{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/24197"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/24197","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the POD Basis","abstract":"In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup.","abstract_html":"In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup.","abstract_has_math":false,"creators":["Unger, Benjamin"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Borggaard, Jeffrey T."],"committee_members":["Elgart, Alexander","Burns, John A."],"year":2013,"date_issued":"2013-11-19","date_published":"2013-11-19","updated_at":"2026-07-22T22:19:01Z","subjects":["nonlinear model reduction","burgers equation","pod","deim","trust region pod"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:1709"],"render_values":[{"text":"vt_gsexam:1709","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/24197","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Borggaard, Jeffrey T."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Elgart, Alexander","Burns, John A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Unger, Benjamin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2013-11-20T09:00:09Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2013-11-20T09:00:09Z"]},{"key":"dc:date.issued","label":"Date","values":["2013-11-19"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["nonlinear model reduction","burgers equation","pod","deim","trust region pod"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:1709"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/24197"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the POD Basis"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Borggaard, Jeffrey T."],"dc:contributor.committeemember":["Elgart, Alexander","Burns, John A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Unger, Benjamin"],"dc:date.accessioned":["2013-11-20T09:00:09Z"],"dc:date.available":["2013-11-20T09:00:09Z"],"dc:date.issued":["2013-11-19"],"dc:description.abstract":["In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:1709"],"dc:identifier.uri":["http://hdl.handle.net/10919/24197"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["nonlinear model reduction","burgers equation","pod","deim","trust region pod"],"dc:title":["Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the POD Basis"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:01Z"}