{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/104752"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/104752","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Exploiting compression in solving discretized linear systems","abstract":"Solving systems of linear algebraic equations is crucial for many computational problems in science and engineering. Numerous techniques are available for solving such linear systems, including direct methods such as Gaussian elimination and iterative methods such as GMRES. This thesis proposes a method for exploiting compression while computing the solution to a given discretized system of linear algebraic equations and investigates both its overall effectiveness in practice and which factors determine its effectiveness. The method is based on computing an approximate solution in a reduced space, and thus we seek a basis in which the solution has a compressed representation and can consequently be computed more efficiently. We address three primary issues: (1) how to compute an approximate solution to the given discretized linear system using a given basis, (2) how to choose a basis that yields significant compression, and (3) how to detect when the basis is of sufficient dimension to provide a satisfactory approximation. While all three aspects have antecedents in previous ideas and methods, we combine, adapt, and extend them in a manner we believe to be novel for the purpose of solving discretized linear systems. We demonstrate that the resulting method can be competitive with, and sometimes outperform, current standard methods and is effective for efficiently solving linear systems resulting from the discretization of major classes of continuous problems.","abstract_html":"Solving systems of linear algebraic equations is crucial for many computational problems in science and engineering. Numerous techniques are available for solving such linear systems, including direct methods such as Gaussian elimination and iterative methods such as GMRES. This thesis proposes a method for exploiting compression while computing the solution to a given discretized system of linear algebraic equations and investigates both its overall effectiveness in practice and which factors determine its effectiveness. The method is based on computing an approximate solution in a reduced space, and thus we seek a basis in which the solution has a compressed representation and can consequently be computed more efficiently. We address three primary issues: (1) how to compute an approximate solution to the given discretized linear system using a given basis, (2) how to choose a basis that yields significant compression, and (3) how to detect when the basis is of sufficient dimension to provide a satisfactory approximation. While all three aspects have antecedents in previous ideas and methods, we combine, adapt, and extend them in a manner we believe to be novel for the purpose of solving discretized linear systems. We demonstrate that the resulting method can be competitive with, and sometimes outperform, current standard methods and is effective for efficiently solving linear systems resulting from the discretization of major classes of continuous problems.","abstract_has_math":false,"creators":["Carrier, Erin Elizabeth"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["Heath, Michael T.","Olson, Luke","Fischer, Paul","Hansen, Per Christian"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-08-23T19:51:31Z","date_published":"2019-08-23T19:51:31Z","updated_at":"2026-07-22T22:24:42Z","subjects":["linear systems","compressed solution","compression basis","projection method","discretized linear system","regularization method"],"languages":["en"],"rights":["Copyright 2019 Erin Elizabeth Carrier"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/104752","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Heath, Michael T.","Olson, Luke","Fischer, Paul","Hansen, Per Christian"]},{"key":"dc:creator","label":"Author","values":["Carrier, Erin Elizabeth"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-08-23T19:51:31Z","2019-03-05","2019-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["linear systems","compressed solution","compression basis","projection method","discretized linear system","regularization method"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 Erin Elizabeth Carrier"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/104752"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Solving systems of linear algebraic equations is crucial for many computational problems in science and engineering. Numerous techniques are available for solving such linear systems, including direct methods such as Gaussian elimination and iterative methods such as GMRES. This thesis proposes a method for exploiting compression while computing the solution to a given discretized system of linear algebraic equations and investigates both its overall effectiveness in practice and which factors determine its effectiveness. The method is based on computing an approximate solution in a reduced space, and thus we seek a basis in which the solution has a compressed representation and can consequently be computed more efficiently. We address three primary issues: (1) how to compute an approximate solution to the given discretized linear system using a given basis, (2) how to choose a basis that yields significant compression, and (3) how to detect when the basis is of sufficient dimension to provide a satisfactory approximation. While all three aspects have antecedents in previous ideas and methods, we combine, adapt, and extend them in a manner we believe to be novel for the purpose of solving discretized linear systems. We demonstrate that the resulting method can be competitive with, and sometimes outperform, current standard methods and is effective for efficiently solving linear systems resulting from the discretization of major classes of continuous problems.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-08-22 without embargo terms","The student, Erin Carrier, accepted the attached license on 2019-03-04 at 16:13.","The student, Erin Carrier, submitted this Dissertation for approval on 2019-03-05 at 10:46.","This Dissertation was approved for publication on 2019-03-05 at 13:28.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13409 on 2019-08-22 at 14:40:30","Made available in DSpace on 2019-08-23T19:51:31Z (GMT). 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This thesis proposes a method for exploiting compression while computing the solution to a given discretized system of linear algebraic equations and investigates both its overall effectiveness in practice and which factors determine its effectiveness. The method is based on computing an approximate solution in a reduced space, and thus we seek a basis in which the solution has a compressed representation and can consequently be computed more efficiently. We address three primary issues: (1) how to compute an approximate solution to the given discretized linear system using a given basis, (2) how to choose a basis that yields significant compression, and (3) how to detect when the basis is of sufficient dimension to provide a satisfactory approximation. While all three aspects have antecedents in previous ideas and methods, we combine, adapt, and extend them in a manner we believe to be novel for the purpose of solving discretized linear systems. We demonstrate that the resulting method can be competitive with, and sometimes outperform, current standard methods and is effective for efficiently solving linear systems resulting from the discretization of major classes of continuous problems.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-08-22 without embargo terms","The student, Erin Carrier, accepted the attached license on 2019-03-04 at 16:13.","The student, Erin Carrier, submitted this Dissertation for approval on 2019-03-05 at 10:46.","This Dissertation was approved for publication on 2019-03-05 at 13:28.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13409 on 2019-08-22 at 14:40:30","Made available in DSpace on 2019-08-23T19:51:31Z (GMT). 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