{"id":"https://openalex.org/W2886895161","doi":"https://doi.org/10.1137/17m113767x","title":"Two-Level Spectral Methods for Nonlinear Elliptic Equations with Multiple Solutions","display_name":"Two-Level Spectral Methods for Nonlinear Elliptic Equations with Multiple Solutions","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2886895161","doi":"https://doi.org/10.1137/17m113767x","mag":"2886895161"},"language":"en","primary_location":{"id":"doi:10.1137/17m113767x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m113767x","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/A5009179403","display_name":"Yingwei Wang","orcid":"https://orcid.org/0000-0001-9257-6389"},"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"]},{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Yingwei Wang","raw_affiliation_strings":["Purdue University","University of Wisconsin-Madison"],"affiliations":[{"raw_affiliation_string":"Purdue University","institution_ids":["https://openalex.org/I219193219"]},{"raw_affiliation_string":"University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032431412","display_name":"Wenrui Hao","orcid":"https://orcid.org/0000-0002-6925-7424"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wenrui Hao","raw_affiliation_strings":["Mathematics"],"affiliations":[{"raw_affiliation_string":"Mathematics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078138445","display_name":"Guang Lin","orcid":"https://orcid.org/0000-0002-0976-1987"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Guang Lin","raw_affiliation_strings":["PURDUE UNIVERSITY"],"affiliations":[{"raw_affiliation_string":"PURDUE UNIVERSITY","institution_ids":["https://openalex.org/I219193219"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5009179403"],"corresponding_institution_ids":["https://openalex.org/I135310074","https://openalex.org/I219193219"],"apc_list":null,"apc_paid":null,"fwci":1.8174,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.87153372,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"40","issue":"4","first_page":"B1180","last_page":"B1205"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9995999932289124,"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":0.9995999932289124,"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/T10514","display_name":"Numerical methods in engineering","score":0.9993000030517578,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8287469148635864},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5924444198608398},{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.4733434021472931},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4705524742603302},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.40914931893348694}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8287469148635864},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5924444198608398},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.4733434021472931},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4705524742603302},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40914931893348694},{"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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/17m113767x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m113767x","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":[{"id":"https://openalex.org/G3165708934","display_name":null,"funder_award_id":"DMS-1555072","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3287271385","display_name":null,"funder_award_id":"51728601","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3357136345","display_name":null,"funder_award_id":"DMS-1736364","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G627674342","display_name":null,"funder_award_id":"17SDG33660722","funder_id":"https://openalex.org/F4320306230","funder_display_name":"American Heart Association"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306230","display_name":"American Heart Association","ror":"https://ror.org/013kjyp64"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320332359","display_name":"Office of Science","ror":"https://ror.org/00mmn6b08"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":40,"referenced_works":["https://openalex.org/W228399867","https://openalex.org/W1525628435","https://openalex.org/W1595513685","https://openalex.org/W1967676806","https://openalex.org/W1980372935","https://openalex.org/W1980455776","https://openalex.org/W1982227707","https://openalex.org/W1984781443","https://openalex.org/W1985843599","https://openalex.org/W1988665632","https://openalex.org/W1989369020","https://openalex.org/W1995464753","https://openalex.org/W2003539068","https://openalex.org/W2006640177","https://openalex.org/W2017786109","https://openalex.org/W2020010349","https://openalex.org/W2028811441","https://openalex.org/W2030004448","https://openalex.org/W2035498182","https://openalex.org/W2038928461","https://openalex.org/W2041657637","https://openalex.org/W2045467508","https://openalex.org/W2046128078","https://openalex.org/W2054563910","https://openalex.org/W2058525241","https://openalex.org/W2069941038","https://openalex.org/W2073602344","https://openalex.org/W2082849992","https://openalex.org/W2117652326","https://openalex.org/W2122069319","https://openalex.org/W2132394231","https://openalex.org/W2148110955","https://openalex.org/W2229505215","https://openalex.org/W2269132390","https://openalex.org/W2334003472","https://openalex.org/W2410898212","https://openalex.org/W2461925557","https://openalex.org/W2523227026","https://openalex.org/W2962828807","https://openalex.org/W4241062739"],"related_works":["https://openalex.org/W2137442321","https://openalex.org/W2049401387","https://openalex.org/W2963460081","https://openalex.org/W2050770628","https://openalex.org/W4313906306","https://openalex.org/W2162316542","https://openalex.org/W1978414406","https://openalex.org/W4280592256","https://openalex.org/W2353181440","https://openalex.org/W2220084361"],"abstract_inverted_index":{"The":[0,63],"present":[1],"paper":[2],"provides":[3],"a":[4,40,57],"two-level":[5,65],"framework":[6],"based":[7],"on":[8],"spectral":[9,66],"methods":[10],"and":[11,46,84,104,111],"homotopy":[12],"continuation":[13],"for":[14,101],"solving":[15,32,48],"second-order":[16],"nonlinear":[17,34,113],"boundary":[18],"value":[19],"problems":[20,35,52],"exhibiting":[21],"multiple":[22,75],"solutions.":[23],"Our":[24],"proposed":[25],"method":[26,67],"consists":[27],"of":[28,43,60,88,108],"two":[29],"steps:":[30],"(i)":[31,72],"the":[33,49,69,78,97],"using":[36],"low-order":[37],"polynomials":[38,55],"or":[39,56],"small":[41],"number":[42,59],"collocation":[44,61],"points,":[45],"(ii)":[47,77],"corresponding":[50],"linearized":[51],"by":[53,96],"high-order":[54,90],"large":[58],"points.":[62],"resulting":[64],"enjoys":[68],"following":[70],"merits:":[71],"it":[73,86],"guarantees":[74],"solutions,":[76],"computational":[79],"cost":[80],"is":[81,87],"relatively":[82],"small,":[83],"(iii)":[85],"proven":[89],"accuracy.":[91],"These":[92],"claims":[93],"are":[94],"supported":[95],"detailed":[98],"error":[99],"estimates":[100],"semilinear":[102,110],"equations":[103],"extensive":[105],"numerical":[106],"experiments":[107],"both":[109],"fully":[112],"equations.":[114]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}