{"id":"https://openalex.org/W2025033760","doi":"https://doi.org/10.1137/080725702","title":"Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media","display_name":"Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2025033760","doi":"https://doi.org/10.1137/080725702","mag":"2025033760"},"language":"en","primary_location":{"id":"doi:10.1137/080725702","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080725702","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/A5009824948","display_name":"Matthias Bollh\u00f6fer","orcid":"https://orcid.org/0000-0002-8093-5812"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Matthias Bollh\u00f6fer","raw_affiliation_strings":["m.bollhoefer@tu-bs.de#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"m.bollhoefer@tu-bs.de#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040323736","display_name":"Marcus J. Grote","orcid":"https://orcid.org/0000-0001-8129-0799"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marcus J. Grote","raw_affiliation_strings":["marcus.grote@unibas.ch#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"marcus.grote@unibas.ch#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5017590193","display_name":"Olaf Schenk","orcid":"https://orcid.org/0000-0001-8636-1023"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Olaf Schenk","raw_affiliation_strings":["olaf.schenk@unibas.ch#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"olaf.schenk@unibas.ch#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.137,"has_fulltext":false,"cited_by_count":90,"citation_normalized_percentile":{"value":0.97084161,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"31","issue":"5","first_page":"3781","last_page":"3805"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9997000098228455,"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T10514","display_name":"Numerical methods in engineering","score":0.9990000128746033,"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/preconditioner","display_name":"Preconditioner","score":0.9475038051605225},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.716364324092865},{"id":"https://openalex.org/keywords/krylov-subspace","display_name":"Krylov subspace","score":0.6259678602218628},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.6231191158294678},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.6081905364990234},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5029053092002869},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.4567722678184509},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.43488195538520813},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.43386387825012207},{"id":"https://openalex.org/keywords/numerical-stability","display_name":"Numerical stability","score":0.42279481887817383},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4025467038154602},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.3260471820831299},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.27495914697647095},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.2357165515422821},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2063586413860321}],"concepts":[{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.9475038051605225},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.716364324092865},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.6259678602218628},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.6231191158294678},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.6081905364990234},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5029053092002869},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.4567722678184509},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.43488195538520813},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.43386387825012207},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.42279481887817383},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4025467038154602},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.3260471820831299},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27495914697647095},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.2357165515422821},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2063586413860321},{"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/080725702","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080725702","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W71570128","https://openalex.org/W175176216","https://openalex.org/W1486210902","https://openalex.org/W1487444668","https://openalex.org/W1506342804","https://openalex.org/W1581078542","https://openalex.org/W1647103664","https://openalex.org/W1965394225","https://openalex.org/W1969526019","https://openalex.org/W1969600280","https://openalex.org/W1976437885","https://openalex.org/W1987152170","https://openalex.org/W1989314419","https://openalex.org/W1993489874","https://openalex.org/W2004668340","https://openalex.org/W2005733061","https://openalex.org/W2005972435","https://openalex.org/W2017431061","https://openalex.org/W2023570228","https://openalex.org/W2024949119","https://openalex.org/W2025124289","https://openalex.org/W2031392034","https://openalex.org/W2037144335","https://openalex.org/W2041785146","https://openalex.org/W2042199958","https://openalex.org/W2064172716","https://openalex.org/W2065983078","https://openalex.org/W2069595221","https://openalex.org/W2070232376","https://openalex.org/W2071928229","https://openalex.org/W2072062192","https://openalex.org/W2072598943","https://openalex.org/W2073555064","https://openalex.org/W2082941919","https://openalex.org/W2084463157","https://openalex.org/W2087170426","https://openalex.org/W2106787546","https://openalex.org/W2115272362","https://openalex.org/W2137541800","https://openalex.org/W2138940359","https://openalex.org/W2141331848","https://openalex.org/W2141461755","https://openalex.org/W2142094322","https://openalex.org/W2142399659","https://openalex.org/W2151478420","https://openalex.org/W2153998033","https://openalex.org/W2155216327","https://openalex.org/W2172085288","https://openalex.org/W2217462822","https://openalex.org/W2222512263","https://openalex.org/W4235317438","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W1492522302","https://openalex.org/W2096454229","https://openalex.org/W2042199958","https://openalex.org/W3046775784","https://openalex.org/W2103149966","https://openalex.org/W2382988560","https://openalex.org/W2177482563","https://openalex.org/W2464766188","https://openalex.org/W2024149451","https://openalex.org/W2016008514"],"abstract_inverted_index":{"An":[0],"algebraic":[1,79,84],"multilevel":[2,19],"(ML)":[3],"preconditioner":[4,35,94],"is":[5,15,95],"presented":[6],"for":[7,40,68,88],"the":[8,25,30,57,61,64,127,132],"Helmholtz":[9,31],"equation":[10],"in":[11],"heterogeneous":[12],"media.":[13],"It":[14],"based":[16],"on":[17],"a":[18,98,105,109],"incomplete":[20],"$LDL^T$":[21],"factorization":[22],"and":[23,42,49,102,114,129],"preserves":[24],"inherent":[26],"(complex)":[27],"symmetry":[28],"of":[29,60,111,131],"equation.":[32],"The":[33,54,92,123],"ML":[34,93,133],"incorporates":[36],"two":[37],"key":[38],"components":[39],"efficiency":[41,130],"numerical":[43,69,124],"stability:":[44],"symmetric":[45],"maximum":[46],"weight":[47],"matchings":[48],"an":[50,78],"inverse-based":[51],"pivoting":[52],"strategy.":[53],"former":[55],"increases":[56],"block-diagonal":[58],"dominance":[59],"system,":[62],"whereas":[63],"latter":[65],"controls":[66],"$\\|L^{-1}\\|$":[67],"stability.":[70],"When":[71],"applied":[72,103],"recursively,":[73],"their":[74],"combined":[75,96],"effect":[76],"yields":[77],"coarsening":[80],"strategy,":[81],"similar":[82],"to":[83,108],"multigrid":[85],"methods,":[86],"even":[87,135],"highly":[89],"indefinite":[90],"matrices.":[91],"with":[97],"Krylov":[99],"subspace":[100],"method":[101],"as":[104],"\u201cblack-box\u201d":[106],"solver":[107],"series":[110],"challenging":[112],"two-":[113],"three-dimensional":[115],"test":[116],"problems,":[117],"mainly":[118],"from":[119],"geophysical":[120],"seismic":[121],"imaging.":[122],"results":[125],"demonstrate":[126],"robustness":[128],"preconditioner,":[134],"at":[136],"higher":[137],"frequency":[138],"regimes.":[139]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":12},{"year":2013,"cited_by_count":11},{"year":2012,"cited_by_count":13}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
