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Karen E. Willcox. Galbally, David Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics 2008-11-07T18:59:45Z 2008-11-07T18:59:45Z 2008 2008 http://hdl.handle.net/1721.1/43079 244382245 Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008. Includes bibliographical references (p. 147-152). There are multiple instances in science and engineering where quantities of interest are evaluated by solving one or several nonlinear partial differential equations (PDEs) that are parametrized in terms of a set of inputs. Even though well-established numerical techniques exist for solving these problems, their computational cost often precludes their use in cases where the outputs of interest must be evaluated repeatedly for different values of the input parameters such as probabilistic analysis applications. In this thesis we present a model reduction methodology that combines efficient representation of the nonlinearities in the governing PDE with an efficient model-constrained, greedy algorithm for sampling the input parameter space. The nonlinearities in the PDE are represented using a coefficient-function approximation that enables the development of an efficient offline-online computational procedure where the online computational cost is independent of the size of the original high-fidelity model. The input space sampling algorithm used for generating the reduced space basis adaptively improves the quality of the reduced order approximation by solving a PDE-constrained continuous optimization problem that targets the output error between the reduced and full order models in order to determine the optimal sampling point at every greedy cycle. The resulting model reduction methodology is applied to a highly nonlinear combustion problem governed by a convection-diffusion-reaction PDE with up to 3 input parameters. The reduced basis approximation developed for this problem is up to 50, 000 times faster to solve than the original high-fidelity finite element model with an average relative error in prediction of outputs of interest of 2.5 - 10-6 over the input parameter space. The reduced order model developed in this thesis is used in a novel probabilistic methodology for solving inverse problems. (cont) The extreme computational cost of the Bayesian framework approach for inferring the values of the inputs that generated a given set of empirically measured outputs often precludes its use in practical applications. In this thesis we show that using a reduced order model for running the Markov by David Galbally. S.M. 152 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 Aeronautics and Astronautics. Nonlinear model reduction for uncertainty quantification in large-scale inverse problems : application to nonlinear convection-diffusion-reaction equation Thesis application/pdf Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="df339353-5954-4e39-aa77-c8b1c40b8d9d"> <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>Nonlinear model reduction for uncertainty quantification in large-scale inverse problems : application to nonlinear convection-diffusion-reaction equation</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2008</PublicationDate> <Authors> <Author> <DisplayName>Galbally, David</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>Aeronautics and Astronautics.</Keyword> <Abstract>There are multiple instances in science and engineering where quantities of interest are evaluated by solving one or several nonlinear partial differential equations (PDEs) that are parametrized in terms of a set of inputs. Even though well-established numerical techniques exist for solving these problems, their computational cost often precludes their use in cases where the outputs of interest must be evaluated repeatedly for different values of the input parameters such as probabilistic analysis applications. In this thesis we present a model reduction methodology that combines efficient representation of the nonlinearities in the governing PDE with an efficient model-constrained, greedy algorithm for sampling the input parameter space. The nonlinearities in the PDE are represented using a coefficient-function approximation that enables the development of an efficient offline-online computational procedure where the online computational cost is independent of the size of the original high-fidelity model. The input space sampling algorithm used for generating the reduced space basis adaptively improves the quality of the reduced order approximation by solving a PDE-constrained continuous optimization problem that targets the output error between the reduced and full order models in order to determine the optimal sampling point at every greedy cycle. The resulting model reduction methodology is applied to a highly nonlinear combustion problem governed by a convection-diffusion-reaction PDE with up to 3 input parameters. The reduced basis approximation developed for this problem is up to 50, 000 times faster to solve than the original high-fidelity finite element model with an average relative error in prediction of outputs of interest of 2.5 - 10-6 over the input parameter space. The reduced order model developed in this thesis is used in a novel probabilistic methodology for solving inverse problems.</Abstract> <Abstract>(cont) The extreme computational cost of the Bayesian framework approach for inferring the values of the inputs that generated a given set of empirically measured outputs often precludes its use in practical applications. In this thesis we show that using a reduced order model for running the Markov</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>