{"id":"https://openalex.org/W2099190740","doi":"https://doi.org/10.1137/130921842","title":"Fast Solvers for Cahn--Hilliard Inpainting","display_name":"Fast Solvers for Cahn--Hilliard Inpainting","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2099190740","doi":"https://doi.org/10.1137/130921842","mag":"2099190740"},"language":"en","primary_location":{"id":"doi:10.1137/130921842","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130921842","pdf_url":null,"source":{"id":"https://openalex.org/S152600803","display_name":"SIAM Journal on Imaging Sciences","issn_l":"1936-4954","issn":["1936-4954"],"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 Imaging Sciences","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/A5042763323","display_name":"Jessica Bosch","orcid":null},"institutions":[{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I149899117"]},{"id":"https://openalex.org/I4210127568","display_name":"Max Planck Institute for Dynamics of Complex Technical Systems","ror":"https://ror.org/030h7k016","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210127568"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Jessica Bosch","raw_affiliation_strings":["Numerical Linear Algebra for Dynamical Systems, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Numerical Linear Algebra for Dynamical Systems, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society","institution_ids":["https://openalex.org/I4210127568","https://openalex.org/I149899117"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037754645","display_name":"David Kay","orcid":"https://orcid.org/0000-0003-4984-3669"},"institutions":[{"id":"https://openalex.org/I4210146410","display_name":"Science Oxford","ror":"https://ror.org/04j8yhy50","country_code":"GB","type":"nonprofit","lineage":["https://openalex.org/I4210146410"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"David Kay","raw_affiliation_strings":["Univeristy of Oxford"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univeristy of Oxford","institution_ids":["https://openalex.org/I4210146410"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055578402","display_name":"Martin Stoll","orcid":"https://orcid.org/0000-0003-0951-4756"},"institutions":[{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I149899117"]},{"id":"https://openalex.org/I4210127568","display_name":"Max Planck Institute for Dynamics of Complex Technical Systems","ror":"https://ror.org/030h7k016","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210127568"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Martin Stoll","raw_affiliation_strings":["Numerical Linear Algebra for Dynamical Systems, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Numerical Linear Algebra for Dynamical Systems, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society","institution_ids":["https://openalex.org/I4210127568","https://openalex.org/I149899117"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5110146377","display_name":"Andrew J. Wathen","orcid":null},"institutions":[{"id":"https://openalex.org/I4210146410","display_name":"Science Oxford","ror":"https://ror.org/04j8yhy50","country_code":"GB","type":"nonprofit","lineage":["https://openalex.org/I4210146410"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Andrew J. Wathen","raw_affiliation_strings":["Univeristy of Oxford"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univeristy of Oxford","institution_ids":["https://openalex.org/I4210146410"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.7966,"has_fulltext":false,"cited_by_count":56,"citation_normalized_percentile":{"value":0.96427885,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"7","issue":"1","first_page":"67","last_page":"97"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9954000115394592,"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"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9954000115394592,"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/T11822","display_name":"Analytic and geometric function theory","score":0.9825999736785889,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9782000184059143,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/inpainting","display_name":"Inpainting","score":0.9502744078636169},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.788643479347229},{"id":"https://openalex.org/keywords/variational-inequality","display_name":"Variational inequality","score":0.6792880296707153},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6189264059066772},{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.5749084949493408},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.5576357245445251},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5327097773551941},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5066364407539368},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.5003924369812012},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.42512691020965576},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.39553770422935486},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.36553409695625305},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27040135860443115},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.262122243642807},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.24287334084510803},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.16138434410095215},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.15472030639648438},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.06624403595924377}],"concepts":[{"id":"https://openalex.org/C11727466","wikidata":"https://www.wikidata.org/wiki/Q1628157","display_name":"Inpainting","level":3,"score":0.9502744078636169},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.788643479347229},{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.6792880296707153},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6189264059066772},{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.5749084949493408},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.5576357245445251},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5327097773551941},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5066364407539368},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.5003924369812012},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.42512691020965576},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.39553770422935486},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.36553409695625305},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27040135860443115},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.262122243642807},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.24287334084510803},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.16138434410095215},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.15472030639648438},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.06624403595924377},{"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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/130921842","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130921842","pdf_url":null,"source":{"id":"https://openalex.org/S152600803","display_name":"SIAM Journal on Imaging Sciences","issn_l":"1936-4954","issn":["1936-4954"],"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 Imaging Sciences","raw_type":"journal-article"},{"id":"pmh:oai:repository.kaust.edu.sa:10754/598319","is_oa":false,"landing_page_url":"http://hdl.handle.net/10754/598319","pdf_url":null,"source":{"id":"https://openalex.org/S4306401596","display_name":"King Abdullah University of Science and Technology Repository (King Abdullah University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I71920554","host_organization_name":"King Abdullah University of Science and Technology","host_organization_lineage":["https://openalex.org/I71920554"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":56,"referenced_works":["https://openalex.org/W92933111","https://openalex.org/W372785527","https://openalex.org/W1506342804","https://openalex.org/W1521182415","https://openalex.org/W1554469977","https://openalex.org/W1558012644","https://openalex.org/W1560263223","https://openalex.org/W1571187512","https://openalex.org/W1572639095","https://openalex.org/W1805381866","https://openalex.org/W1961194538","https://openalex.org/W1966227249","https://openalex.org/W1973728860","https://openalex.org/W1975393729","https://openalex.org/W1981632256","https://openalex.org/W2002168069","https://openalex.org/W2025944155","https://openalex.org/W2026365342","https://openalex.org/W2026993378","https://openalex.org/W2031832209","https://openalex.org/W2032257836","https://openalex.org/W2044355921","https://openalex.org/W2046528357","https://openalex.org/W2050206349","https://openalex.org/W2054225369","https://openalex.org/W2056217824","https://openalex.org/W2064309976","https://openalex.org/W2070584352","https://openalex.org/W2079482881","https://openalex.org/W2081160575","https://openalex.org/W2083629653","https://openalex.org/W2085048703","https://openalex.org/W2085259167","https://openalex.org/W2095195575","https://openalex.org/W2102295719","https://openalex.org/W2103559027","https://openalex.org/W2114487471","https://openalex.org/W2117199631","https://openalex.org/W2128204029","https://openalex.org/W2130604180","https://openalex.org/W2130698343","https://openalex.org/W2131831293","https://openalex.org/W2133248904","https://openalex.org/W2138026128","https://openalex.org/W2140153041","https://openalex.org/W2140865211","https://openalex.org/W2148395215","https://openalex.org/W2155216327","https://openalex.org/W2161620664","https://openalex.org/W2165736859","https://openalex.org/W2168815698","https://openalex.org/W2295936755","https://openalex.org/W2413948113","https://openalex.org/W2498157954","https://openalex.org/W2883237090","https://openalex.org/W4245101459"],"related_works":["https://openalex.org/W2380775572","https://openalex.org/W2213520135","https://openalex.org/W4242046654","https://openalex.org/W1504109132","https://openalex.org/W3134074939","https://openalex.org/W2117562399","https://openalex.org/W3174923100","https://openalex.org/W2244018504","https://openalex.org/W4298074124","https://openalex.org/W2003548600"],"abstract_inverted_index":{"The":[0,41],"solution":[1,59,68],"of":[2,7,16,60,69,79],"Cahn--Hilliard":[3,103],"variational":[4],"inequalities":[5],"is":[6],"interest":[8],"in":[9],"many":[10],"applications.":[11],"We":[12,52],"discuss":[13],"the":[14,48,57,61,66,70,77,89,99,108],"use":[15],"them":[17],"as":[18,38],"a":[19,93],"tool":[20],"for":[21,35,56,88],"binary":[22],"image":[23],"inpainting.":[24],"This":[25],"has":[26],"been":[27],"done":[28],"before":[29],"using":[30,92],"double-well":[31,94],"potentials":[32,37],"but":[33],"not":[34],"nonsmooth":[36,102],"considered":[39],"here.":[40],"existing":[42],"bound":[43],"constraints":[44],"are":[45,86],"incorporated":[46],"via":[47],"Moreau--Yosida":[49,71],"regularization":[50],"technique.":[51],"develop":[53],"effective":[54],"preconditioners":[55],"efficient":[58],"Newton":[62],"steps":[63],"associated":[64],"with":[65],"fast":[67],"regularized":[72],"problem.":[73],"Numerical":[74],"results":[75],"illustrate":[76],"efficiency":[78],"our":[80],"approach.":[81],"Moreover,":[82],"precise":[83],"eigenvalue":[84],"intervals":[85],"given":[87],"preconditioned":[90],"system":[91],"potential.":[95],"A":[96],"comparison":[97],"between":[98],"smooth":[100],"and":[101],"inpainting":[104],"models":[105],"shows":[106],"that":[107],"latter":[109],"achieves":[110],"better":[111],"results.":[112]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
