{"id":"https://openalex.org/W2963196150","doi":"https://doi.org/10.1137/16m1077611","title":"Exploring the Locally Low Dimensional Structure in Solving Random Elliptic PDEs","display_name":"Exploring the Locally Low Dimensional Structure in Solving Random Elliptic PDEs","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2963196150","doi":"https://doi.org/10.1137/16m1077611","mag":"2963196150"},"language":"en","primary_location":{"id":"doi:10.1137/16m1077611","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1077611","pdf_url":null,"source":{"id":"https://openalex.org/S4357572","display_name":"Multiscale Modeling and Simulation","issn_l":"1540-3459","issn":["1540-3459","1540-3467"],"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":"Multiscale Modeling &amp; Simulation","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/A5113999511","display_name":"Thomas Y. Hou","orcid":null},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Thomas Y. Hou","raw_affiliation_strings":["Applied and Computational Math, Caltech, Pasadena, CA 91125"],"affiliations":[{"raw_affiliation_string":"Applied and Computational Math, Caltech, Pasadena, CA 91125","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100438891","display_name":"Qin Li","orcid":"https://orcid.org/0000-0001-9210-8948"},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Qin Li","raw_affiliation_strings":["Mathematics Department, UW-Madison, Madison, WI 53705"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, UW-Madison, Madison, WI 53705","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059735251","display_name":"Pengchuan Zhang","orcid":"https://orcid.org/0000-0003-1155-9507"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Pengchuan Zhang","raw_affiliation_strings":["Applied and Computational Math, Caltech, Pasadena, CA 91125"],"affiliations":[{"raw_affiliation_string":"Applied and Computational Math, Caltech, Pasadena, CA 91125","institution_ids":["https://openalex.org/I122411786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5113999511"],"corresponding_institution_ids":["https://openalex.org/I122411786"],"apc_list":null,"apc_paid":null,"fwci":1.3627,"has_fulltext":false,"cited_by_count":23,"citation_normalized_percentile":{"value":0.83865146,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"15","issue":"2","first_page":"661","last_page":"695"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9991999864578247,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9980000257492065,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.597034215927124},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5959463715553284},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5759924054145813},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5759297609329224},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5722564458847046},{"id":"https://openalex.org/keywords/elliptic-partial-differential-equation","display_name":"Elliptic partial differential equation","score":0.5573110580444336},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5524519681930542},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.5189608931541443},{"id":"https://openalex.org/keywords/stochastic-optimization","display_name":"Stochastic optimization","score":0.47250401973724365},{"id":"https://openalex.org/keywords/stochastic-partial-differential-equation","display_name":"Stochastic partial differential equation","score":0.43396836519241333},{"id":"https://openalex.org/keywords/random-field","display_name":"Random field","score":0.42964786291122437},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.4154040813446045},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.386226624250412},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.32932454347610474},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.28511396050453186},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15741324424743652},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.11900997161865234},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07416278123855591}],"concepts":[{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.597034215927124},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5959463715553284},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5759924054145813},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5759297609329224},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5722564458847046},{"id":"https://openalex.org/C54067925","wikidata":"https://www.wikidata.org/wiki/Q3607962","display_name":"Elliptic partial differential equation","level":3,"score":0.5573110580444336},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5524519681930542},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5189608931541443},{"id":"https://openalex.org/C194387892","wikidata":"https://www.wikidata.org/wiki/Q1747770","display_name":"Stochastic optimization","level":2,"score":0.47250401973724365},{"id":"https://openalex.org/C84629840","wikidata":"https://www.wikidata.org/wiki/Q16979017","display_name":"Stochastic partial differential equation","level":3,"score":0.43396836519241333},{"id":"https://openalex.org/C130402806","wikidata":"https://www.wikidata.org/wiki/Q5361768","display_name":"Random field","level":2,"score":0.42964786291122437},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.4154040813446045},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.386226624250412},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32932454347610474},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.28511396050453186},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15741324424743652},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.11900997161865234},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07416278123855591},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C502989409","wikidata":"https://www.wikidata.org/wiki/Q11425","display_name":"Animation","level":2,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"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":2,"locations":[{"id":"doi:10.1137/16m1077611","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1077611","pdf_url":null,"source":{"id":"https://openalex.org/S4357572","display_name":"Multiscale Modeling and Simulation","issn_l":"1540-3459","issn":["1540-3459","1540-3467"],"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":"Multiscale Modeling &amp; Simulation","raw_type":"journal-article"},{"id":"pmh:oai:authors.library.caltech.edu:76556","is_oa":false,"landing_page_url":"https://resolver.caltech.edu/CaltechAUTHORS:20170413-142311390","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"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":[{"id":"https://openalex.org/G2331882365","display_name":null,"funder_award_id":"DMS-1613861","funder_id":"https://openalex.org/F4320337380","funder_display_name":"Division of Mathematical Sciences"},{"id":"https://openalex.org/G92415911","display_name":null,"funder_award_id":"DMS-1318377","funder_id":"https://openalex.org/F4320337380","funder_display_name":"Division of Mathematical Sciences"}],"funders":[{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W1646309082","https://openalex.org/W1965130545","https://openalex.org/W1966480196","https://openalex.org/W1966606705","https://openalex.org/W1970476229","https://openalex.org/W1971155154","https://openalex.org/W1972576201","https://openalex.org/W1972649461","https://openalex.org/W1975900269","https://openalex.org/W1982421072","https://openalex.org/W1983156129","https://openalex.org/W1984320340","https://openalex.org/W1986474611","https://openalex.org/W1992066335","https://openalex.org/W2011224005","https://openalex.org/W2018159038","https://openalex.org/W2019797126","https://openalex.org/W2024300082","https://openalex.org/W2029918260","https://openalex.org/W2030373618","https://openalex.org/W2033721384","https://openalex.org/W2034705487","https://openalex.org/W2036766781","https://openalex.org/W2049222546","https://openalex.org/W2051708793","https://openalex.org/W2052275102","https://openalex.org/W2052690618","https://openalex.org/W2055460625","https://openalex.org/W2056558085","https://openalex.org/W2060368109","https://openalex.org/W2083845086","https://openalex.org/W2086018594","https://openalex.org/W2087800106","https://openalex.org/W2101948461","https://openalex.org/W2117756735","https://openalex.org/W2126028850","https://openalex.org/W2135459060","https://openalex.org/W2143591652","https://openalex.org/W2146552914","https://openalex.org/W2163715525","https://openalex.org/W2170798819","https://openalex.org/W2615253071","https://openalex.org/W2963066768","https://openalex.org/W2963529021","https://openalex.org/W2963634130","https://openalex.org/W2963833528","https://openalex.org/W2963876936","https://openalex.org/W3021293439","https://openalex.org/W3103355665"],"related_works":["https://openalex.org/W184885699","https://openalex.org/W1723150922","https://openalex.org/W3214840295","https://openalex.org/W2027111064","https://openalex.org/W1668360133","https://openalex.org/W63747332","https://openalex.org/W2949311065","https://openalex.org/W4224283474","https://openalex.org/W94839203","https://openalex.org/W2582405793"],"abstract_inverted_index":{"We":[0,53],"propose":[1,54],"a":[2,17,68,96,126,187],"stochastic":[3,19,29,43,47,74,163],"multiscale":[4,107],"finite":[5],"element":[6],"method":[7,66,71,94,98],"(StoMsFEM)":[8],"to":[9,25,88,170],"solve":[10],"random":[11,37,90],"elliptic":[12],"partial":[13],"differential":[14],"equations":[15],"with":[16,86,121],"high":[18,79,182],"dimension.":[20,164],"The":[21,64,92],"key":[22],"idea":[23],"is":[24,67,95,142,154,160],"simultaneously":[26],"upscale":[27],"the":[28,32,41,46,73,78,82,89,101,106,122,129,136,143,146,149,155,161,172,177,181],"solutions":[30],"in":[31,72],"physical":[33,156],"space":[34,75],"for":[35,58],"all":[36],"samples":[38],"and":[39,148,158,174],"explore":[40],"low":[42,102],"dimensions":[44],"of":[45,81,105,138,176,189],"solution":[48],"within":[49,110],"each":[50,111],"local":[51,62,83,162],"patch.":[52,114],"two":[55],"effective":[56],"methods":[57],"achieving":[59],"this":[60],"simultaneous":[61],"upscaling.":[63],"first":[65],"high-order":[69],"interpolation":[70],"that":[76,99],"explores":[77,100],"regularity":[80],"upscaled":[84],"quantities":[85],"respect":[87],"variables.":[91],"second":[93],"reduced-order":[97],"rank":[103],"property":[104],"basis":[108],"functions":[109],"coarse":[112,147],"grid":[113,151],"Our":[115],"complexity":[116],"analysis":[117],"shows":[118],"that,":[119],"compared":[120],"standard":[123],"FEM":[124],"on":[125,135],"fine":[127,150],"grid,":[128],"StoMsFEM":[130],"can":[131],"achieve":[132],"computational":[133],"savings":[134],"order":[137],"$(H/h)^{d}/(\\log(H/h))^k$,":[139],"where":[140],"$H/h$":[141],"ratio":[144],"between":[145],"sizes,":[152],"$d$":[153],"dimension,":[157],"$k$":[159],"Several":[165],"numerical":[166],"examples":[167],"are":[168],"presented":[169],"demonstrate":[171],"accuracy":[173],"effectiveness":[175],"proposed":[178],"methods.":[179],"In":[180],"contrast":[183],"example,":[184],"we":[185],"observe":[186],"factor":[188],"2000":[190],"speed-up.":[191]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
