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

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dc.contributor.advisorHeinkenschloss, Matthias
dc.contributor.committeeMemberSorensen, Danny C.
dc.contributor.committeeMemberRiviere, Beatrice M.
dc.contributor.committeeMemberCox, Dennis D.
dc.creatorKouri, Drew
dc.date.accessioned2012-09-05T23:57:26Z
dc.date.accessioned2012-09-05T23:57:29Z
dc.date.available2012-09-05T23:57:26Z
dc.date.available2012-09-05T23:57:29Z
dc.date.created2012-05
dc.date.issued2012-09-05
dc.date.submittedMay 2012
dc.date.updated2012-09-05T23:57:29Z
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.
dc.format.mimetypeapplication/pdf
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>.
dc.identifier.slug123456789/ETD-2012-05-60
dc.identifier.urihttps://hdl.handle.net/1911/64617
dc.language.isoeng
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.
dc.subjectApplied mathematics
dc.subjectPDE constrained optimization
dc.subjectUncertainty Quantification
dc.subjectTrust regions
dc.subjectAdaptivity
dc.subjectSparse grids
dc.titleAn Approach for the Adaptive Solution of Optimization Problems Governed by Partial Differential Equations with Uncertain Coefficients
dc.typeThesis
dc.type.materialText
thesis.degree.departmentComputational and Applied Mathematics
thesis.degree.disciplineEngineering
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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