Title page for ETD etd-05102012-163445

Type of Document Dissertation
Author van Wyk, Hans-Werner
Author's Email Address hvanwyk@vt.edu
URN etd-05102012-163445
Title A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Borggaard, Jeffrey T. Committee Chair
Day, Martin V. Committee Member
Herdman, Terry L. Committee Member
Zietsman, Lizette Committee Member
  • uncertainty quantification
  • parameter identification
  • elliptic systems
  • stochastic collocation methods
Date of Defense 2012-04-26
Availability unrestricted
As 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.
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