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Jamie Peraire. Men, Han (Han Abby) Massachusetts Institute of Technology. Computation for Design and Optimization Program. Massachusetts Institute of Technology. Computation for Design and Optimization Program 2007-10-19T20:32:05Z 2007-10-19T20:32:05Z 2006 2006 http://hdl.handle.net/1721.1/39214 85844161 Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006. Includes bibliographical references (p. 61-65). In this thesis, the reduced-basis method is applied to nonlinear time-dependent convection-diffusion parameterized partial differential equations (PDEs). A proper orthogonal decomposition (POD) procedure is used for the construction of reduced-basis approximation for the field variables. In the presence of highly nonlinear terms, conventional reduced-basis would be inefficient and no longer superior to classical numerical approaches using advanced iterative techniques. To recover the computational advantage of the reduced-basis approach, an empirical interpolation approximation method is employed to define the coefficient-function approximation of the nonlinear terms. Next, the coefficient-function approximation is incorporated into the reduced-basis method to obtain a reduced-order model of nonlinear time-dependent parameterized convection-diffusion PDEs. Two formulations for the reduced-order models are proposed, which construct the reduced-basis space for the nonlinear functions and residual vector respectively. Finally, an offline-online procedure for rapid and inexpensive evaluation of the reduced-order model solutions and outputs, as well as associated asymptotic a posterior error estimators are developed. (cont.) The operation count for the online stage depends only on the dimension of our reduced-basis approximation space and the dimension of our coefficient-function approximation space. The extension of the reduced-order model to a system of equations is also explored. by Han Men. S.M. 65 p. eng Massachusetts Institute of Technology M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 Computation for Design and Optimization Program. Efficient reduced-basis approximation of scalar nonlinear time-dependent convection-diffusion problems, and extension to compressible flow problems Thesis application/pdf false Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="41ff9e5f-780c-4cac-8cb7-8fd985910aec"> <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>Efficient reduced-basis approximation of scalar nonlinear time-dependent convection-diffusion problems, and extension to compressible flow problems</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2006</PublicationDate> <Authors> <Author> <DisplayName>Men, Han (Han Abby)</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>Computation for Design and Optimization Program.</Keyword> <Abstract>In this thesis, the reduced-basis method is applied to nonlinear time-dependent convection-diffusion parameterized partial differential equations (PDEs). A proper orthogonal decomposition (POD) procedure is used for the construction of reduced-basis approximation for the field variables. In the presence of highly nonlinear terms, conventional reduced-basis would be inefficient and no longer superior to classical numerical approaches using advanced iterative techniques. To recover the computational advantage of the reduced-basis approach, an empirical interpolation approximation method is employed to define the coefficient-function approximation of the nonlinear terms. Next, the coefficient-function approximation is incorporated into the reduced-basis method to obtain a reduced-order model of nonlinear time-dependent parameterized convection-diffusion PDEs. Two formulations for the reduced-order models are proposed, which construct the reduced-basis space for the nonlinear functions and residual vector respectively. Finally, an offline-online procedure for rapid and inexpensive evaluation of the reduced-order model solutions and outputs, as well as associated asymptotic a posterior error estimators are developed.</Abstract> <Abstract>(cont.) The operation count for the online stage depends only on the dimension of our reduced-basis approximation space and the dimension of our coefficient-function approximation space. The extension of the reduced-order model to a system of equations is also explored.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>