{"id":"https://openalex.org/W2788259854","doi":"https://doi.org/10.1137/16m108361x","title":"On the Spectrum of Deflated Matrices with Applications to the Deflated Shifted Laplace Preconditioner for the Helmholtz Equation","display_name":"On the Spectrum of Deflated Matrices with Applications to the Deflated Shifted Laplace Preconditioner for the Helmholtz Equation","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2788259854","doi":"https://doi.org/10.1137/16m108361x","mag":"2788259854"},"language":"en","primary_location":{"id":"doi:10.1137/16m108361x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m108361x","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/A5006388685","display_name":"Luis Garc\u00eda Ramos","orcid":"https://orcid.org/0000-0001-5301-8085"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luis Garc\u00eda Ramos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5082240312","display_name":"Reinhard Nabben","orcid":"https://orcid.org/0000-0001-5076-9895"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Reinhard Nabben","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.287,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.81605012,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"39","issue":"1","first_page":"262","last_page":"286"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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":1.0,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"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":0.9972000122070312,"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/preconditioner","display_name":"Preconditioner","score":0.8483978509902954},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7210257053375244},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.6810317039489746},{"id":"https://openalex.org/keywords/krylov-subspace","display_name":"Krylov subspace","score":0.6550403833389282},{"id":"https://openalex.org/keywords/generalized-minimal-residual-method","display_name":"Generalized minimal residual method","score":0.60191810131073},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5427058935165405},{"id":"https://openalex.org/keywords/hermitian-matrix","display_name":"Hermitian matrix","score":0.47950559854507446},{"id":"https://openalex.org/keywords/deflation","display_name":"Deflation","score":0.47273436188697815},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.4631255567073822},{"id":"https://openalex.org/keywords/laplace-transform","display_name":"Laplace transform","score":0.4542841911315918},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.4500298500061035},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.42005470395088196},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3527544140815735},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.12590166926383972}],"concepts":[{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.8483978509902954},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7210257053375244},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.6810317039489746},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.6550403833389282},{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.60191810131073},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5427058935165405},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.47950559854507446},{"id":"https://openalex.org/C8112001","wikidata":"https://www.wikidata.org/wiki/Q161081","display_name":"Deflation","level":3,"score":0.47273436188697815},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.4631255567073822},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.4542841911315918},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.4500298500061035},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.42005470395088196},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3527544140815735},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.12590166926383972},{"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},{"id":"https://openalex.org/C556758197","wikidata":"https://www.wikidata.org/wiki/Q580018","display_name":"Monetary economics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C126285488","wikidata":"https://www.wikidata.org/wiki/Q178476","display_name":"Monetary policy","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/16m108361x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m108361x","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"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.7699999809265137,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"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":35,"referenced_works":["https://openalex.org/W649715400","https://openalex.org/W1673309917","https://openalex.org/W1833809025","https://openalex.org/W1969745760","https://openalex.org/W1971226336","https://openalex.org/W1974565581","https://openalex.org/W1987710178","https://openalex.org/W1987737035","https://openalex.org/W1993993976","https://openalex.org/W1995078768","https://openalex.org/W2019434176","https://openalex.org/W2025124289","https://openalex.org/W2028989647","https://openalex.org/W2038958847","https://openalex.org/W2041226551","https://openalex.org/W2041660628","https://openalex.org/W2042199958","https://openalex.org/W2052579356","https://openalex.org/W2056749336","https://openalex.org/W2067028470","https://openalex.org/W2079308547","https://openalex.org/W2092886819","https://openalex.org/W2111268954","https://openalex.org/W2114535880","https://openalex.org/W2115272362","https://openalex.org/W2128836543","https://openalex.org/W2137008096","https://openalex.org/W2139940635","https://openalex.org/W2169549054","https://openalex.org/W2289699240","https://openalex.org/W2469069581","https://openalex.org/W2597181770","https://openalex.org/W2963726548","https://openalex.org/W3098036980","https://openalex.org/W3123875103"],"related_works":["https://openalex.org/W2042846224","https://openalex.org/W3157581606","https://openalex.org/W2113830538","https://openalex.org/W2358905935","https://openalex.org/W2942315554","https://openalex.org/W2804461272","https://openalex.org/W2040391781","https://openalex.org/W2382988560","https://openalex.org/W2024149451","https://openalex.org/W4322732133"],"abstract_inverted_index":{"The":[0,20,147,218],"deflation":[1,35,49,89,123,203,299],"technique":[2],"for":[3,9,38,101,111,133,154,168,273,300],"accelerating":[4],"Krylov":[5,72,295],"subspace":[6],"iterative":[7],"methods":[8,73,296],"the":[10,31,46,70,80,119,134,143,164,169,172,186,191,194,207,223,229,240,243,249,254,290,301],"solution":[11],"of":[12,24,82,114,122,136,145,166,171,174,193,209,225,228,242,276,293],"linear":[13,43,182,256],"systems":[14,183,257],"has":[15,50,233],"long":[16],"been":[17,51,234],"well":[18,98],"established.":[19],"first":[21],"landmark":[22],"papers":[23],"Nicolaides":[25],"and":[26,53,60,77,79,129,216,258,280],"Dost\u00e1l":[27],"date":[28],"back":[29],"to":[30,64,181,268,288],"late":[32],"1980s,":[33],"where":[34],"is":[36,206],"used":[37,52],"Hermitian":[39],"positive":[40],"definite":[41],"(HPD)":[42],"systems.":[44],"In":[45],"last":[47],"decade":[48],"analyzed":[54],"in":[55,86,99,125,236],"combination":[56,192],"with":[57,200,278],"domain":[58],"decomposition":[59],"multigrid":[61],"methods,":[62],"leading":[63],"very":[65,97],"efficient":[66],"algorithms.":[67],"Examples":[68],"are":[69,91,108,259,266],"multilevel":[71,88,211,294],"introduced":[74,213],"by":[75,214],"Erlangga":[76,215],"Nabben,":[78],"works":[81],"Sheikh":[83],"et":[84],"al.":[85],"which":[87,205],"techniques":[90],"presented.":[92],"Although":[93],"these":[94,270],"algorithms":[95],"work":[96],"practice":[100],"non-Hermitian":[102],"problems,":[103],"not":[104],"many":[105],"theoretical":[106],"results":[107,153,180,272,286],"yet":[109],"known":[110],"this":[112],"class":[113,275],"problems.":[115],"We":[116,177],"study":[117],"here":[118,220],"general":[120],"case":[121,224],"operators":[124],"arbitrary":[126,138],"inner":[127],"products,":[128],"give":[130,159],"inclusion":[131,149],"regions":[132,150],"spectrum":[135],"an":[137],"deflated":[139,175,244],"matrix":[140],"based":[141,162,297],"on":[142,163,190,248,298],"field":[144,165],"values.":[146],"new":[148],"generalize":[151],"previous":[152],"HPD":[155],"matrices.":[156],"Moreover,":[157],"we":[158,265],"a":[160,201,210,274],"bound":[161],"values":[167],"norms":[170],"residuals":[173],"GMRES.":[176],"apply":[178],"our":[179],"arising":[184],"from":[185,262],"Helmholtz":[187,302],"equation,":[188],"focusing":[189],"complex":[195],"shifted":[196,260],"Laplace":[197],"(CSL)":[198],"preconditioner":[199],"two-level":[202],"preconditioner,":[204],"basis":[208],"method":[212],"Nabben.":[217],"analysis":[219],"only":[221],"covers":[222],"exact":[226,250],"inversion":[227],"CSL":[230],"preconditioner.":[231],"It":[232],"observed":[235],"numerical":[237],"experiments":[238],"that":[239],"eigenvalues":[241],"CSL-preconditioned":[245,255],"system":[246],"lie":[247],"same":[251],"disks":[252],"as":[253],"away":[261],"zero.":[263],"Here":[264],"able":[267],"prove":[269],"surprising":[271],"problems":[277],"Dirichlet":[279],"first-order":[281],"Sommerfeld":[282],"boundary":[283],"conditions.":[284],"Our":[285],"help":[287],"explain":[289],"high":[291],"performance":[292],"equation.":[303]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}