https://openalex.org/W2105970221","https://openalex.org/W2123426570","https://openalex.org/W2140489561","https://openalex.org/W2151317657","https://openalex.org/W2188157585","https://openalex.org/W2210625349","https://openalex.org/W2328262485","https://openalex.org/W2336372639","https://openalex.org/W2594128612","https://openalex.org/W3102123643"],"related_works":["https://openalex.org/W2058965144","https://openalex.org/W2164382479","https://openalex.org/W2146343568","https://openalex.org/W2573021234","https://openalex.org/W2106610492","https://openalex.org/W2383355899","https://openalex.org/W1524119935","https://openalex.org/W3011358647","https://openalex.org/W2155760131","https://openalex.org/W2161759000"],"abstract_inverted_index":{"The":[0,90,119,148],"need":[1],"for":[2,33,57,80,192,216],"parallelism":[3,41,59],"in":[4,13,42,48],"the":[5,43,96,124,139,154,167,217,241],"time":[6,35,44,49,71,164],"dimension":[7],"is":[8,51,95,106,123,152],"being":[9],"driven":[10],"by":[11,60],"changes":[12],"computer":[14],"architectures,":[15],"where":[16,104,126],"performance":[17,111],"increases":[18],"are":[19],"now":[20],"provided":[21],"through":[22],"greater":[23],"concurrency,":[24],"not":[25],"faster":[26],"clock":[27],"speeds.":[28],"This":[29],"creates":[30],"a":[31,66,130,193,203],"bottleneck":[32],"sequential":[34],"marching":[36],"schemes":[37],"because":[38],"they":[39],"lack":[40],"dimension.":[45],"Multigrid":[46],"reduction":[47,63],"(MGRIT)":[50],"an":[52,115,180,189,212],"iterative":[53],"procedure":[54],"that":[55,153,198,225],"allows":[56],"temporal":[58],"utilizing":[61],"multigrid":[62,175],"techniques":[64],"and":[65,135,208,221],"multilevel":[67],"hierarchy":[68],"of":[69,85,92,99,183,205],"coarse":[70],"grids.":[72],"MGRIT":[73,206],"has":[74,177],"been":[75],"shown":[76],"to":[77,87,114,129,138,187,232,240],"be":[78,233],"effective":[79],"linear":[81,117,141,146,243],"problems,":[82],"with":[83,102,235],"speedups":[84],"up":[86],"50":[88],"times.":[89],"goal":[91],"this":[93,200],"work":[94],"efficient":[97],"solution":[98],"nonlinear":[100,121,132,155,218,229],"problems":[101,231],"MGRIT,":[103],"efficiency":[105],"defined":[107],"as":[108,166],"achieving":[109],"similar":[110],"when":[112],"compared":[113],"equivalent":[116],"problem.":[118,147,244],"benchmark":[120,145],"problem":[122,195],"$p$-Laplacian,":[125],"p=4":[127],"corresponds":[128,137],"well-known":[131],"diffusion":[133,142],"equation":[134],"$p=2$":[136],"standard":[140],"operator,":[143],"our":[144],"key":[149],"difficulty":[150],"encountered":[151],"time-step":[156,168,219],"solver":[157,191,220],"becomes":[158],"progressively":[159],"more":[160],"expensive":[161],"on":[162],"coarser":[163],"levels":[165],"size":[169],"increases.":[170],"To":[171,197],"overcome":[172],"such":[173],"difficulties,":[174],"research":[176],"historically":[178],"targeted":[179],"accumulated":[181],"body":[182],"experience":[184],"regarding":[185],"how":[186],"choose":[188],"appropriate":[190],"specific":[194],"type.":[196],"end,":[199],"paper":[201],"develops":[202],"library":[204],"optimizations":[207],"modifications,":[209],"most":[210],"important":[211],"alternate":[213],"initial":[214],"guess":[215],"delayed":[222],"spatial":[223],"coarsening,":[224],"will":[226],"allow":[227],"many":[228],"parabolic":[230],"solved":[234],"parallel":[236],"scaling":[237],"behavior":[238],"comparable":[239],"corresponding":[242]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":8}],"updated_date":"2026-04-14T08:04:32.555800","created_date":"2025-10-10T00:00:00"}