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Klaus-Jürgen Bathe and Mavis Driscoll. Wilkins, Bryce Daniel. Massachusetts Institute of Technology. Department of Mechanical Engineering. Massachusetts Institute of Technology. Department of Mechanical Engineering 2019-09-17T19:49:27Z 2019-09-17T19:49:27Z 2019 2019 https://hdl.handle.net/1721.1/122238 1119389008 Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2019 Cataloged from PDF version of thesis. Includes bibliographical references (pages 91-93). Since its development in 1971, the Bathe subspace iteration method has been widely-used to solve the generalized symmetric-definite eigenvalue problem. The method is particularly useful for solving large eigenvalue problems when only a few of the least dominant eigenpairs are sought. In reference [18], an enriched subspace iteration method was proposed that accelerated the convergence of the basic method by replacing some of the iteration vectors with more effective turning vectors. In this thesis, we build upon this recent acceleration effort and further enrich the subspace of each iteration by replacing additional iteration vectors with our new turning-of-turning vectors. We begin by reviewing the underpinnings of the subspace iteration methodology. Then, we present the steps of our new algorithm, which we refer to as the Enriched- Enriched (E2 ) Bathe subspace iteration method. This is followed by a tabulation of the number of floating point operations incurred during a general iteration of the E2 algorithm. Additionally, we perform a simplified convergence analysis showing that the E2 method converges asymptotically at a faster rate than the enriched method. Finally, we examine the results from several test problems that were used to illustrate the E2 method and to assess its potential computational savings compared to the enriched method. The sample results for the E2 method are consistent with the theoretical asymptotic convergence rate that was obtained in our convergence analysis. Further, the results from the CPU time tests suggest that the E2 method can often provide a useful reduction in computational effort compared to the enriched method, particularly when relatively few iteration vectors are used in comparison with the number of eigenpairs that are sought. by Bryce Daniel Wilkins. S.M. S.M. Massachusetts Institute of Technology, Department of Mechanical Engineering 93 pages eng Massachusetts Institute of Technology MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. http://dspace.mit.edu/handle/1721.1/7582 Mechanical Engineering. The E² Bathe subspace iteration method Enriched-Enriched Bathe subspace iteration method Thesis application/pdf 2019-09-17T19:49:25Z Publication Master MechE unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="552b9841-4a10-4e0e-b515-a6a0de5bdc4d"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Language>eng</Language> <Title>The E² Bathe subspace iteration method</Title> <Subtitle>Enriched-Enriched Bathe subspace iteration method</Subtitle> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2019</PublicationDate> <Authors> <Author> <DisplayName>Wilkins, Bryce Daniel.</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://dspace.mit.edu/handle/1721.1/7582</License> <Keyword>Mechanical Engineering.</Keyword> <Abstract>Since its development in 1971, the Bathe subspace iteration method has been widely-used to solve the generalized symmetric-definite eigenvalue problem. The method is particularly useful for solving large eigenvalue problems when only a few of the least dominant eigenpairs are sought. In reference [18], an enriched subspace iteration method was proposed that accelerated the convergence of the basic method by replacing some of the iteration vectors with more effective turning vectors. In this thesis, we build upon this recent acceleration effort and further enrich the subspace of each iteration by replacing additional iteration vectors with our new turning-of-turning vectors. We begin by reviewing the underpinnings of the subspace iteration methodology. Then, we present the steps of our new algorithm, which we refer to as the Enriched- Enriched (E2 ) Bathe subspace iteration method. This is followed by a tabulation of the number of floating point operations incurred during a general iteration of the E2 algorithm. Additionally, we perform a simplified convergence analysis showing that the E2 method converges asymptotically at a faster rate than the enriched method. Finally, we examine the results from several test problems that were used to illustrate the E2 method and to assess its potential computational savings compared to the enriched method. The sample results for the E2 method are consistent with the theoretical asymptotic convergence rate that was obtained in our convergence analysis. Further, the results from the CPU time tests suggest that the E2 method can often provide a useful reduction in computational effort compared to the enriched method, particularly when relatively few iteration vectors are used in comparison with the number of eigenpairs that are sought.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>