An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Differential Equations with Uncertain Coefficients

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dc.contributor.advisorHeinkenschloss, Matthiasen_US
dc.contributor.committeeMemberSorensen, Danny C.en_US
dc.contributor.committeeMemberRiviere, Beatrice M.en_US
dc.contributor.committeeMemberCox, Dennis D.en_US
dc.creatorKouri, Drewen_US
dc.date.accessioned2012-09-05T23:57:26Zen_US
dc.date.accessioned2012-09-05T23:57:29Zen_US
dc.date.available2012-09-05T23:57:26Zen_US
dc.date.available2012-09-05T23:57:29Zen_US
dc.date.created2012-05en_US
dc.date.issued2012-09-05en_US
dc.date.submittedMay 2012en_US
dc.date.updated2012-09-05T23:57:29Zen_US
dc.description.abstractUsing derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationKouri, Drew. "An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Differential Equations with Uncertain Coefficients." (2012) Diss., Rice University. <a href="https://hdl.handle.net/1911/64617">https://hdl.handle.net/1911/64617</a>.en_US
dc.identifier.slug123456789/ETD-2012-05-60en_US
dc.identifier.urihttps://hdl.handle.net/1911/64617en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectApplied mathematicsen_US
dc.subjectPDE constrained optimizationen_US
dc.subjectUncertainty Quantificationen_US
dc.subjectTrust regionsen_US
dc.subjectAdaptivityen_US
dc.subjectSparse gridsen_US
dc.titleAn Approach for the Adaptive Solution of Optimization Problems Governed by Partial Differential Equations with Uncertain Coefficientsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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