{"id":"https://openalex.org/W2150175265","doi":"https://doi.org/10.1137/070705817","title":"Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data","display_name":"Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2150175265","doi":"https://doi.org/10.1137/070705817","mag":"2150175265"},"language":"en","primary_location":{"id":"doi:10.1137/070705817","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070705817","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/A5084624008","display_name":"Oliver G. Ernst","orcid":"https://orcid.org/0000-0002-0176-7321"},"institutions":[{"id":"https://openalex.org/I61893789","display_name":"TU Bergakademie Freiberg","ror":"https://ror.org/031vc2293","country_code":"DE","type":"education","lineage":["https://openalex.org/I61893789"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"O.G. Ernst","raw_affiliation_strings":["Institut f\u00fcr Numerische Mathematik und Optimierung, Technische Universit\u00e4t Bergakademie Freiberg, D-09596, Freiberg, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Numerische Mathematik und Optimierung, Technische Universit\u00e4t Bergakademie Freiberg, D-09596, Freiberg, Germany","institution_ids":["https://openalex.org/I61893789"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073050362","display_name":"Catherine E. Powell","orcid":"https://orcid.org/0000-0003-3605-6797"},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"C.E. Powell","raw_affiliation_strings":["School of Mathematics, University of Manchester, Manchester, M139PL, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Manchester, Manchester, M139PL, United Kingdom","institution_ids":["https://openalex.org/I28407311"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007116796","display_name":"David J. Silvester","orcid":"https://orcid.org/0000-0002-0652-5444"},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"D.J. Silvester","raw_affiliation_strings":["School of Mathematics, University of Manchester, Manchester, M139PL, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Manchester, Manchester, M139PL, United Kingdom","institution_ids":["https://openalex.org/I28407311"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015636859","display_name":"Elisabeth Ullmann","orcid":"https://orcid.org/0000-0002-5699-7394"},"institutions":[{"id":"https://openalex.org/I61893789","display_name":"TU Bergakademie Freiberg","ror":"https://ror.org/031vc2293","country_code":"DE","type":"education","lineage":["https://openalex.org/I61893789"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"E. Ullmann","raw_affiliation_strings":["Institut f\u00fcr Numerische Mathematik und Optimierung, Technische Universit\u00e4t Bergakademie Freiberg, D-09596, Freiberg, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Numerische Mathematik und Optimierung, Technische Universit\u00e4t Bergakademie Freiberg, D-09596, Freiberg, Germany","institution_ids":["https://openalex.org/I61893789"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.112,"has_fulltext":false,"cited_by_count":61,"citation_normalized_percentile":{"value":0.95214463,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"31","issue":"2","first_page":"1424","last_page":"1447"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9919999837875366,"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/T10396","display_name":"Fatigue and fracture mechanics","score":0.9793999791145325,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/mathematics","display_name":"Mathematics","score":0.816753625869751},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7532557845115662},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.6100011467933655},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5816249847412109},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.5660679340362549},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.5385343432426453},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.4874563217163086},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.4861859679222107},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.4259510040283203},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3914330005645752},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.27670198678970337},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0923311710357666}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.816753625869751},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7532557845115662},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.6100011467933655},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5816249847412109},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.5660679340362549},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.5385343432426453},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.4874563217163086},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.4861859679222107},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.4259510040283203},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3914330005645752},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27670198678970337},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0923311710357666},{"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/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}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1137/070705817","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070705817","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:eprints.maths.manchester.ac.uk.MIMS.EPrints:1229","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.108.4891","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.108.4891","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/862/01/covered/MIMS_ep2007_126.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.219.1796","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.219.1796","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mathe.tu-freiberg.de/~ernst/PubArchive/ernstEtAl2009.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.72.3606","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.72.3606","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mathe.tu-freiberg.de/~ernst/PubArchive/partIf.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/a2196efb-8345-4551-9393-1f3ed433a6f8","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/a2196efb-8345-4551-9393-1f3ed433a6f8","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Ernst, O G, Powell, C E, Silvester, D J & Ullmann, E 2008, 'Efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data', SIAM Journal on Scientific Computing, vol. 31, no. 2, pp. 1424-1447. https://doi.org/10.1137/070705817","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:pure.atira.dk:publications/a2196efb-8345-4551-9393-1f3ed433a6f8","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/efficient-solvers-for-a-linear-stochastic-galerkin-mixed-formulation-of-diffusion-problems-with-random-data(a2196efb-8345-4551-9393-1f3ed433a6f8).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Ernst, O G, Powell, C E, Silvester, D J & Ullmann, E 2008, 'Efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data', SIAM Journal on Scientific Computing, vol. 31, no. 2, pp. 1424-1447. https://doi.org/10.1137/070705817","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320875","display_name":"Deutscher Akademischer Austauschdienst","ror":"https://ror.org/039djdh30"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1538934584","https://openalex.org/W1966827182","https://openalex.org/W1971746901","https://openalex.org/W1983156129","https://openalex.org/W1999331305","https://openalex.org/W1999723468","https://openalex.org/W2014490899","https://openalex.org/W2015605368","https://openalex.org/W2018850201","https://openalex.org/W2033721384","https://openalex.org/W2044390636","https://openalex.org/W2048791667","https://openalex.org/W2055460625","https://openalex.org/W2058362163","https://openalex.org/W2062993069","https://openalex.org/W2064332562","https://openalex.org/W2074860140","https://openalex.org/W2088117085","https://openalex.org/W2102031020","https://openalex.org/W2113517083","https://openalex.org/W2121613024","https://openalex.org/W2143591652","https://openalex.org/W3145008516","https://openalex.org/W3149667697","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2006251942","https://openalex.org/W2364741597","https://openalex.org/W1492103595","https://openalex.org/W1864774435","https://openalex.org/W946352265","https://openalex.org/W3020787026","https://openalex.org/W2334479858","https://openalex.org/W2799209613","https://openalex.org/W2055696851","https://openalex.org/W4367320189"],"abstract_inverted_index":{"We":[0,58],"introduce":[1],"a":[2],"stochastic":[3,35],"Galerkin":[4],"mixed":[5],"formulation":[6],"of":[7,19,34,72],"the":[8,15,20,54,63,70,73,81],"steady-state":[9],"diffusion":[10,47],"equation":[11],"and":[12,67],"focus":[13],"on":[14,42,69],"efficient":[16],"iterative":[17],"solution":[18],"saddle-point":[21],"systems":[22],"obtained":[23],"by":[24],"combining":[25],"standard":[26],"finite":[27],"element":[28],"discretizations":[29],"with":[30,53,77],"two":[31],"distinct":[32],"types":[33],"basis":[36],"functions.":[37],"So-called":[38],"mean-based":[39],"preconditioners,":[40],"based":[41],"fast":[43],"solvers":[44],"for":[45,51,62],"scalar":[46],"problems,":[48],"are":[49],"introduced":[50],"use":[52],"minimum":[55],"residual":[56],"method.":[57],"derive":[59],"eigenvalue":[60],"bounds":[61],"preconditioned":[64],"system":[65],"matrices":[66],"report":[68],"efficiency":[71],"chosen":[74],"preconditioning":[75],"schemes":[76],"respect":[78],"to":[79],"all":[80],"discretization":[82],"parameters.":[83]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":5}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}
