{"id":"https://openalex.org/W2022442138","doi":"https://doi.org/10.1137/s1064827598339785","title":"Multiscale Linear Solvers for Very Large Systems Derived from PDES","display_name":"Multiscale Linear Solvers for Very Large Systems Derived from PDES","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W2022442138","doi":"https://doi.org/10.1137/s1064827598339785","mag":"2022442138"},"language":"en","primary_location":{"id":"doi:10.1137/s1064827598339785","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827598339785","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/A5040701303","display_name":"Klaus S. Lackner","orcid":"https://orcid.org/0000-0001-5509-6327"},"institutions":[{"id":"https://openalex.org/I1343871089","display_name":"Los Alamos National Laboratory","ror":"https://ror.org/01e41cf67","country_code":"US","type":"facility","lineage":["https://openalex.org/I1330989302","https://openalex.org/I1343871089","https://openalex.org/I198811213","https://openalex.org/I4210120050"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Klaus Lackner","raw_affiliation_strings":["Los Alamos National Laboratory,Theoretical Division"],"affiliations":[{"raw_affiliation_string":"Los Alamos National Laboratory,Theoretical Division","institution_ids":["https://openalex.org/I1343871089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008194905","display_name":"Ralph Menikoff","orcid":"https://orcid.org/0000-0002-3338-1597"},"institutions":[{"id":"https://openalex.org/I1343871089","display_name":"Los Alamos National Laboratory","ror":"https://ror.org/01e41cf67","country_code":"US","type":"facility","lineage":["https://openalex.org/I1330989302","https://openalex.org/I1343871089","https://openalex.org/I198811213","https://openalex.org/I4210120050"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ralph Menikoff","raw_affiliation_strings":["Los Alamos National Laboratory,Theoretical Division"],"affiliations":[{"raw_affiliation_string":"Los Alamos National Laboratory,Theoretical Division","institution_ids":["https://openalex.org/I1343871089"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5040701303"],"corresponding_institution_ids":["https://openalex.org/I1343871089"],"apc_list":null,"apc_paid":null,"fwci":0.5528,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.67586996,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"21","issue":"5","first_page":"1950","last_page":"1968"},"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9998999834060669,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9980999827384949,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.8511480093002319},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.8438372015953064},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6849645376205444},{"id":"https://openalex.org/keywords/conjugate-residual-method","display_name":"Conjugate residual method","score":0.5887473225593567},{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.5277707576751709},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5164802074432373},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.4549115002155304},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.42354536056518555},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42160624265670776},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.4183313846588135},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.4149321913719177},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4138477146625519},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.30593788623809814},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2514934539794922},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18291416764259338},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.14994359016418457}],"concepts":[{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.8511480093002319},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.8438372015953064},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6849645376205444},{"id":"https://openalex.org/C91394653","wikidata":"https://www.wikidata.org/wiki/Q5161159","display_name":"Conjugate residual method","level":4,"score":0.5887473225593567},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.5277707576751709},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5164802074432373},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4549115002155304},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.42354536056518555},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42160624265670776},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.4183313846588135},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.4149321913719177},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4138477146625519},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30593788623809814},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2514934539794922},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18291416764259338},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.14994359016418457},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/s1064827598339785","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827598339785","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:CiteSeerX.psu:10.1.1.234.4355","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.4355","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/9809043v1.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1492522302","https://openalex.org/W1506342804","https://openalex.org/W1542622005","https://openalex.org/W1546004968","https://openalex.org/W1575701986","https://openalex.org/W1964747175","https://openalex.org/W1971920652","https://openalex.org/W1974385426","https://openalex.org/W2006453807","https://openalex.org/W2017085690","https://openalex.org/W2048357515"],"related_works":["https://openalex.org/W2081820376","https://openalex.org/W2091325992","https://openalex.org/W2129625121","https://openalex.org/W2375597358","https://openalex.org/W2059625160","https://openalex.org/W2356365850","https://openalex.org/W2006453807","https://openalex.org/W3194092090","https://openalex.org/W2159515580","https://openalex.org/W2980404785"],"abstract_inverted_index":{"We":[0,356],"present":[1],"a":[2,40,53,62,84,160,185,188,195,234,241,367,407],"novel":[3],"linear":[4,91,103,391],"solver":[5,208,318,360],"that":[6,43,109,180,209,301,389],"works":[7,396],"well":[8],"for":[9,20,146,233,267],"large":[10],"systems":[11],"obtained":[12,199,238],"from":[13,96,106,383],"discretizing":[14],"PDEs.":[15,166],"It":[16,93],"is":[17,37,61,83,94,172,221,230,273,300,307,340,351],"robust":[18],"and,":[19],"the":[21,25,31,57,59,69,72,75,78,90,97,100,117,124,129,133,137,147,165,168,176,181,201,211,215,219,228,249,257,264,268,271,276,280,285,289,298,302,311,315,326,333,337,352,362,390,400],"examples":[22],"we":[23,369],"studied,":[24],"computational":[26,338],"effort":[27,265,339],"scales":[28,328,335],"linearly":[29],"with":[30,67,240],"number":[32,286,353],"of":[33,71,77,89,99,128,203,263,279,287,297,314,354,381,409],"equations.":[34,355],"The":[35,81,120,141,154,190,224,244,394],"algorithm":[36,60,73,177,213,258,272,299,395],"based":[38,111],"on":[39,112,116,325,332,361,373],"wavelength":[41,126,139],"decomposition":[42],"combines":[44],"conjugate":[45,65,142,206,250,316],"gradient,":[46],"multiscaling,":[47],"and":[48,105,151,336,387],"iterative":[49],"splitting":[50,115],"methods":[51],"into":[52],"single":[54],"approach.":[55],"On":[56,366],"surface,":[58],"simple":[63],"preconditioned":[64,205],"gradient":[66,143,207,251,317],"all":[68],"sophistication":[70],"in":[74,376],"choice":[76],"preconditioning":[79],"matrix.":[80],"preconditioner":[82,182,290],"very":[85,259],"good":[86],"approximate":[87,281],"inverse":[88,98,130,171,192,282],"operator.":[92],"constructed":[95],"coarse":[101,121,155,169,191,269,305],"grained":[102,156,170],"operator":[104,114,135,157],"smoothing":[107,134],"operators":[108],"are":[110],"an":[113],"fine":[118],"grid.":[119],"graining":[122],"captures":[123,136],"long":[125,150],"behavior":[127],"operator,":[131],"while":[132],"short":[138,152],"behavior.":[140],"iteration":[144],"accounts":[145],"coupling":[148],"between":[149,304],"wavelengths.":[153],"corresponds":[158],"to":[159,164,187,194,214,236,292,342,385],"lower":[161],"resolution":[162],"approximation":[163],"While":[167],"not":[173],"known":[174],"explicitly,":[175],"requires":[178],"only":[179],"can":[183,197],"be":[184,198,237,293],"applied":[186,193],"vector.":[189],"vector":[196],"as":[200],"solution":[202,235],"another":[204],"applies":[210],"same":[212],"smaller":[216],"problem.":[217],"Thus,":[218],"method":[220],"naturally":[222],"recursive.":[223],"recursion":[225],"ends":[226],"when":[227,399],"matrix":[229],"sufficiently":[231],"small":[232],"efficiently":[239],"standard":[242],"solver.":[243],"local":[245],"feedback":[246],"provided":[247],"by":[248,310],"step":[252],"at":[253,319],"every":[254],"level":[255],"makes":[256],"robust.":[260],"In":[261],"spite":[262],"required":[266],"inverse,":[270],"efficient":[274],"because":[275],"increased":[277],"quality":[278],"greatly":[283],"reduces":[284],"times":[288],"needs":[291],"evaluated.":[294],"A":[295],"feature":[296],"transition":[303],"grids":[306,374],"determined":[308],"dynamically":[309],"accuracy":[312],"requirement":[313],"each":[320],"level.":[321],"Typically,":[322],"later":[323],"iterations":[324,331],"finer":[327],"need":[329],"fewer":[330],"coarser":[334],"proportional":[341],"N":[343,346,350],"rather":[344],"than":[345],"log":[347],"N,":[348],"where":[349],"have":[357,370],"tested":[358],"our":[359],"porous":[363],"flow":[364],"equation.":[365],"workstation":[368],"solved":[371],"problems":[372],"ranging":[375],"dimension":[377],"over":[378],"three":[379],"orders":[380],"magnitude,":[382],"103":[384],"106,":[386],"found":[388],"scaling":[392],"holds.":[393],"well,":[397],"even":[398],"permeability":[401],"tensor":[402],"has":[403],"spatial":[404],"variations":[405],"exceeding":[406],"factor":[408],"109.":[410]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}