{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/32026"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/32026","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Nodal Reordering Strategies to Improve Preconditioning for Finite Element Systems","abstract":"The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. However, new algorithms need to be developed to take advantage of the new computer architecture (in particular, distributed memory clusters). Since the solution of linear systems still demands most of the computational effort in many problems (such as the approximation of partial differential equation models) iterative methods and, in particular, efficient preconditioners need to be developed. In this study, we consider application of incomplete LU (ILU) preconditioners for finite element models to partial differential equations. Since finite elements lead to large, sparse systems, reordering the node numbers can have a substantial influence on the effectiveness of these preconditioners. We study two implementations of the ILU preconditioner: a stucturebased method and a threshold-based method. The main emphasis of the thesis is to test a variety of breadth-first ordering strategies on the convergence properties of the preconditioned systems. These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. Finally, we perform a preliminary study of the multi-processor case and observe the importance of partitioning quality and the parallel ILU reordering strategy.","abstract_html":"The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. However, new algorithms need to be developed to take advantage of the new computer architecture (in particular, distributed memory clusters). Since the solution of linear systems still demands most of the computational effort in many problems (such as the approximation of partial differential equation models) iterative methods and, in particular, efficient preconditioners need to be developed. In this study, we consider application of incomplete LU (ILU) preconditioners for finite element models to partial differential equations. Since finite elements lead to large, sparse systems, reordering the node numbers can have a substantial influence on the effectiveness of these preconditioners. We study two implementations of the ILU preconditioner: a stucturebased method and a threshold-based method. The main emphasis of the thesis is to test a variety of breadth-first ordering strategies on the convergence properties of the preconditioned systems. These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. Finally, we perform a preliminary study of the multi-processor case and observe the importance of partitioning quality and the parallel ILU reordering strategy.","abstract_has_math":false,"creators":["Hou, Peter S."],"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":["Iliescu, Traian","Gugercin, Serkan"],"year":2005,"date_issued":"2005-04-27","date_published":"2005-04-27","updated_at":"2026-07-22T22:19:22Z","subjects":["Iterative Solver","Scientific Computing","Unstructured Mesh","Finite element method","Preconditioner","Nodal Reordering Strategy"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04292005-144622"],"render_values":[{"text":"etd-04292005-144622","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/32026","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":["Iliescu, Traian","Gugercin, Serkan"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Hou, Peter S."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:34:35Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:34:35Z","2005-05-05"]},{"key":"dc:date.issued","label":"Date","values":["2005-04-27"]},{"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":["Iterative Solver","Scientific Computing","Unstructured Mesh","Finite element method","Preconditioner","Nodal Reordering Strategy"]}]},{"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":["etd-04292005-144622"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/32026"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. 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These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. Finally, we perform a preliminary study of the multi-processor case and observe the importance of partitioning quality and the parallel ILU reordering strategy."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:title","label":"Title","values":["Nodal Reordering Strategies to Improve Preconditioning for Finite Element Systems"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Borggaard, Jeffrey T."],"dc:contributor.committeemember":["Iliescu, Traian","Gugercin, Serkan"],"dc:contributor.department":["Mathematics"],"dc:creator":["Hou, Peter S."],"dc:date.accessioned":["2014-03-14T20:34:35Z"],"dc:date.available":["2014-03-14T20:34:35Z","2005-05-05"],"dc:date.issued":["2005-04-27"],"dc:description.abstract":["The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. 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These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. 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