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Pablo A. Parrilo. Sethuraman, Sandeep Massachusetts Institute of Technology. Computation for Design and Optimization Program. Massachusetts Institute of Technology. Computation for Design and Optimization Program 2007-10-19T20:31:58Z 2007-10-19T20:31:58Z 2006 2006 http://hdl.handle.net/1721.1/39213 85844111 Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006. Includes bibliographical references (leaf 95). In this thesis, we present a modification of an existing methodology to obtain a hierarchy of lower and upper bounds on moments of solutions of linear differential equations. The motivation for change is to obtain tighter bounds by solving smaller semidefinite problems. The modification we propose involves partitioning the domain and normalizing each partition to ensure numerical stability. Using the adjoint operator, linear constraints involving the boundary conditions and moments of the solution are developed for each partition. Semidefinite constraints are imposed on the moments, and an optimization problem is solved to obtain the bounds. We have demonstrated the algorithm by calculating bounds on moments of various one-dimensional case differential equations including the Bessel ODE, and Legendre polynomials. In the two-dimensional case we have demonstrated the algorithm by calculating bounds on various PDEs including the Helmholtz equation, and heat equation. In both cases, the results were encouraging with tighter bounds on moments being obtained by solving smaller problems with domain partitioning. by Sandeep Sethuraman. S.M. 95 leaves 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. Domain partitioning to bound moments of differential equations using semidefinite optimization Thesis application/pdf false Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="702655f8-7a73-4a1b-bc43-2e1dc7476f87"> <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>Domain partitioning to bound moments of differential equations using semidefinite optimization</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2006</PublicationDate> <Authors> <Author> <DisplayName>Sethuraman, Sandeep</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, we present a modification of an existing methodology to obtain a hierarchy of lower and upper bounds on moments of solutions of linear differential equations. The motivation for change is to obtain tighter bounds by solving smaller semidefinite problems. The modification we propose involves partitioning the domain and normalizing each partition to ensure numerical stability. Using the adjoint operator, linear constraints involving the boundary conditions and moments of the solution are developed for each partition. Semidefinite constraints are imposed on the moments, and an optimization problem is solved to obtain the bounds. We have demonstrated the algorithm by calculating bounds on moments of various one-dimensional case differential equations including the Bessel ODE, and Legendre polynomials. In the two-dimensional case we have demonstrated the algorithm by calculating bounds on various PDEs including the Helmholtz equation, and heat equation. In both cases, the results were encouraging with tighter bounds on moments being obtained by solving smaller problems with domain partitioning.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>