{"id":"https://openalex.org/W4388977060","doi":"https://doi.org/10.1007/s10589-023-00540-w","title":"The continuous stochastic gradient method: part II\u2013application and numerics","display_name":"The continuous stochastic gradient method: part II\u2013application and numerics","publication_year":2023,"publication_date":"2023-11-24","ids":{"openalex":"https://openalex.org/W4388977060","doi":"https://doi.org/10.1007/s10589-023-00540-w"},"language":"en","primary_location":{"id":"doi:10.1007/s10589-023-00540-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-023-00540-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00540-w.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00540-w.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031898082","display_name":"Max Grieshammer","orcid":null},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Max Grieshammer","raw_affiliation_strings":["Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069223642","display_name":"Lukas Pflug","orcid":"https://orcid.org/0000-0001-8001-5832"},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Lukas Pflug","raw_affiliation_strings":["Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany","FAU Competence Center Scientific Computing, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany","institution_ids":["https://openalex.org/I181369854"]},{"raw_affiliation_string":"FAU Competence Center Scientific Computing, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013459854","display_name":"Michael Stingl","orcid":"https://orcid.org/0000-0002-3626-0723"},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Michael Stingl","raw_affiliation_strings":["Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5000112468","display_name":"Andrian Uihlein","orcid":"https://orcid.org/0000-0002-0650-3747"},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Andrian Uihlein","raw_affiliation_strings":["Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany"],"raw_orcid":"https://orcid.org/0000-0002-0650-3747","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Chair of Applied Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Erlangen, Germany","institution_ids":["https://openalex.org/I181369854"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5031898082"],"corresponding_institution_ids":["https://openalex.org/I181369854"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.3618,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.85153462,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"87","issue":"3","first_page":"977","last_page":"1008"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9915000200271606,"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/T11115","display_name":"Topology Optimization in Engineering","score":0.9704999923706055,"subfield":{"id":"https://openalex.org/subfields/2205","display_name":"Civil and Structural 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":0.6457406282424927},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5977282524108887},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5821185111999512},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5355678796768188},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5224101543426514},{"id":"https://openalex.org/keywords/stochastic-optimization","display_name":"Stochastic optimization","score":0.5195108652114868},{"id":"https://openalex.org/keywords/gradient-method","display_name":"Gradient method","score":0.513978898525238},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.49039316177368164},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4611968994140625},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.42765650153160095},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.42715027928352356},{"id":"https://openalex.org/keywords/topology-optimization","display_name":"Topology optimization","score":0.4262523353099823},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27324172854423523},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.26186680793762207},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12149965763092041},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.0637446939945221}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6457406282424927},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5977282524108887},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5821185111999512},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5355678796768188},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5224101543426514},{"id":"https://openalex.org/C194387892","wikidata":"https://www.wikidata.org/wiki/Q1747770","display_name":"Stochastic optimization","level":2,"score":0.5195108652114868},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.513978898525238},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.49039316177368164},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4611968994140625},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.42765650153160095},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.42715027928352356},{"id":"https://openalex.org/C189216461","wikidata":"https://www.wikidata.org/wiki/Q2443456","display_name":"Topology optimization","level":3,"score":0.4262523353099823},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27324172854423523},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26186680793762207},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12149965763092041},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.0637446939945221},{"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","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},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"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":4,"locations":[{"id":"doi:10.1007/s10589-023-00540-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-023-00540-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00540-w.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:coopap:v:87:y:2024:i:3:d:10.1007_s10589-023-00540-w","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10589-023-00540-w","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"},{"id":"pmh:oai:econstor.eu:10419/312463","is_oa":true,"landing_page_url":"https://hdl.handle.net/10419/312463","pdf_url":null,"source":{"id":"https://openalex.org/S4306401696","display_name":"Econstor (Econstor)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"doc-type:article"},{"id":"pmh:oai:econstor.eu:10419/317927","is_oa":true,"landing_page_url":"https://hdl.handle.net/10419/317927","pdf_url":null,"source":{"id":"https://openalex.org/S4306401696","display_name":"Econstor (Econstor)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"doc-type:article"}],"best_oa_location":{"id":"doi:10.1007/s10589-023-00540-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-023-00540-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00540-w.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3106994977","display_name":null,"funder_award_id":"416229255","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G5764230684","display_name":null,"funder_award_id":"CRC 1411","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G7841707202","display_name":null,"funder_award_id":"416229225","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4388977060.pdf"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W1494085563","https://openalex.org/W1660964545","https://openalex.org/W1964614150","https://openalex.org/W1994616650","https://openalex.org/W2011994939","https://openalex.org/W2017449002","https://openalex.org/W2036581304","https://openalex.org/W2045276693","https://openalex.org/W2080492562","https://openalex.org/W2094817952","https://openalex.org/W2125542949","https://openalex.org/W2126092109","https://openalex.org/W2147708754","https://openalex.org/W2151084831","https://openalex.org/W2162246342","https://openalex.org/W2487619204","https://openalex.org/W2505244635","https://openalex.org/W2622515460","https://openalex.org/W2755862218","https://openalex.org/W2763058577","https://openalex.org/W2801548858","https://openalex.org/W2963134136","https://openalex.org/W2963156201","https://openalex.org/W2963433607","https://openalex.org/W2991249485","https://openalex.org/W3032328388","https://openalex.org/W4295339154","https://openalex.org/W4312126127","https://openalex.org/W4360877661","https://openalex.org/W4388943625"],"related_works":["https://openalex.org/W2046491619","https://openalex.org/W2389142552","https://openalex.org/W2239929624","https://openalex.org/W2883804989","https://openalex.org/W1974803732","https://openalex.org/W2008556190","https://openalex.org/W3203799340","https://openalex.org/W2785358337","https://openalex.org/W65522538","https://openalex.org/W2313335380"],"abstract_inverted_index":{"Abstract":[0],"In":[1,24,98],"this":[2,99],"contribution,":[3],"we":[4],"present":[5],"a":[6,51,78,84,91,120,141],"numerical":[7,134,172],"analysis":[8],"of":[9,73,103,123,136,171],"the":[10,58,62,66,71,74,96,101,133,169],"continuous":[11],"stochastic":[12,28,86],"gradient":[13,29,37,60,93],"(CSG)":[14],"method,":[15],"including":[16],"applications":[17,110],"from":[18,39,111],"topology":[19,112],"optimization":[20,30,124,153],"and":[21,88,126,166],"convergence":[22,160],"rates.":[23],"contrast":[25],"to":[26,49,57,159],"standard":[27],"schemes,":[31],"CSG":[32,76,104],"does":[33],"not":[34,147],"discard":[35],"old":[36],"samples":[38],"previous":[40],"iterations.":[41],"Instead,":[42],"design":[43],"dependent":[44],"integration":[45],"weights":[46],"are":[47,116,164],"calculated":[48],"form":[50],"convex":[52],"combination":[53],"as":[54],"an":[55,127],"approximation":[56,67],"true":[59],"at":[61],"current":[63],"design.":[64],"As":[65],"error":[68],"vanishes":[69],"in":[70,95,140],"course":[72],"iterations,":[75],"represents":[77],"hybrid":[79],"approach,":[80],"starting":[81],"off":[82],"like":[83,90],"purely":[85],"method":[87,154],"behaving":[89],"full":[92],"scheme":[94],"limit.":[97],"work,":[100],"efficiency":[102],"is":[105],"demonstrated":[106],"for":[107],"practically":[108],"relevant":[109],"optimization.":[113],"These":[114],"settings":[115],"characterized":[117],"by":[118,150],"both,":[119],"large":[121],"number":[122],"variables":[125],"objective":[128],"function,":[129],"whose":[130],"evaluation":[131],"requires":[132],"computation":[135],"multiple":[137],"integrals":[138],"concatenated":[139],"nonlinear":[142],"fashion.":[143],"Such":[144],"problems":[145],"could":[146],"be":[148],"solved":[149],"any":[151],"existing":[152],"before.":[155],"Lastly,":[156],"with":[157,168],"regards":[158],"rates,":[161],"first":[162],"estimates":[163],"provided":[165],"confirmed":[167],"help":[170],"experiments.":[173]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-04-25T08:17:42.794288","created_date":"2025-10-10T00:00:00"}
