{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/81180"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/81180","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Krylov Projection Methods for Model Reduction","abstract":"This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reduced-order models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete model reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.","abstract_html":"This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reduced-order models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete model reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.","abstract_has_math":false,"creators":["Grimme, Eric James"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Electrical Engineering","degree_department":null,"school":null,"contributors":["Kyle Gallivan","Paul Van Dooren"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-25T20:09:57Z","date_published":"2015-09-25T20:09:57Z","updated_at":"2026-07-22T22:26:15Z","subjects":["Engineering, Electronics and Electrical"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9737124"],"render_values":[{"text":"(MiAaPQ)AAI9737124","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/81180","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kyle Gallivan","Paul Van Dooren"]},{"key":"dc:creator","label":"Author","values":["Grimme, Eric James"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-25T20:09:57Z","10000-01-01","1997"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Electrical Engineering"]},{"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":["Engineering, Electronics and Electrical"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/81180","(MiAaPQ)AAI9737124"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. 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Projections onto unions of Krylov subspaces lead to a class of reduced-order models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete model reduction package. 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