A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems

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dc.contributor.authorvan Wyk, Hans-Werneren
dc.contributor.committeechairBorggaard, Jeffrey T.en
dc.contributor.committeememberHerdman, Terry L.en
dc.contributor.committeememberZietsman, Lizetteen
dc.contributor.committeememberDay, Martin V.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:11:49Zen
dc.date.adate2012-05-25en
dc.date.available2014-03-14T20:11:49Zen
dc.date.issued2012-04-26en
dc.date.rdate2012-05-25en
dc.date.sdate2012-05-10en
dc.description.abstractAs simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. However, the statistical characterization of the model parameters is rarely known. In this thesis, we propose a variational approach to solve the stochastic inverse problem of obtaining a statistical description of the diffusion coefficient in an elliptic partial differential equation, based noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations (via a truncated Karhunen-Loeve expansion) allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called 'finite noise' problem, in the space of functions with bounded mixed derivatives. We prove convergence of 'finite noise' minimizers to the appropriate infinite dimensional ones, and devise a gradient based, as well as a sampling based strategy for locating these numerically. Lastly, we illustrate our methods by means of numerical examples.en
dc.description.degreePh. D.en
dc.identifier.otheretd-05102012-163445en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-05102012-163445/en
dc.identifier.urihttp://hdl.handle.net/10919/27635en
dc.publisherVirginia Techen
dc.relation.haspartVanWyk_HW_D_2012.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectuncertainty quantificationen
dc.subjectparameter identificationen
dc.subjectelliptic systemsen
dc.subjectstochastic collocation methodsen
dc.titleA Variational Approach to Estimating Uncertain Parameters in Elliptic Systemsen
dc.typeDissertationen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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