{"id":"https://openalex.org/W2120072386","doi":"https://doi.org/10.1137/130932144","title":"Multiscale Approximation and Reproducing Kernel Hilbert Space Methods","display_name":"Multiscale Approximation and Reproducing Kernel Hilbert Space Methods","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2120072386","doi":"https://doi.org/10.1137/130932144","mag":"2120072386"},"language":"en","primary_location":{"id":"doi:10.1137/130932144","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130932144","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","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/A5028074195","display_name":"Michael Griebel","orcid":"https://orcid.org/0000-0002-3147-6680"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Michael Griebel","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110275232","display_name":"Christian Rieger","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Christian Rieger","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5009623277","display_name":"Barbara Zwicknagl","orcid":"https://orcid.org/0000-0002-6394-0775"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Barbara Zwicknagl","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5028074195"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.633,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.93316187,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"53","issue":"2","first_page":"852","last_page":"873"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9984999895095825,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9984999895095825,"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/T11558","display_name":"Composite Material Mechanics","score":0.9843999743461609,"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"}},{"id":"https://openalex.org/T11366","display_name":"Elasticity and Material Modeling","score":0.9840999841690063,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"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":0.613502025604248},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5974996089935303},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.5236858129501343},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.519786536693573},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.4566750228404999},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.41544023156166077},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2878437042236328},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22990933060646057},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.14403346180915833},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09340840578079224}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.613502025604248},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5974996089935303},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5236858129501343},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.519786536693573},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.4566750228404999},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.41544023156166077},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2878437042236328},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22990933060646057},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14403346180915833},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09340840578079224},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/130932144","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130932144","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:bonndoc.ulb.uni-bonn.de:20.500.11811/11928","is_oa":false,"landing_page_url":"https://hdl.handle.net/20.500.11811/11928","pdf_url":null,"source":{"id":"https://openalex.org/S4306402493","display_name":"bonndoc (University of Bonn)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I135140700","host_organization_name":"University of Bonn","host_organization_lineage":["https://openalex.org/I135140700"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"doc-type:preprint"},{"id":"pmh:oai:fraunhofer.de:N-358547","is_oa":false,"landing_page_url":"http://publica.fraunhofer.de/documents/N-358547.html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400801","display_name":"Publikationsdatenbank der Fraunhofer-Gesellschaft (Fraunhofer-Gesellschaft)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4923324","host_organization_name":"Fraunhofer-Gesellschaft","host_organization_lineage":["https://openalex.org/I4923324"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Fraunhofer SCAI","raw_type":"Journal Article"},{"id":"pmh:oai:publica.fraunhofer.de:publica/240941","is_oa":false,"landing_page_url":"https://publica.fraunhofer.de/handle/publica/240941","pdf_url":null,"source":{"id":"https://openalex.org/S4306400318","display_name":"Fraunhofer-Publica (Fraunhofer-Gesellschaft)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4923324","host_organization_name":"Fraunhofer-Gesellschaft","host_organization_lineage":["https://openalex.org/I4923324"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"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":25,"referenced_works":["https://openalex.org/W19451620","https://openalex.org/W66049266","https://openalex.org/W1486164486","https://openalex.org/W1491507172","https://openalex.org/W1504602158","https://openalex.org/W1597405158","https://openalex.org/W1601740268","https://openalex.org/W1605660994","https://openalex.org/W1988473367","https://openalex.org/W1989908085","https://openalex.org/W2001388931","https://openalex.org/W2024697390","https://openalex.org/W2041099140","https://openalex.org/W2043244576","https://openalex.org/W2063894334","https://openalex.org/W2070330278","https://openalex.org/W2076699062","https://openalex.org/W2089245375","https://openalex.org/W2090536661","https://openalex.org/W2108280101","https://openalex.org/W2123805128","https://openalex.org/W2150695134","https://openalex.org/W2161591817","https://openalex.org/W3133571595","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W238609530","https://openalex.org/W2734192830","https://openalex.org/W3016395638","https://openalex.org/W1636163329","https://openalex.org/W4292947165","https://openalex.org/W3045121788","https://openalex.org/W3201655533","https://openalex.org/W3134442314","https://openalex.org/W2281248943","https://openalex.org/W3102538728"],"abstract_inverted_index":{"We":[0,53],"consider":[1],"reproducing":[2],"kernels":[3,14,114],"$K:\\Omega\\times":[4],"\\Omega":[5],"\\to":[6],"\\mathbb{R}$":[7],"in":[8],"multiscale":[9],"series":[10,113],"expansion":[11],"form,":[12],"i.e.,":[13],"of":[15,59,75,98,115],"the":[16,45,76,84,92,116,123,133,156],"form":[17,117],"$K\\left(\\boldsymbol{x},\\boldsymbol{y}\\right)=\\sum_{\\ell\\in\\mathbb{N}}\\lambda_\\ell\\sum_{j\\in":[18],"I_\\ell}\\phi_{\\ell,j}\\left(\\boldsymbol{x}\\right)\\phi_{\\ell,j}\\left(\\boldsymbol{y}\\right)$":[19],"with":[20,32,81],"weights":[21],"$\\lambda_\\ell$":[22],"and":[23,70,141],"structurally":[24],"simple":[25],"basis":[26,33,63,157],"functions":[27,34,64,158],"$\\left\\{\\phi_{\\ell,i}\\right\\}$.":[28],"Here,":[29],"we":[30],"deal":[31],"such":[35],"as":[36],"polynomials":[37],"or":[38,51],"frame":[39],"systems,":[40],"where,":[41],"for":[42,145,152],"$\\ell\\in":[43],"\\mathbb{N}$,":[44],"index":[46,86,125],"set":[47,94,135],"$I_\\ell$":[48],"is":[49,88,120,127],"finite":[50],"countable.":[52],"derive":[54],"relations":[55],"between":[56],"approximation":[57,102],"properties":[58],"spaces":[60,71,105,147,153],"based":[61,109,154],"on":[62,110,132,155],"$\\{\\phi_{\\ell,j}":[65,159],":1\\leq\\ell\\leq":[66,160],"L,":[67,161],"j\\in":[68,162],"I_\\ell\\}$":[69],"spanned":[72],"by":[73],"translates":[74],"kernel":[77],"span":[78,106],"$\\{K(\\boldsymbol{x}_1,\\cdot),\\dots,":[79],"K(\\boldsymbol{x}_N,\\cdot)\\}$":[80],"$X_{N}:=\\{\\boldsymbol{x}_1,\\dots,\\boldsymbol{x}_N\\}\\subset\\Omega$":[82],"if":[83],"truncation":[85,124],"$L$":[87,126],"appropriately":[89],"coupled":[90],"to":[91],"discrete":[93],"$X_{N}$.":[95,136],"An":[96],"analysis":[97],"a":[99],"numerically":[100],"feasible":[101],"from":[103,150],"trial":[104],"$\\{K^L(\\boldsymbol{x}_1,\\cdot),\\dots,":[107],"K^L(\\boldsymbol{x}_N,\\cdot)\\}$":[108],"finitely":[111],"truncated":[112],"$K^L\\left(\\boldsymbol{x},\\boldsymbol{y}\\right):=\\sum_{\\ell=1}^L\\lambda_\\ell\\sum_{j\\in":[118],"I_\\ell}\\phi_{\\ell,j}(\\boldsymbol{x})\\phi_{\\ell,j}(\\boldsymbol{y})$":[119],"provided,":[121],"where":[122],"chosen":[128],"sufficiently":[129],"large":[130],"depending":[131],"point":[134],"Furthermore,":[137],"Bernstein-type":[138],"inverse":[139],"estimates":[140,151],"derivative-free":[142],"sampling":[143],"inequalities":[144],"kernel-based":[146],"are":[148],"obtained":[149],"I_\\ell\\}$.":[163]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}