The focus of this thesis was to investigate which baseball metrics are most conducive to run creation and prevention. Stepwise regression and Liu estimation were used to formulate two models for the dependent variables and also used for cross validation. Finally, the predicted values were fed into the Pythagorean Expectation formula to predict a team’s most important goal: winning. Each model fit strongly and collinearity amongst offensive predictors was considered using variance inflation factors. Hits, walks, and home runs allowed, infield putouts, errors, defense-independent earned run average ratio, defensive efficiency ratio, saves, runners left on base, shutouts, and walks per nine innings were significant defensive predictors. Doubles, home runs, walks, batting average, and runners left on base were significant offensive regressors. Both models produced error rates below 3% for run prediction and together they did an excellent job of estimating a team’s per-season win ratio.

Electrical propagation in excitable tissue, such as nerve fibers and heart muscle, is described by a nonlinear diffusion-reaction parabolic partial differential equation for the transmembrane voltage $V(x,t)$, known as the cable equation. This equation involves a highly nonlinear source term, representing the total ionic current across the membrane, governed by a Hodgkin-Huxley type ionic model, and requires the solution of a system of ordinary differential equations. Thus, the model consists of a PDE (in 1-, 2- or 3-dimensions) coupled to a system of ODEs, and it is very expensive to solve, especially in 2 and 3 dimensions. In order to solve this equation numerically, we develop an algorithm, extended from the Parareal Algorithm, to efficiently incorporate space-parallelized solvers into the framework of the Parareal algorithm, to achieve time-and-space parallelization. Numerical results and comparison of the performance of several serial, space-parallelized and time-and-space-parallelized time-stepping numerical schemes in one-dimension and in two-dimensions are also presented.

Model reduction is a powerful and ubiquitous tool used to reduce the complexity of a dynamical system while preserving the input-output behavior. It has been applied throughout many different disciplines, including controls, fluid and structural dynamics. Model reduction via proper orthogonal decomposition (POD) is utilized for of control of partial differential equations. In this thesis, the analytical expressions of POD modes are derived for the heat equation. The autocorrelation function of the latter is viewed as the kernel of a self adjoint compact operator, and the POD modes and corresponding eigenvalues are computed by solving homogeneous integral equations of the second kind. The computed POD modes are compared to the modes obtained from snapshots for both the one-dimensional and two-dimensional heat equation. Boundary feedback control is obtained through reduced-order POD models of the heat equation and the effectiveness of reduced-order control is compared to the full-order control. Moreover, the explicit computation of the POD modes and eigenvalues are shown to allow the computation of different n-widths approximations for the heat equation, including the linear, Kolmogorov, Gelfand, and Bernstein n-widths.

Mathematical Biology has found recent success applying Computational Fluid Dynamics (CFD) to model airflow in the human lung. Detailed modeling of flow patterns in the alveoli, where the oxygen-carbon dioxide gas exchange occurs, has provided data that is useful in treating illnesses and designing drug-delivery systems. Unfortunately, many CFD software packages have high licensing fees that are out of reach for independent researchers. This thesis uses three open-source software packages, Gmsh, OpenFOAM, and ParaView, to design a mesh, create a simulation, and visualize the results of an idealized terminal alveolar sac model. This model successfully demonstrates that OpenFOAM can be used to model airflow in the acinar region of the lung under biologically relevant conditions.

Spatial-temporal distribution of soil moisture in the vadose zone is an important aspect of the hydrological cycle that plays a fundamental role in water resources management, including modeling of water flow and mass transport. The vadose zone is a critical transfer and storage compartment, which controls the partitioning of energy and mass linked to surface runoff, evapotranspiration and infiltration. This dissertation focuses on integrating hydraulic characterization methods with remote sensing technologies to estimate the soil moisture distribution by modeling the spatial coverage of soil moisture in the horizontal and vertical dimensions with high temporal resolution. The methodology consists of using satellite images with an ultrafine 3-m resolution to estimate soil surface moisture content that is used as a top boundary condition in the hydrologic model, SWAP, to simulate transport of water in the vadose zone. To demonstrate the methodology, herein developed, a number of model simulations were performed to forecast a range of possible moisture distributions in the Everglades National Park (ENP) vadose zone. Intensive field and laboratory experiments were necessary to prepare an area of interest (AOI) and characterize the soils, and a framework was developed on ArcGIS platform for organizing and processing of data applying a simple sequential data approach, in conjunction with SWAP. An error difference of 3.6% was achieved when comparing radar backscatter coefficient (σ0) to surface Volumetric Water Content (VWC); this result was superior to the 6.1% obtained by Piles during a 2009 NASA SPAM campaign. A registration error (RMSE) of 4% was obtained between model and observations. These results confirmed the potential use of SWAP to simulate transport of water in the vadose zone of the ENP. Future work in the ENP must incorporate the use of preferential flow given the great impact of macropore on water and solute transport through the vadose zone. Among other recommendations, there is a need to develop procedures for measuring the ENP peat shrinkage characteristics due to changes in moisture content in support of the enhanced modeling of soil moisture distribution.

This thesis starts with the fundamentals of matrix theory and ends with applications of the matrix singular value decomposition (SVD). The background matrix theory coverage includes unitary and Hermitian matrices, and matrix norms and how they relate to matrix SVD. The matrix condition number is discussed in relationship to the solution of linear equations. Some inequalities based on the trace of a matrix, polar matrix decomposition, unitaries and partial isometies are discussed. Among the SVD applications discussed are the method of least squares and image compression. Expansion of a matrix as a linear combination of rank one partial isometries is applied to image compression by using reduced rank matrix approximations to represent greyscale images. MATLAB results for approximations of JPEG and .bmp images are presented. The results indicate that images can be represented with reasonable resolution using low rank matrix SVD approximations.

This thesis is intended to provide fundamental information for the construction and analysis of rotordynamic theoretical models, and their comparison the experimental systems. Finite Element Method (FEM) is used to construct models using Timoshenko beam elements with viscous and hysteretic internal damping. Eigenvalues and eigenvectors of state space equations are used to perform stability analysis, produce critical speed maps, and visualize mode shapes. Frequency domain analysis of theoretical models is used to provide Bode diagrams and in experimental data full spectrum cascade plots. Experimental and theoretical model analyses are used to optimize the control algorithm for an Active Magnetic Bearing on an overhung rotor.

The dissertation addresses the research topics of machine learning outlined below. We developed the theory about traditional first-order algorithms from convex opti- mization and provide new insights in nonconvex objective functions from machine learning. Based on the theory analysis, we designed and developed new algorithms to overcome the difficulty of nonconvex objective and to accelerate the speed to obtain the desired result. In this thesis, we answer the two questions: (1) How to design a step size for gradient descent with random initialization? (2) Can we accelerate the current convex optimization algorithms and improve them into nonconvex objective? For application, we apply the optimization algorithms in sparse subspace clustering. A new algorithm, CoCoSSC, is proposed to improve the current sample complexity under the condition of the existence of noise and missing entries. Gradient-based optimization methods have been increasingly modeled and inter- preted by ordinary differential equations (ODEs). Existing ODEs in the literature are, however, inadequate to distinguish between two fundamentally different meth- ods, Nesterov’s acceleration gradient method for strongly convex functions (NAG-SC) and Polyak’s heavy-ball method. In this paper, we derive high-resolution ODEs as more accurate surrogates for the two methods in addition to Nesterov’s acceleration gradient method for general convex functions (NAG-C), respectively. These novel ODEs can be integrated into a general framework that allows for a fine-grained anal- ysis of the discrete optimization algorithms through translating properties of the amenable ODEs into those of their discrete counterparts. As a first application of this framework, we identify the effect of a term referred to as gradient correction in NAG-SC but not in the heavy-ball method, shedding deep insight into why the for- mer achieves acceleration while the latter does not. Moreover, in this high-resolution ODE framework, NAG-C is shown to boost the squared gradient norm minimization at the inverse cubic rate, which is the sharpest known rate concerning NAG-C itself. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates as NAG-C for minimizing convex functions.

On-orbit satellite servicing is critical to maximizing space utilization and sustainability and is of growing interest for commercial, civil, and defense applications. Reliance on astronauts or anchored robotic arms for the servicing of next-generation large, complex space structures operating beyond Low Earth Orbit is impractical. Substantial literature has investigated the mission design and analysis of robotic servicing missions that utilize a single servicing satellite to approach and service a single target satellite. This motivates the present research to investigate a fleet of servicing satellites performing several operations for a large, central space structure. This research leverages a distributed control approach, implemented using the Horizon Simulation Framework (HSF), to develop a tool capable of integrated mission modeling and task scheduling for a servicing satellite fleet. HSF is a modeling and simulation framework for verification of system level requirements with an emphasis on state representations, modularity, and event scheduling. HSF consists of two major modules: the main scheduling algorithm and the system model. The distributed control architecture allocates processing and decision making for this multi-agent cooperative control problem across multiple subsystem models and the main HSF scheduling algorithm itself. Models were implemented with a special emphasis on the dynamics, control, trajectory constraints, and trajectory optimization for the servicing satellite fleet. The integrated mission modeling and scheduling tool was applied to a sample scenario in which a fleet of 3 servicing assets is tasked with performing 12 servicing activities for a large satellite in Geostationary Orbit. The tool was able to successfully determine a schedule in which all 12 servicing activities were completed in under 32 hours, subject to the fuel and trajectory constraints of the servicing assets.

<p>The graphite structure in compacted graphite iron (CGI) is more coral-like and interconnected only within each eutectic cell. The irregular surface of the graphite-matrix interface has blunt edges which results in the intimate adhesion of the graphite particles to the metal matrix producing more resistance to crack initiation and more vermicular paths arrest crack propagation. Furthermore, the coral-like graphite particles, which are characterized with round edges, also do not promote crack propagation and serve as crack arrestors once cracks are initiated. This unique morphology of graphite in CGI, thereafter, pays off in a higher tensile strength and modulus of elasticity while possessing reasonable thermal conductivity.</p> <p>This work is divided into two phases. The first phase establishes a foundation of a microstructure modeling technique which will be then applied to model CGI in machining. Modeling is being done to shift the approach away from trial and error as is currently being done to a more physics based approach. As machining is conceptually a controlled fracture process, this stage comprehensively studies and models the initiation and propagation of fracture in compacted graphite iron.</p> <p>The second phase serves as an application of the previously built model to capture the more complex scenario involving machining of CGI at different cutting speeds and feeds. The finite element modeling of CGI in machining provides an as of yet unavailable procedure on which future optimization techniques can be performed. The study of chip formation, cutting insert wear, and force measurements are performed in parallel with the modeling process and are employed as means to validate the FE model. Validation of both work phases has been completed to support the model developed in this thesis that captures the critical aspects of machining CGI under different operating scenarios.</p> / Doctor of Philosophy (PhD)