{"id":"https://openalex.org/W2067590156","doi":"https://doi.org/10.1109/mmar.2014.6957404","title":"A Hopf-Lax formula for the level-set equation and applications to PDE-constrained shape optimisation","display_name":"A Hopf-Lax formula for the level-set equation and applications to PDE-constrained shape optimisation","publication_year":2014,"publication_date":"2014-09-01","ids":{"openalex":"https://openalex.org/W2067590156","doi":"https://doi.org/10.1109/mmar.2014.6957404","mag":"2067590156"},"language":"en","primary_location":{"id":"doi:10.1109/mmar.2014.6957404","is_oa":false,"landing_page_url":"https://doi.org/10.1109/mmar.2014.6957404","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2014 19th International Conference on Methods and Models in Automation and Robotics (MMAR)","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://unipub.uni-graz.at/obvugroa/content/titleinfo/4376274/full.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5090228551","display_name":"Daniel Kraft","orcid":"https://orcid.org/0000-0002-0862-5350"},"institutions":[{"id":"https://openalex.org/I15766117","display_name":"University of Graz","ror":"https://ror.org/01faaaf77","country_code":"AT","type":"education","lineage":["https://openalex.org/I15766117"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Daniel Kraft","raw_affiliation_strings":["University of Graz Institute of Mathematics, Graz, Austria","University of Graz, Institute of Mathematics, NAWI Graz, Universit\u00e4tsplatz 3, 8010 Graz, Austria"],"affiliations":[{"raw_affiliation_string":"University of Graz Institute of Mathematics, Graz, Austria","institution_ids":["https://openalex.org/I15766117"]},{"raw_affiliation_string":"University of Graz, Institute of Mathematics, NAWI Graz, Universit\u00e4tsplatz 3, 8010 Graz, Austria","institution_ids":["https://openalex.org/I15766117"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5090228551"],"corresponding_institution_ids":["https://openalex.org/I15766117"],"apc_list":null,"apc_paid":null,"fwci":1.2874,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.79348244,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":null,"issue":null,"first_page":"498","last_page":"503"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.994700014591217,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.994700014591217,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.986299991607666,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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":0.9825999736785889,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6503943800926208},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5790559649467468},{"id":"https://openalex.org/keywords/level-set","display_name":"Level set (data structures)","score":0.5261670351028442},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.48636963963508606},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4858691096305847},{"id":"https://openalex.org/keywords/solution-set","display_name":"Solution set","score":0.451660692691803},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.440099835395813},{"id":"https://openalex.org/keywords/hamiltonian","display_name":"Hamiltonian (control theory)","score":0.4315780997276306},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.429842084646225},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3518820106983185},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3269191384315491},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.25748783349990845},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.19149550795555115}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6503943800926208},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5790559649467468},{"id":"https://openalex.org/C153008295","wikidata":"https://www.wikidata.org/wiki/Q6535093","display_name":"Level set (data structures)","level":2,"score":0.5261670351028442},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.48636963963508606},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4858691096305847},{"id":"https://openalex.org/C60033838","wikidata":"https://www.wikidata.org/wiki/Q1751685","display_name":"Solution set","level":3,"score":0.451660692691803},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.440099835395813},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.4315780997276306},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.429842084646225},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3518820106983185},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3269191384315491},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25748783349990845},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.19149550795555115},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/mmar.2014.6957404","is_oa":false,"landing_page_url":"https://doi.org/10.1109/mmar.2014.6957404","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2014 19th International Conference on Methods and Models in Automation and Robotics (MMAR)","raw_type":"proceedings-article"},{"id":"pmh:oai:unipub.uni-graz.at:4376274","is_oa":true,"landing_page_url":"https://resolver.obvsg.at/urn:nbn:at:at-ubg:3-6544","pdf_url":"https://unipub.uni-graz.at/obvugroa/content/titleinfo/4376274/full.pdf","source":{"id":"https://openalex.org/S4306400359","display_name":"Unipub UB Graz (Universit\u00e4t Graz)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I15766117","host_organization_name":"University of Graz","host_organization_lineage":["https://openalex.org/I15766117"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"pmh:oai:unipub.uni-graz.at:4376274","is_oa":true,"landing_page_url":"https://resolver.obvsg.at/urn:nbn:at:at-ubg:3-6544","pdf_url":"https://unipub.uni-graz.at/obvugroa/content/titleinfo/4376274/full.pdf","source":{"id":"https://openalex.org/S4306400359","display_name":"Unipub UB Graz (Universit\u00e4t Graz)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I15766117","host_organization_name":"University of Graz","host_organization_lineage":["https://openalex.org/I15766117"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5721084645","display_name":null,"funder_award_id":"IGDK 1754","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"},{"id":"https://openalex.org/F4320325582","display_name":"Karl-Franzens-Universit\u00e4t Graz","ror":"https://ror.org/01faaaf77"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2067590156.pdf","grobid_xml":"https://content.openalex.org/works/W2067590156.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W261580265","https://openalex.org/W1532111732","https://openalex.org/W1593038947","https://openalex.org/W1595065160","https://openalex.org/W1939243400","https://openalex.org/W1991113069","https://openalex.org/W1999244633","https://openalex.org/W2003166268","https://openalex.org/W2032316144","https://openalex.org/W2481372533","https://openalex.org/W4236632459","https://openalex.org/W4245654886","https://openalex.org/W4293183906","https://openalex.org/W6640445486"],"related_works":["https://openalex.org/W2075972694","https://openalex.org/W3080476861","https://openalex.org/W1983586807","https://openalex.org/W4238581249","https://openalex.org/W2223158884","https://openalex.org/W2024959962","https://openalex.org/W2037624026","https://openalex.org/W2027510160","https://openalex.org/W2044047017","https://openalex.org/W4224441902"],"abstract_inverted_index":{"Level-sets":[0],"are":[1],"a":[2,13,21,29,54,66,79],"flexible":[3],"method":[4],"to":[5,12,36,59,106,124],"describe":[6],"geometries":[7,49],"and":[8,91,109,132],"their":[9],"changes":[10],"according":[11],"speed":[14,63],"field.":[15],"This":[16,51],"can":[17,33,85,102],"be":[18,34,86,103],"used":[19,35,87],"in":[20,128],"wide":[22],"variety":[23],"of":[24,40,56,62,75,113],"applications.":[25],"We":[26,117],"will":[27,118],"present":[28,133],"Hopf-Lax":[30],"formula":[31,52,84],"that":[32],"represent":[37],"the":[38,41,47,60,95,98,114,121],"solution":[39],"level-set":[42,122],"equation":[43,73],"as":[44,46],"well":[45],"described":[48],"directly.":[50],"is":[53,74],"generalisation":[55],"existing":[57],"results":[58],"case":[61],"fields":[64],"without":[65],"uniform,":[67],"positive":[68],"lower":[69],"bound.":[70],"The":[71],"corresponding":[72],"Hamilton-Jacobi":[76],"type":[77],"with":[78],"non-convex":[80],"Hamiltonian.":[81],"Our":[82],"representation":[83],"both":[88],"for":[89],"theoretical":[90],"numerical":[92,111,134],"purposes.":[93],"In":[94],"latter":[96],"case,":[97],"Fast":[99],"Marching":[100],"Method":[101],"applied,":[104],"leading":[105],"very":[107],"efficient":[108],"robust":[110],"calculations":[112],"geometry":[115],"evolutions.":[116],"also":[119],"apply":[120],"framework":[123],"an":[125],"illustrative":[126],"problem":[127],"PDE-constrained":[129],"shape":[130],"optimisation,":[131],"results.":[135]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}