{"id":"https://openalex.org/W884468685","doi":"https://doi.org/10.1137/15m1021349","title":"Null-Space Preconditioners for Saddle Point Systems","display_name":"Null-Space Preconditioners for Saddle Point Systems","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W884468685","doi":"https://doi.org/10.1137/15m1021349","mag":"884468685"},"language":"en","primary_location":{"id":"doi:10.1137/15m1021349","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1021349","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","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/A5062815653","display_name":"Jennifer Pestana","orcid":"https://orcid.org/0000-0003-1527-3178"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jennifer Pestana","raw_affiliation_strings":["Mathematics And Statistics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics And Statistics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032539803","display_name":"Tyrone Rees","orcid":"https://orcid.org/0000-0003-0476-2259"},"institutions":[{"id":"https://openalex.org/I162524378","display_name":"Science and Technology Facilities Council","ror":"https://ror.org/057g20z61","country_code":"GB","type":"government","lineage":["https://openalex.org/I162524378","https://openalex.org/I4210087105"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Tyrone Rees","raw_affiliation_strings":["SCIENCE AND TECHNOLOGY FACILITIES COUNCIL"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"SCIENCE AND TECHNOLOGY FACILITIES COUNCIL","institution_ids":["https://openalex.org/I162524378"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0814,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.78634788,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"37","issue":"3","first_page":"1103","last_page":"1128"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","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/T11416","display_name":"Numerical methods for differential equations","score":0.9987999796867371,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.996999979019165,"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/saddle-point","display_name":"Saddle point","score":0.8255143165588379},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7344116568565369},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.6883188486099243},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.6603886485099792},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.5234379768371582},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.5060307383537292},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.5005965232849121},{"id":"https://openalex.org/keywords/null","display_name":"Null (SQL)","score":0.48609471321105957},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.46268004179000854},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.45221415162086487},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4270758330821991},{"id":"https://openalex.org/keywords/complement","display_name":"Complement (music)","score":0.4167252480983734},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.31723499298095703},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.263272225856781},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1922326385974884},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18180790543556213},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1297883689403534}],"concepts":[{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.8255143165588379},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7344116568565369},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.6883188486099243},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.6603886485099792},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.5234379768371582},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5060307383537292},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.5005965232849121},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.48609471321105957},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.46268004179000854},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.45221415162086487},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4270758330821991},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.4167252480983734},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.31723499298095703},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.263272225856781},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1922326385974884},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18180790543556213},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1297883689403534},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","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/C188082640","wikidata":"https://www.wikidata.org/wiki/Q1780899","display_name":"Complementation","level":4,"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/C127716648","wikidata":"https://www.wikidata.org/wiki/Q104053","display_name":"Phenotype","level":3,"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"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":5,"locations":[{"id":"doi:10.1137/15m1021349","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1021349","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:strathprints.strath.ac.uk:56709","is_oa":false,"landing_page_url":"https://strathprints.strath.ac.uk/view/author/890806.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:CiteSeerX.psu:10.1.1.699.7393","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.699.7393","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.numerical.rl.ac.uk/people/rees/pdf/PestanaRees15.pdf","raw_type":"text"},{"id":"pmh:oai:purl.org/net/epubs:work/18043896","is_oa":false,"landing_page_url":"http://purl.org/net/epubs/work/18043896","pdf_url":null,"source":{"id":"https://openalex.org/S4306400334","display_name":"Science and Technology Facilities Council","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:purl.org/net/epubs:work/30691672","is_oa":false,"landing_page_url":"http://purl.org/net/epubs/work/30691672","pdf_url":null,"source":{"id":"https://openalex.org/S4306400334","display_name":"Science and Technology Facilities Council","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1162350429","display_name":null,"funder_award_id":"EP/M025179/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6004452866","display_name":"Linear Algebra and Optimization: Structure, Sparsity, Algorithms and Software","funder_award_id":"EP/I013067/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6757365500","display_name":null,"funder_award_id":"EP/I005293","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G877243714","display_name":null,"funder_award_id":"EP/I013067/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W1518039036","https://openalex.org/W1524015806","https://openalex.org/W1966886095","https://openalex.org/W1969012295","https://openalex.org/W1977061454","https://openalex.org/W1980053021","https://openalex.org/W1992234740","https://openalex.org/W1993553082","https://openalex.org/W1993671884","https://openalex.org/W1996982555","https://openalex.org/W2009926490","https://openalex.org/W2015295891","https://openalex.org/W2015891870","https://openalex.org/W2021402826","https://openalex.org/W2022852340","https://openalex.org/W2026079992","https://openalex.org/W2031691614","https://openalex.org/W2032748809","https://openalex.org/W2035080386","https://openalex.org/W2050206349","https://openalex.org/W2052956836","https://openalex.org/W2053050827","https://openalex.org/W2053893326","https://openalex.org/W2053994227","https://openalex.org/W2054034220","https://openalex.org/W2057565823","https://openalex.org/W2058696262","https://openalex.org/W2061702008","https://openalex.org/W2065977427","https://openalex.org/W2070419067","https://openalex.org/W2073191407","https://openalex.org/W2078843646","https://openalex.org/W2080622320","https://openalex.org/W2082314805","https://openalex.org/W2089024363","https://openalex.org/W2099359799","https://openalex.org/W2113795148","https://openalex.org/W2118431207","https://openalex.org/W2118469889","https://openalex.org/W2123963295","https://openalex.org/W2133248904","https://openalex.org/W2140153041","https://openalex.org/W2141870784","https://openalex.org/W2152309118","https://openalex.org/W2155216327","https://openalex.org/W2160429522","https://openalex.org/W2161327317","https://openalex.org/W2162221417","https://openalex.org/W2164805844","https://openalex.org/W2165622925","https://openalex.org/W2319362186","https://openalex.org/W2963697667"],"related_works":["https://openalex.org/W2028160813","https://openalex.org/W4236459141","https://openalex.org/W4205304778","https://openalex.org/W2020252434","https://openalex.org/W73248859","https://openalex.org/W2584253892","https://openalex.org/W2350324449","https://openalex.org/W119381072","https://openalex.org/W2034033896","https://openalex.org/W2094891943"],"abstract_inverted_index":{"The":[0],"null-space":[1,55],"method":[2,30,82],"is":[3,102,110],"a":[4,15,20,35,53,85,122,128],"technique":[5],"that":[6],"has":[7],"been":[8],"used":[9],"for":[10],"many":[11],"years":[12],"to":[13,19,23,72],"reduce":[14],"saddle":[16],"point":[17],"system":[18],"smaller,":[21],"easier":[22],"solve,":[24],"symmetric":[25],"positive":[26],"definite":[27],"system.":[28,40],"This":[29,89],"can":[31],"be":[32],"understood":[33],"as":[34],"block":[36],"factorization":[37,56],"of":[38,46,52,116,124,130],"the":[39,44,62,74,79,95,99,114,117],"Here":[41],"we":[42],"explore":[43],"use":[45],"preconditioners":[47,76,120],"based":[48,66],"on":[49,121],"incomplete":[50],"versions":[51],"particular":[54],"and":[57,98],"compare":[58],"their":[59],"performance":[60],"with":[61,84,94],"equivalent":[63],"Schur":[64],"complement":[65],"preconditioners.":[67],"We":[68,112],"also":[69],"describe":[70],"how":[71],"apply":[73],"nonsymmetric":[75],"proposed":[77,119],"using":[78],"conjugate":[80],"gradient":[81],"(CG)":[83],"nonstandard":[86],"inner":[87],"product.":[88],"requires":[90],"an":[91],"exact":[92],"solve":[93],"(1,1)":[96],"block,":[97],"resulting":[100],"algorithm":[101],"applicable":[103],"in":[104],"other":[105],"cases":[106,126],"where":[107],"Bramble--Pasciak":[108],"CG":[109],"used.":[111],"verify":[113],"efficiency":[115],"newly":[118],"number":[123],"test":[125],"from":[127],"range":[129],"applications.":[131]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
