{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20281"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20281","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Incomplete factorization preconditioning for linear least squares problems","abstract":"A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, $min\\Vert Ax-b\\Vert\\sb2$, is proposed. The family is based on incomplete Gram-Schmidt (IGS) factorizations of A. Particular attention has been given to the following members of the family: Incomplete Classical Gram-Schmidt (ICGS), Incomplete Modified Gram-Schmidt (IMGS) and Compressed Incomplete Modified Gram-Schmidt (CIMGS) factorizations. The numerical properties of each of these methods have been considered as well as the relationships between the methods concerning the preservation of sparsity, computational efficiency and the quality of the preconditioner. The implementation of these methods has been investigated and all of the important family members have been coded. One of the important topics in this portion of the dissertation is the careful symbolic analysis of the production of the preconditioner and its use during the incomplete factorization phase to avoid excessive unnecessary work.","abstract_html":"A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, $min\\Vert Ax-b\\Vert\\sb2$, is proposed. The family is based on incomplete Gram-Schmidt (IGS) factorizations of A. Particular attention has been given to the following members of the family: Incomplete Classical Gram-Schmidt (ICGS), Incomplete Modified Gram-Schmidt (IMGS) and Compressed Incomplete Modified Gram-Schmidt (CIMGS) factorizations. The numerical properties of each of these methods have been considered as well as the relationships between the methods concerning the preservation of sparsity, computational efficiency and the quality of the preconditioner. The implementation of these methods has been investigated and all of the important family members have been coded. One of the important topics in this portion of the dissertation is the careful symbolic analysis of the production of the preconditioner and its use during the incomplete factorization phase to avoid excessive unnecessary work.","abstract_has_math":true,"creators":["Wang, Xiaoge"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["Gallivan, Kyle A."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:34:44Z","date_published":"2011-05-07T12:34:44Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics","Computer Science"],"languages":["eng"],"rights":["Copyright 1994 Wang, Xiaoge"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9416449","(UMI)AAI9416449"],"render_values":[{"text":"AAI9416449","href":null,"code":true},{"text":"(UMI)AAI9416449","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20281","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Gallivan, Kyle A."]},{"key":"dc:creator","label":"Author","values":["Wang, Xiaoge"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:34:44Z","10000-01-01","1994"]},{"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":["Mathematics","Computer Science"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Wang, Xiaoge"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9416449","(UMI)AAI9416449","http://hdl.handle.net/2142/20281"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, $min\\Vert Ax-b\\Vert\\sb2$, is proposed. The family is based on incomplete Gram-Schmidt (IGS) factorizations of A. Particular attention has been given to the following members of the family: Incomplete Classical Gram-Schmidt (ICGS), Incomplete Modified Gram-Schmidt (IMGS) and Compressed Incomplete Modified Gram-Schmidt (CIMGS) factorizations. The numerical properties of each of these methods have been considered as well as the relationships between the methods concerning the preservation of sparsity, computational efficiency and the quality of the preconditioner. The implementation of these methods has been investigated and all of the important family members have been coded. One of the important topics in this portion of the dissertation is the careful symbolic analysis of the production of the preconditioner and its use during the incomplete factorization phase to avoid excessive unnecessary work.","We have also analyzed some of the implications of viewing the incomplete Cholesky (IC) factorization as a limit case of the IGS family. When certain conditions are imposed on the sparsity pattern of the preconditioning matrix, IC is identical to IMGS. Therefore, a sufficient condition on the sparsity pattern for a stable IC factorization can be derived. Based on this condition, two algorithms for modifying a sparsity pattern for which IC may not succeed have been designed. The altered IC method with the pattern modifications (ICPM) has been studied and its performance evaluated. These methods differ from previously proposed methods to guarantee the existence and improve the stability of the IC factorization since they do not require numerical information. Of course, for a particular matrix we can then use numerical information in addition to the pattern modification techniques to further improve the preconditioner.","Numerical experiments illustrating the capabilities of the preconditioners are also presented. These include matrices from the standard Harwell-Boeing collection and various test matrices from different practical application problems.","Made available in DSpace on 2011-05-07T12:34:44Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9416449.pdf: 9718058 bytes, checksum: 2043dcae7c5a89a8858c938e71b767e3 (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:50Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:40-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Incomplete factorization preconditioning for linear least squares problems"]}]}],"canonical_facts":{"dc:contributor":["Gallivan, Kyle A."],"dc:creator":["Wang, Xiaoge"],"dc:date":["2011-05-07T12:34:44Z","10000-01-01","1994"],"dc:description":["A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, $min\\Vert Ax-b\\Vert\\sb2$, is proposed. The family is based on incomplete Gram-Schmidt (IGS) factorizations of A. Particular attention has been given to the following members of the family: Incomplete Classical Gram-Schmidt (ICGS), Incomplete Modified Gram-Schmidt (IMGS) and Compressed Incomplete Modified Gram-Schmidt (CIMGS) factorizations. The numerical properties of each of these methods have been considered as well as the relationships between the methods concerning the preservation of sparsity, computational efficiency and the quality of the preconditioner. The implementation of these methods has been investigated and all of the important family members have been coded. One of the important topics in this portion of the dissertation is the careful symbolic analysis of the production of the preconditioner and its use during the incomplete factorization phase to avoid excessive unnecessary work.","We have also analyzed some of the implications of viewing the incomplete Cholesky (IC) factorization as a limit case of the IGS family. When certain conditions are imposed on the sparsity pattern of the preconditioning matrix, IC is identical to IMGS. Therefore, a sufficient condition on the sparsity pattern for a stable IC factorization can be derived. Based on this condition, two algorithms for modifying a sparsity pattern for which IC may not succeed have been designed. The altered IC method with the pattern modifications (ICPM) has been studied and its performance evaluated. These methods differ from previously proposed methods to guarantee the existence and improve the stability of the IC factorization since they do not require numerical information. Of course, for a particular matrix we can then use numerical information in addition to the pattern modification techniques to further improve the preconditioner.","Numerical experiments illustrating the capabilities of the preconditioners are also presented. These include matrices from the standard Harwell-Boeing collection and various test matrices from different practical application problems.","Made available in DSpace on 2011-05-07T12:34:44Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9416449.pdf: 9718058 bytes, checksum: 2043dcae7c5a89a8858c938e71b767e3 (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:50Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:40-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9416449","(UMI)AAI9416449","http://hdl.handle.net/2142/20281"],"dc:language":["eng"],"dc:rights":["Copyright 1994 Wang, Xiaoge"],"dc:subject":["Mathematics","Computer Science"],"dc:title":["Incomplete factorization preconditioning for linear least squares problems"],"dc:type":["text"],"thesis:degree_discipline":["Computer Science"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}