Browsing by Author "H. Thomas Banks, Committee Chair"

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    Modeling, Analysis, and Estimation of an in vitro HIV Infection Using Functional Differential Equations
    (2002-09-05) Bortz, David Matthew; H. Thomas Banks, Committee Chair; Marie Davidian, Committee Member; Kazufumi Ito, Committee Member; Hien T. Tran, Committee Member
    This dissertation focuses on developing mathematical and computational tools for use as an aid in understanding the cellular population dynamics of an in vitro HIV experiment. We carefully develop a functional differential equation model which incorporates mathematical mechanisms that account for both the biological delays and the parameter uncertainty inherent in the system. We present the theoretical foundations for our methodology which then allow us to develop a numerical approximation scheme and perform parameter identifications (even on the delay distributions) and sensitivity analyses. We summarize the results of a numerical investigation of the delays followed by the results from the nonlinear least squares inverse problem. We then present a statistical significance argument for the importance of the delay mechanism as well as the results of a sample sensitivity analysis of the system with respect to select parameters.
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    Terahertz-Based Electromagnetic Interrogation Techniques for Damage Detection
    (2004-06-24) Gibson, Nathan Louis; H. Thomas Banks, Committee Chair; Hien T. Tran, Committee Member; Kazufumi Ito, Committee Member; Negash G. Medhin, Committee Member
    We apply an inverse problem formulation to determine characteristics of a defect from a perturbed electromagnetic interrogating signal. A defect (gap) inside of a dielectric material causes a disruption, via reflections and refractions at the material interfaces, of the windowed interrogating signal. We model these electromagnetic waves inside the material with Maxwell's equations. In order to resolve the dimensions and location of the defect, we use simulations as forward solves in our Newton-based, iterative scheme which optimizes an innovative cost functional appropriate for reflected waves where phase differences can produce ill-posedness in the inverse problem when one uses the usual ordinary least squares criterion. Our choice of terahertz frequency allows good resolution of desired gap widths without significant attenuation. Numerical results are given in tables and plots, standard errors are calculated, and computational issues are addressed. An inverse problem formulation is also developed for the determination of polarization parameters in heterogeneous Debye materials with multiple polarization mechanisms. For the case in which a distribution of mechanisms is present we show continuous dependence of the solutions on the probability distribution of polarization parameters in the sense of the Prohorov metric. This in turn implies well-posedness of the corresponding inverse problem, which we attempt to solve numerically for a simple uniform distribution. Lastly we address an alternate approach to modeling electromagnetic waves inside of materials with highly oscillating dielectric parameters which involves the technique of homogenization. We formulate our model in such a way that homogenization may be applied, and demonstrate the necessary equations to be solved.