Browsing by Author "C. T. Kelley, Committee Chair"

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    Nonsmooth Nonlinearities in Applications from Hydrology
    (2003-07-28) Kavanagh, Kathleen Rose; P.A. Gremaud, Committee Member; S.E. Howington, Committee Member; C.T. Miller, Committee Member; C. T. Kelley, Committee Chair; C.D. Meyer, Committee Member
    This work has two parts; simulation of unsaturated flow and optimization of remediation problems. For the unsaturated flow simulation, we propose an adaptive time stepping scheme based on error control for Richards' equation, a model for flow in unsaturated porous media. The motivation for this work is a ground and surface water simulator being developed by the U.S. Engineering Research Development Center called the ADaptive Hydrology Model. ADH uses unstructured, adaptive finite elements. ADH advances in time implicitly, solving the nonlinear equations with an inexact---Newton method with a two-level domain decomposition preconditioner. The nonlinearity in Richards' Equation can be non-Lipschitz and nonsmooth. Standard theory for temporal integration may not apply for certain physical parameters. We consider a method for error estimation and control for temporal adaption. In the optimization section, we investigate a suite of test problems from the literature that are intended for benchmarking purposes and comparison of optimization algorithms. The objective functions can be nonsmooth, nonconvex, or have several minima that may trap standard gradient based methods. We apply the implicit filtering algorithm to some such problems.
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    Numerical Methods for the Wigner-Poisson Equations
    (2005-10-06) Lasater, Matthew; M. Shearer, Committee Member; R. H. Martin, Committee Member; C. T. Kelley, Committee Chair; D. L. Woolard, Committee Member; P. A. Gremaud, Committee Member
    This thesis applies modern numerical methods to solve the Wigner-Poisson equations for simulating quantum mechanical electron transport in nanoscale semiconductor devices, in particular, a resonant tunneling diode (RTD). The goal of this dissertation is to provide engineers with a simulation tool that will verify earlier numerical results as well as improve upon the computational efficiency and resolution of older simulations. Iterative methods are applied to the linear and nonlinear problems in these simulations to reduce the computational memory and time required to calculate solutions. Initially the focus of the research involved updating time-integration techniques, but this switched to implementing continuation methods for finding steady-state solutions to the equations as the applied voltage drop across the device varied. This method requires the solution to eigenvalue problems to produce information on the RTD's time-dependent behavior such as the development of current oscillation at a particular applied voltage drop. The continuation algorithms/eigensolving capabilities were provided by Sandia National Laboratories' software library LOCA (Library of Continuation Algorithms). The RTD simulator was parallelized, and a preconditioner was developed to speed-up the iterative linear solver. This allowed us to use finer computational meshes to fully resolve the physics. We also theoretically analyze the steady-state solutions of the Wigner-Poisson equations by noting that the solutions to the steady-state problems are also solutions to a fixed point problem. By analyzing the fixed point map, we are able to prove some regularity of the steady-state solutions as well provide a theoretical explanation for the mesh-independence of the preconditioned linear solver.
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    Parameter Identification in Lumped Compartment Cardiorespiratory Models
    (2009-04-13) Pope, Scott R.; C. T. Kelley, Committee Chair; Mette Olufsen, Committee Member; Shu-Cherng Fang, Committee Member; Ilse Ipsen, Committee Member
    The parameter identification problem attempts to find parameter values that cause the solution of a predictive model to match data. In this work, parameters in cardiovascular and respiratory models are identified. This work’s main contribution is in its application of gradient based optimization techniques and insight into methods to identify parameters that can be estimated given subject specific data. The models presented in this paper are lumped compartment models of the cardiovascular and respiratory systems. Lumped compartment models treat the cardiovascular and respiratory systems as collections of interconnected compartments transporting blood and exchanging oxygen and carbon dioxide. Using these compartments, a system of ordinary differential equations (ODE) is generated that incorporates several physiological parameters representing vascular resistances, compliances, and tissue metabolic rates. The solution to this ODE system is used to predict cerebral blood flow, systemic arterial blood pressure, and expired carbon dioxide partial pressures, which are then compared to subject data. Minimizing the two-norm difference between between the result of the predictive model and the experimental data is a non-linear least squares problem. Although the least squares problem is overdetermined, the data do not contain enough information to determine all model parameters. A combination of sensitivity analysis, expert knowledge, and subset selection techniques reduce the number of model parameters estimated.