Timothy L. Johnson. / Bibliography: p. 7. / "April, 1981"--report documentation page. "20th CDC." / U.S. Air Force Office of Scientific Research contract F49620-80-C-0002

Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines and develops discrete homotopy theory for metric spaces, which can be thought of as a discrete analog of classical path-homotopy and covering space theory. Given a metric space, X, this leads to the construction of a collection of covering spaces of X - and corresponding covering groups - parameterized by the positive real numbers, which we call the [epsilon]-covers and the [epsilon]-groups. These covers and groups evolve dynamically as the parameter decreases, changing topological type at specific parameter values which depend on the topology and local geometry of X. This leads to the definition of a critical spectrum for metric spaces, which is the set of all values at which the topological type of the covers change. Several results are proved regarding the critical spectrum and its connections to topology and local geometry, particularly in the context of geodesic spaces, refinable spaces, and Gromov-Hausdorff limits of compact metric spaces. We investigate the relationship between the critical spectrum and covering spectrum in the case when X is geodesic, connections between the geometry of the [epsilon]-groups and the metric and topological structure of the [epsilon]-covers, as well as the behavior of the [epsilon]-covers and critical values under Gromov-Hausdorff convergence.

The objective of this body of work was to produce a code system capable of processing boundary angular flux data from discrete ordinates calculations in 2D and 3D Cartesian and cylindrical geometries into cumulative probability density functions that can be used with a Monte Carlo radiation transport code to define neutron and photon initial positions, directions, and energies. In order to accomplish this goal, the DISCO (DetermInistic-Stochastic Coupling Operation) code was created to interface between the DORT and TORT deterministic radiation transport codes and the MCNP stochastic radiation transport code. DISCO introduces new methods to use the boundary angular flux data, along with information regarding the deterministic quadrature sets and spatial mesh structure, to create cumulative probability density functions that are passed to MCNP for sampling within the source.F90 subroutine that was also generated as part of this work. Operating in concert, DISCO and the MCNP source.F90 subroutine create a source term according to the discrete ordinates angular flux information. In order to validate the work described herein, 24 test cases were created to exercise the different geometries and execution modes available. The results of these test cases confirm that the methodology and corresponding implementation is appropriate and functioning correctly. Furthermore, this work incorporates several novel features such as compatibility with all 2D and 3D Cartesian and cylindrical geometries, an angular and spatial indexing scheme to reduce random sampling operations, a streamlining of process execution, and the ability for the resulting Monte Carlo code to operate in either serial and parallel mode.

Ph.D. / Electrical and Computer Engineering / Numerous applications require either the estimation or prediction of a noisy time-series. Examples include speech enhancement, economic forecasting, and geophysical modeling. A noisy time-series can be described in terms of a probabilistic model, which accounts for both the deterministic and stochastic components of the dynamics. Such a model can be used with a Kalman filter (or extended Kalman filter) to estimate and predict the time-series from noisy measurements. When the model is unknown, it must be estimated as well; dual estimation refers to the problem of estimating both the time-series, and its underlying probabilistic model, from noisy data. The majority of dual estimation techniques in the literature are for signals described by linear models, and many are restricted to off-line application domains. Using a probabilistic approach to dual estimation, this work unifies many of the approaches in the literature within a common theoretical and algorithmic framework, and extends their capabilities to include sequential dual estimation of both linear and nonlinear signals. The dual Kalman filtering method is developed as a method for minimizing a variety of dual estimation cost functions, and is shown to be an effective general method for estimating the signal, model parameters, and noise variances in both on-line and off-line environments.

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Spontaneous imbibition plays a very important role in the displacement mechanism of non-wetting fluid in naturally fractured reservoirs. We developed a new 2D two-phase finite element numerical model, as available commercial simulators cannot be used to model small-scale experiments with different boundary conditions as well as complex boundary conditions such as fractures and vugs. Starting with the basic equation of fluid flow, we derived the non-linear diffusion saturation equation. This equation cannot be put in weighted-integral weak variational form and hence Rayleigh-Ritz finite element method (FEM) cannot be applied. Traditionally, the way around it is to use higher order interpolation functions and use Galerkin FEM or reduce the differentiability requirement and use Mixed FEM formulation. Other FEM methods can also be used, but iterative nature of those methods makes them unsuitable for solving large-scale field problems. But if we truncate the non-linear terms and decouple the dependent variables, from the spatial as well as the temporal domains of the primary variable to solve them analytically, the non-linear FEM problem reduces to a simple weighted integral form, which can be put into its corresponding weak form. The advantage of using Rayleigh-Ritz method is that it has immediate effect on the computation time required to solve a particular problem apart from incorporating complex boundary conditions. We compared our numerical models with the analytical solution of this diffusion equation. We validated the FDM numerical model using X-Ray Tomography (CT) experimental data from the single-phase spontaneous imbibition experiment, where two simultaneously varying parameters of weight gain and CT water saturation were used and then went ahead and compared the results of FEM model to that of FDM model. A two-phase field size example was taken and results from a commercial simulator were compared to the FEM model to bring out the limitations of this approach.

Aggregate strength, gradation, and shape play a vital role in controlling asphalt mixture performance. Many studies have demonstrated the effects of these factors on asphalt mixture performance in terms of resistance to fatigue cracking and rutting. This study introduces numerical and analytical approaches supported with imaging techniques for studying the interrelated effects of aggregate strength, gradation, and shape on resistance of asphalt mixtures to fracture. The numerical approach relies on the discrete element method (DEM). The main advantage of this approach is the ability to account for the interaction between the internal structure distribution and aggregate properties in the analysis of asphalt mixture response and performance. The analytical approach combines aggregate strength variability and internal force distribution in an asphalt mixture to predict the probability of aggregate fracture. The numerical and analytical approaches were calibrated and verified using laboratory tests on various aggregate types and mixtures. Consequently these approaches were used to: (1) determine the resistance of various mixture types with different aggregate properties to fracture, (2) study the effects of aggregate strength variability on fracture, (3) quantify the influence of blending different types of aggregate on mixture strength, (4) develop a mathematical expression for calculating the probability of aggregate fracture within asphalt mixture, and (5) relate cracking patterns (cohesive: aggregate - aggregate and matrix - matrix, and adhesive: aggregate - matrix) in an asphalt mixture to internal structure distribution and aggregate properties. The results of this dissertation established numerical and analytical techniques that are useful for developing a virtual testing environment of asphalt mixtures. Such a virtual testing environment would be capable of relating the microscopic response of asphalt mixtures to the properties of the mixture constituents and internal structure distribution. The virtual testing environment would be an inexpensive mean to evaluate the influence of changing different material and design factors on the mixture response.

This work is concerned with the study of nonlinear nonconvex stochastic programming, in particular in the context of trust-region approaches. We first explore how to exploit the structure of multistage stochastic nonlinear programs with linear constraints, in the framework of primal-dual interior point methods. We next study consistency of sample average approximations (SAA) for general nonlinear stochastic programs. We also develop a new algorithm to solve the SAA problem, using the statistical inference information to reduce numercial costs, by means of an internal variable sample size strategy. We finally assess the numerical efficiency of the proposed method for the estimation of discrete choice models, more precisely mixed logit models, using our software AMLET, written for this purpose.