{"id":"https://openalex.org/W1428340059","doi":"https://doi.org/10.1137/15m102887x","title":"Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method","display_name":"Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W1428340059","doi":"https://doi.org/10.1137/15m102887x","mag":"1428340059"},"language":"en","primary_location":{"id":"doi:10.1137/15m102887x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m102887x","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":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1605.04419","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024462955","display_name":"Victorita Dolean","orcid":"https://orcid.org/0000-0002-5885-1903"},"institutions":[{"id":"https://openalex.org/I4210112930","display_name":"Laboratoire Jean-Alexandre Dieudonn\u00e9","ror":"https://ror.org/0274zdr66","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I201841394","https://openalex.org/I4210112930","https://openalex.org/I4210141950"]},{"id":"https://openalex.org/I181647926","display_name":"University of Strathclyde","ror":"https://ror.org/00n3w3b69","country_code":"GB","type":"education","lineage":["https://openalex.org/I181647926"]}],"countries":["FR","GB"],"is_corresponding":true,"raw_author_name":"V. Dolean","raw_affiliation_strings":["Laboratoire Jean Alexandre Dieudonn\u00e9","University of Strathclyde [Glasgow]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire Jean Alexandre Dieudonn\u00e9","institution_ids":["https://openalex.org/I4210112930"]},{"raw_affiliation_string":"University of Strathclyde [Glasgow]","institution_ids":["https://openalex.org/I181647926"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089456729","display_name":"Martin J. Gander","orcid":"https://orcid.org/0000-0001-8450-9223"},"institutions":[{"id":"https://openalex.org/I3019441195","display_name":"Laboratoire de Math\u00e9matiques d'Orsay","ror":"https://ror.org/03ab0zs98","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I277688954","https://openalex.org/I3019441195"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"M. J. Gander","raw_affiliation_strings":["Section de math\u00e9matiques [Gen\u00e8ve]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Section de math\u00e9matiques [Gen\u00e8ve]","institution_ids":["https://openalex.org/I3019441195"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024240882","display_name":"Walid Kheriji","orcid":null},"institutions":[{"id":"https://openalex.org/I4210112930","display_name":"Laboratoire Jean-Alexandre Dieudonn\u00e9","ror":"https://ror.org/0274zdr66","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I201841394","https://openalex.org/I4210112930","https://openalex.org/I4210141950"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"W. Kheriji","raw_affiliation_strings":["Laboratoire Jean Alexandre Dieudonn\u00e9"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire Jean Alexandre Dieudonn\u00e9","institution_ids":["https://openalex.org/I4210112930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007796851","display_name":"F\u00e9lix Kwok","orcid":"https://orcid.org/0000-0002-8360-6328"},"institutions":[{"id":"https://openalex.org/I114457229","display_name":"University of Geneva","ror":"https://ror.org/01swzsf04","country_code":"CH","type":"education","lineage":["https://openalex.org/I114457229"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"F. Kwok","raw_affiliation_strings":["Universit\u00e9 de Gen\u00e8ve = University of Geneva"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 de Gen\u00e8ve = University of Geneva","institution_ids":["https://openalex.org/I114457229"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5033676643","display_name":"Roland Masson","orcid":"https://orcid.org/0009-0006-8713-3069"},"institutions":[{"id":"https://openalex.org/I4210112930","display_name":"Laboratoire Jean-Alexandre Dieudonn\u00e9","ror":"https://ror.org/0274zdr66","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I201841394","https://openalex.org/I4210112930","https://openalex.org/I4210141950"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"R. Masson","raw_affiliation_strings":["Laboratoire Jean Alexandre Dieudonn\u00e9"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire Jean Alexandre Dieudonn\u00e9","institution_ids":["https://openalex.org/I4210112930"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5024462955"],"corresponding_institution_ids":["https://openalex.org/I181647926","https://openalex.org/I4210112930"],"apc_list":null,"apc_paid":null,"fwci":4.6864,"has_fulltext":false,"cited_by_count":80,"citation_normalized_percentile":{"value":0.94809048,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"38","issue":"6","first_page":"A3357","last_page":"A3380"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.7700999975204468,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.7700999975204468,"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":0.1396999955177307,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.027400000020861626,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.9598128795623779},{"id":"https://openalex.org/keywords/schwarz-alternating-method","display_name":"Schwarz alternating method","score":0.8322222232818604},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.7807289361953735},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7057331204414368},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.597204864025116},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5913582444190979},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5911339521408081},{"id":"https://openalex.org/keywords/newtons-method-in-optimization","display_name":"Newton's method in optimization","score":0.5756327509880066},{"id":"https://openalex.org/keywords/additive-schwarz-method","display_name":"Additive Schwarz method","score":0.5693376064300537},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.5691832304000854},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4629747271537781},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.45763400197029114},{"id":"https://openalex.org/keywords/precondition","display_name":"Precondition","score":0.4205033779144287},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4014989733695984},{"id":"https://openalex.org/keywords/local-convergence","display_name":"Local convergence","score":0.3718763589859009},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.24741750955581665},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.20210248231887817},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.17973178625106812},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.1479613184928894}],"concepts":[{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.9598128795623779},{"id":"https://openalex.org/C31278502","wikidata":"https://www.wikidata.org/wiki/Q7433186","display_name":"Schwarz alternating method","level":4,"score":0.8322222232818604},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.7807289361953735},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7057331204414368},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.597204864025116},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5913582444190979},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5911339521408081},{"id":"https://openalex.org/C179127668","wikidata":"https://www.wikidata.org/wiki/Q17086396","display_name":"Newton's method in optimization","level":4,"score":0.5756327509880066},{"id":"https://openalex.org/C174819683","wikidata":"https://www.wikidata.org/wiki/Q4681340","display_name":"Additive Schwarz method","level":4,"score":0.5693376064300537},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.5691832304000854},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4629747271537781},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.45763400197029114},{"id":"https://openalex.org/C192465680","wikidata":"https://www.wikidata.org/wiki/Q1413450","display_name":"Precondition","level":2,"score":0.4205033779144287},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4014989733695984},{"id":"https://openalex.org/C117898588","wikidata":"https://www.wikidata.org/wiki/Q6664310","display_name":"Local convergence","level":3,"score":0.3718763589859009},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.24741750955581665},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.20210248231887817},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.17973178625106812},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.1479613184928894},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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":7,"locations":[{"id":"doi:10.1137/15m102887x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m102887x","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:strathprints.strath.ac.uk:59253","is_oa":false,"landing_page_url":"https://strathprints.strath.ac.uk/view/author/922517.html>","pdf_url":null,"source":{"id":"https://openalex.org/S4306402226","display_name":"Strathprints: The University of Strathclyde institutional repository (University of Strathclyde)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I181647926","host_organization_name":"University of Strathclyde","host_organization_lineage":["https://openalex.org/I181647926"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},{"id":"pmh:oai:arXiv.org:1605.04419","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1605.04419","pdf_url":"https://arxiv.org/pdf/1605.04419","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":"","raw_type":"text"},{"id":"pmh:oai:HAL:hal-01171167v1","is_oa":true,"landing_page_url":"https://hal.science/hal-01171167","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"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":"SIAM Journal on Scientific Computing, 2016","raw_type":"Journal articles"},{"id":"pmh:oai:repository.hkbu.edu.hk:hkbu_staff_publication-7933","is_oa":false,"landing_page_url":"https://repository.hkbu.edu.hk/hkbu_staff_publication/6909","pdf_url":null,"source":{"id":"https://openalex.org/S4377196872","display_name":"HKBU Institutional Repository (Hong Kong Baptist University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I141568987","host_organization_name":"Hong Kong Baptist University","host_organization_lineage":["https://openalex.org/I141568987"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"HKBU Staff Publication","raw_type":"text"},{"id":"pmh:oai:unige.ch:aou:unige:169499","is_oa":true,"landing_page_url":"https://archive-ouverte.unige.ch/unige:169499","pdf_url":null,"source":{"id":"https://openalex.org/S4306402259","display_name":"Archive ouverte UNIGE (University of Geneva)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I114457229","host_organization_name":"University of Geneva","host_organization_lineage":["https://openalex.org/I114457229"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN: 1064-8275","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.1605.04419","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1605.04419","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1605.04419","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1605.04419","pdf_url":"https://arxiv.org/pdf/1605.04419","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":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1460442162","display_name":null,"funder_award_id":"ECS/22300115","funder_id":"https://openalex.org/F4320321592","funder_display_name":"Research Grants Council, University Grants Committee"},{"id":"https://openalex.org/G4598626517","display_name":null,"funder_award_id":"11501483","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321592","display_name":"Research Grants Council, University Grants Committee","ror":"https://ror.org/00djwmt25"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W73123583","https://openalex.org/W86450251","https://openalex.org/W105972517","https://openalex.org/W593406250","https://openalex.org/W1164247298","https://openalex.org/W1482145699","https://openalex.org/W1488582317","https://openalex.org/W1530872699","https://openalex.org/W1604096666","https://openalex.org/W1961414889","https://openalex.org/W1970659509","https://openalex.org/W1972684149","https://openalex.org/W1975113089","https://openalex.org/W1980455776","https://openalex.org/W1986812736","https://openalex.org/W1995852016","https://openalex.org/W1995897527","https://openalex.org/W2005390260","https://openalex.org/W2007289139","https://openalex.org/W2009575526","https://openalex.org/W2031956460","https://openalex.org/W2046409434","https://openalex.org/W2075087492","https://openalex.org/W2082187902","https://openalex.org/W2092422489","https://openalex.org/W2122824057","https://openalex.org/W2130756427","https://openalex.org/W2151718183","https://openalex.org/W2163831147","https://openalex.org/W2549356488","https://openalex.org/W2593356793","https://openalex.org/W3142140897","https://openalex.org/W4234108706","https://openalex.org/W4248161126","https://openalex.org/W4300000299"],"related_works":["https://openalex.org/W2902154408","https://openalex.org/W2159100661","https://openalex.org/W2000290814","https://openalex.org/W2028890215","https://openalex.org/W3839471","https://openalex.org/W2034767642","https://openalex.org/W2169562629","https://openalex.org/W2182634988","https://openalex.org/W1428340059","https://openalex.org/W2953099512"],"abstract_inverted_index":{"For":[0,74],"linear":[1],"problems,":[2,56],"domain":[3,57,174],"decomposition":[4,58,175],"methods":[5,59],"can":[6,60],"be":[7,61],"used":[8,62,128],"directly":[9,64,113],"as":[10,15,65,69,129],"iterative":[11,66,117,131,149,173],"solvers":[12,67],"but":[13,109],"also":[14,53,151],"preconditioners":[16,70],"for":[17,54,71,145],"Krylov":[18,22,30],"methods.":[19],"In":[20],"practice,":[21],"acceleration":[23],"is":[24,100],"almost":[25],"always":[26],"used,":[27],"since":[28],"the":[29,39,75,79,116,121,161,192],"method":[31,176],"finds":[32],"a":[33,88,141,157,169,178,196],"much":[34,45],"better":[35,143],"residual":[36],"polynomial":[37],"than":[38],"stationary":[40],"iteration":[41],"and":[42,137,177,195],"thus":[43,139],"converges":[44,126],"faster.":[46],"We":[47,184],"show":[48,84],"in":[49,133],"this":[50],"paper":[51],"that":[52,85,123],"nonlinear":[55,172,182,197],"either":[63],"or":[68],"Newton's":[72,146],"method.":[73,118,147],"concrete":[76],"case":[77],"of":[78],"parallel":[80],"Schwarz":[81,95,105],"method,":[82],"we":[83,86,90,138],"obtain":[87],"preconditioner":[89,144],"call":[91],"RASPEN":[92,124,181],"(restricted":[93],"additive":[94],"preconditioned":[96,106],"exact":[97],"Newton),":[98],"which":[99,166],"similar":[101],"to":[102,135,154,168],"ASPIN":[103],"(additive":[104],"inexact":[107],"Newton)":[108],"with":[110,188],"all":[111],"components":[112],"defined":[114],"by":[115],"This":[119],"has":[120],"advantage":[122],"already":[125],"when":[127],"an":[130],"solver,":[132],"contrast":[134],"ASPIN,":[136],"get":[140],"substantially":[142],"The":[148],"construction":[150],"allows":[152],"us":[153],"naturally":[155],"define":[156],"coarse":[158],"correction":[159],"using":[160],"multigrid":[162],"full":[163],"approximation":[164],"scheme,":[165],"leads":[167],"convergent":[170],"two-level":[171],"two":[179],"level":[180],"preconditioner.":[183],"illustrate":[185],"our":[186],"findings":[187],"numerical":[189],"results":[190],"on":[191],"Forchheimer":[193],"equation":[194],"diffusion":[198],"problem.":[199]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":13},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":13},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2}],"updated_date":"2026-05-01T08:36:08.643496","created_date":"2019-06-27T00:00:00"}
