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Modifying Some Iterative Methods for Solving Quadratic Eigenvalue Problems
Master of Science (MS), Wright State University, 2017, Mathematics
In this thesis, we are investigating the solutions λ of a typical quadratic eigenvalue problem (QEP). Indeed, solutions λ of a QEP of the form Q(λ)=λ2M+λD+S that satisfy Q(λ)=0, can be obtained iteratively and without linearizing the problem. However, many iterative methods can only find some of the solutions λ. Therefore, we are going to modify a method based on Newton iterations in order to find all of the solutions λ, that are known also as the eigenvalues of the QEP. In addition, we will investigate how the proposed method compares with standard iterative methods from the literature. Moreover, we will provide a method for finding an upper bound for the number of the eigenvalues of the QEP, and apply this in our method for the purpose of finding all solutions λ.

#### Committee:

Sara Pollock, Ph.D. (Advisor); Yuqing Chen, Ph.D. (Committee Member); Weifu Fang, Ph.D. (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics

#### Keywords:

Quadratic Eigenvalue problem; Matrix Polynomial Problem; Nonlinear Eigenvalue Problem; Newton Iteration; Generalized Eigenvalue Problem; Newton Maehly Method; Newton Maehly Iteration; Newton Correction; QEP; NLEP; NLEVP; MPP; GEP

Using the Non-Uniform Dynamic Mode Decomposition to Reduce the Storage Required for PDE Simulations
Master of Mathematical Sciences, The Ohio State University, 2017, Mathematical Sciences
Partial Differential Equation simulations can produce large amounts of data that are very slow to transfer. There have been many model reduction techniques that have been proposed and utilized over the past three decades. Two popular techniques Proper Orthogonal Decomposition and Dynamic Mode Decomposition have some hindrances. Non-Uniform Dynamic Mode Decomposition (NU-DMD), which was introduced in 2015 by Gueniat et al., that overcomes some of these hindrances. In this thesis, the NU-DMD's mathematics are explained in detail, and three versions of the NU-DMD's algorithm are outlined. Furthermore, different numerical experiments were performed on the NU-DMD to ascertain its behavior with repect to errors, memory usage, and computational efficiency. It was shown that the NU-DMD could reduce an advection-diffusion simulation to 6.0075% of its original memory storage size. The NU-DMD was also applied to a computational fluid dynamics simulation of a NASA single-stage compressor rotor, which resulted in a reduced model of the simulation (using only three of the five simulation variables) that used only about 4.67% of the full simulation's storage with an overall average percent error of 8.90%. It was concluded that the NU-DMD, if used appropriately, could be used to possibly reduce a model that uses 400GB of memory to a model that uses as little as 18.67GB with less than 9% error. Further conclusions were made about how to best implement the NU-DMD.

#### Committee:

Ching-Shan Chou (Advisor); Jen-Ping Chen (Committee Member)

#### Subjects:

Aerospace Engineering; Applied Mathematics; Computer Science; Mathematics; Mechanical Engineering

#### Keywords:

Fluid Dynamics; Fluid Flow; Model Reduction; Partial Differential Equations; reducing memory; Dynamic Mode Decomposition; Decomposition; memory; Non-Uniform Dynamic Mode Decomposition

Galvanic and Pitting Corrosion of a Fastener Assembly
Master of Science, University of Akron, 2018, Applied Mathematics
This research focuses on coupled galvanic/pitting corrosion of AA7075 when combined with stainless steel in a fastener assembly. A one-dimensional mathematical model of a well-mixed electrolyte is developed to predict the damage profile of the AA7075 surface when its protective coating is damaged. The damage exposes the galvanic couple. A time dependent system of partial differential equations for potential, chloride concentration, aluminum ion concentration, and damage is developed and solved numerically. Four approaches to calculate the current density within aluminum pits are discussed. The first is a current balance of the system. This reflects the local chemistry that drives pit growth early on. The second approach uses the potential calculated at the bottom of each initiated pit with a polarization curve relevant to the pit chemistry. The third approach assumes the entire aluminum surface is active and thus uses the bulk polarization curves to determine the current density. The last approach is a combination of the above three approaches to simulate the different mechanisms during the corrosion of the aluminum surface. This approach reflects the growth of pits during the formation of oxide that leads to repassivation. Hence, the model developed describes high local pit current densities and damage done to the AA7075 surface over time.

#### Committee:

Curtis Clemons, Dr. (Advisor); Kevin Kreider, Dr. (Committee Member); Gerald Young, Dr. (Committee Member)

#### Subjects:

Applied Mathematics; Chemical Engineering; Civil Engineering; Engineering

#### Keywords:

Galvanic corrosion; pitting corrosion; AA7075 and steel; pit current density; mathematical model

GAUSS-TYPE QUADRATURE RULES, WITH APPLICATIONS IN LINEAR ALGEBRA
PHD, Kent State University, 2018, College of Arts and Sciences / Department of Mathematical Science
Golub and Meurant describe how pairs of Gauss and Gauss–Radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real-valued matrix functionals defined by a symmetric matrix. However, there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds. In this situation, it may be possible to determine upper and lower bounds by evaluating pairs of Gauss and anti-Gauss rules. Unfortunately, it is difficult to ascertain whether the values determined by Gauss and antiGauss rules bracket the value of the given real-valued matrix functional. Therefore, generalizations of anti-Gauss rules have recently been described, such that pairs of Gauss and generalized anti-Gauss rules may determine upper and lower bounds for real-valued matrix functionals also when pairs of Gauss and (standard) anti-Gauss rules do not. The available generalization requires the matrix that defines the functional to be real and symmetric. The present paper extends generalized anti-Gauss rules in several ways: The real-valued matrix functional may be defined by a nonsymmetric matrix. Moreover, extensions that can be applied to matrix-valued functions are presented. Estimates of element-wise upper and lower bounds then are determined. Modifications that yield simpler formulas are described.The remaining numerical methods presented rely on multiple orthogonal polynomials, which generalize standard orthogonal polynomials by requiring orthogonality with respect to several inner products or bilinear forms.

#### Committee:

Lothar Reichel (Advisor); Jun Li (Committee Member); Jing Li (Committee Member); Arden Ruttan (Committee Member); Austin Melton (Committee Chair)

#### Subjects:

Applied Mathematics

A Comparative Analysis of an Interior-point Method and a Sequential Quadratic Programming Method for the Markowitz Portfolio Management Problem
BA, Oberlin College, 2016, Mathematics
In this paper, I give a brief introduction of the general optimization problem as well as the convex optimization problem. The portfolio selection problem, as a typical type of convex optimization problem, can be easily solved in polynomial time. However, when the number of available stocks in the portfolio becomes large, there might be a significant difference in the running time of different polynomial-time solving methods. In this paper, I perform a comparative analysis of two different solving methods and discuss the characteristics and differences.

#### Subjects:

Applied Mathematics; Industrial Engineering; Mathematics; Operations Research

#### Keywords:

optimization;interior-point method;portfolio optimization;convex optimization;sequential quadratic programming;

Fault Detection for Rolling Element Bearings Using Model-Based Technique
Master of Sciences (Engineering), Case Western Reserve University, 2015, EECS - System and Control Engineering
In this research, we focus on fault detection of rolling element bearings by using a model-based technique. Vibration data are obtained from the mathematical model developed by Michael Louis Adams in [1]. The model is a time-varying nonlinear model with 29 degrees of freedom (DOF), derived by determining the energy terms of the Lagrange equations. Fault frequencies and fault types include outer race (OR), Ball and Inner race (IR). In order to determine the model for performing fault detection, we use the Hammerstein – Wiener model for observer design. The results of system identification for each of the fault types are provided. The Hammerstein – Wiener models were used for observer design and implementation, and we exploited the cross-correlation between observer residuals and temporal features of OR and IR faults for detection and diagnosis.

#### Committee:

Kenneth Loparo, Ph.D. (Advisor); Vira Chankong, Ph.D. (Committee Member); Richard Kolacinski, Ph.D. (Committee Member)

#### Subjects:

Applied Mathematics; Electrical Engineering; Engineering; Mechanical Engineering; Systems Design; Systems Science

#### Keywords:

Fault Detection;Rolling Element Bearing;Hammerstein-Wiener model;Observer Design;Model Based Method;Cross-Correlation;Analytical Redundancy;System Identification

Bayesian Parameter Estimation and Inference Across Scales
Doctor of Philosophy, Case Western Reserve University, 2016, Applied Mathematics
In the analysis of complex phenomena arising in biology, medicine, physics, economics or the social sciences, it is not uncommon to employ mathematical models at different scales. Microscopic models tend to be better suited to capture the fine-scale details of a complex system and are typically stochastic in nature, while macroscopic models, sometimes arising as mean-field approximations to microscopic models, are typically used to describe the overall, population-level dynamics of the system. Typically, both coarse- and fine-scale models depend on parameters whose values are unknown or poorly known, hence need to be estimated. The estimation of the parameters of larger scale models tends to be more straightforward and approachable with standard optimization-based tools. In several important applications, the microscopic model parameters are of greater interest and importance, as they may be interpreted as characterizing the generative process underlying the large-scale model. Due to the stochastic nature of these types of models, the unknown parameters are difficult, if not impossible, to estimate directly. Ideally, we would like to estimate the parameters of the coarse-scale model and relate them to those of the fine-scale model. This connection, however, may be all but trivial to establish, due to the fact that models at different scales may depend on parameters that do not exhibit a one-to-one correspondence. In this work, we propose a Bayesian approach for inferring on the values of the microscopic model parameters, based on estimates of the parameters of the associated macroscopic model. Our methodology is based on connecting model parameters across scales via the posterior probability density, and the subsequent approximation of this density using random samples. We illustrate the viability of our technique with several computed examples, ranging from simple to fairly complex, with a focus on applications in the life sciences.

#### Committee:

Erkki Somersalo, PhD (Advisor); Daniela Calvetti, PhD (Advisor); Peter Thomas, PhD (Committee Member); Jenny Brynjarsdottir, PhD (Committee Member); Giuseppe Strangi, PhD (Committee Member)

#### Subjects:

Applied Mathematics

#### Keywords:

Bayesian Statistics, Parameter Estimation, Multiscale Modeling

Mass Transport Enhancement in Copper Electrodeposition due to Gas Co-Evolution
Doctor of Philosophy, Case Western Reserve University, 2015, Chemical Engineering
Metal electrodeposition is often associated with simultaneous hydrogen co-evolution. The presence of bubbles complicates the design and control of electrodeposition processes. This is particularly relevant to the electrodeposition from aqueous electrolytes of numerous metals with standard potentials that are negative to hydrogen. As shown in this study, hydrogen co-evolution enhances the transport rates of the metal deposition reaction beyond those predicted by the classical, steady-state mass transport model. Available models addressing transport in the presence of gas co-evolution are based on free convection that is enhanced by the rising bubble cloud. However, there are no models that address mass transfer enhancement by bubbles under forced convection, such as analyzed here for the commonly used, facing-down rotating disk electrode (RDE). This study characterizes experimentally the phenomenon and introduces a model for quantifying it. Experimental data was collected in plating copper at high cathodic overpotentials (-0.4 to -1.0V vs SHE) from acidified copper sulfate on a RDE. The transport enhancement (~2-6 fold) was determined by measuring the copper deposition by gravimetry. Pulse experiments, where the current decay was measured following a short bubble generation confirmed the linkage between the current enhancement and the presence of bubbles. A model based on fresh electrolyte replenishing the volume vacated by the translating bubbles and thus subjecting regions of the electrode to enhanced transient currents has been derived. The model correlates the experimental data indicating higher transport enhancement with increasing cathodic polarization and dependence of the enhancement on the rotation rate and on the bulk copper concentration.

#### Committee:

Uziel Landau (Committee Chair); Rohan Akolkar (Committee Member); Donald Feke (Committee Member); Daniel Scherson (Committee Member); Mohan Sankaran (Committee Member)

#### Subjects:

Applied Mathematics; Chemical Engineering; Chemistry; Engineering; Materials Science; Mechanical Engineering; Physics; Technology

#### Keywords:

surface renewal, copper electrodeposition, mass transfer enhancement, limiting current density, Levich, penetration model, hydrogen co-evolution, von Karman velocities, Rotating Disk Electrode, electrode area correction, surface roughness, bubbles

Modeling Monitoring of An Industry In A Game-Theorectic Framework with Imperfect Information
Master of Science, University of Akron, 2015, Applied Mathematics
We model the interaction of a regulator monitoring an industry as an extensive- game with imperfect information, i.e., the players in the game make their moves in a sequential manner and are not aware of the other players’ previous actions when it is their turn. We use a published model that allows for both a regulator and a firm to help control pollution, and modify it by adding a choice for the regulator to monitor the industry or not. We show that as the probability that firms in an industry provide optional controls of pollution increases, the acceptable level of monitoring is non-increasing for a fixed cost of monitoring.

#### Committee:

Stefan Forcey, Dr. (Advisor); Francesco Renna, Dr. (Advisor); Curtis Clemons, Dr. (Committee Member)

#### Subjects:

Applied Mathematics; Economics

#### Keywords:

game theory, applied mathematics, pollution, monitoring, inspections

Developing A Protocol For Describing Problem-Solving Instruction
Master of Education (MEd), Bowling Green State University, 2015, Curriculum and Teaching
This study uses a qualitative approach to examine problem-solving instruction when implementing the Common Core State Standards for Mathematics (CCSS). Problem-solving instruction has previously been discussed using three approaches: teaching about, for, and through problem solving (TAPS, TFPS, TTPS). Fourteen teachers who participated in a CCSSM professional development program (PD) were studied. Specifically, their lesson plans and videos after the PD were used to describe and classify features of each approach. Results indicate that there are commonalities and differences between TAPS, TFPS, and TTPS. The elements that are consistent in all lessons observed are pre-assessment, group work, and routine problem application. The crucial difference in TAPS is the focus on learning of heuristics and/or processes of problem solving; whereas, the foci on teaching for problem solving are in helping students to understand concepts and procedures in problem solving as well as to use the knowledge gained to solve problems. The teacher’s aim in teaching through problem solving is having students learn a concept or procedure through an experience engaged in problem solving; moreover, the teacher provides a focused learning environment for students to discover several ways to solve a problem and promotes discourse in the mathematics classroom.

#### Committee:

Tracy Huziak-Clark, Dr (Advisor); Jonathan Bostic, Dr (Committee Member); Nancy Patterson, Dr (Committee Member)

#### Subjects:

Applied Mathematics; Education; Mathematics; Teacher Education; Teaching

#### Keywords:

Problem-solving instruction; problem solving; Mathematics; Upper elementary levels

On the Electromagnetic Scattering from Small Grooves in a Conical Surface
Master of Science, The Ohio State University, 2011, Electrical and Computer Engineering
Rapid generation of radar signatures is of particular interest for the missile defense community. These radar signautres are normally found through predictive methods. The two classes of predictive methods are Full-Wave Numerical Solutions and High-Frequency Asymptotic Solutions. Full-Wave Numerical Solutions solve problems in a numerically exact fashion, but are inefficient for electrically large targets. High-Frequency solutions work quickly but rely on the assumption that the target is made up of simple canonical shapes, which is not often the case. Real world objects contain many small features that greatly contribute to the radar signature. One such small feature is a groove on a cone. In this thesis, the scattered fields of a groove around a conical surface is derived. In the high frequency case, the scattering from the groove will reduce to an analytic solution of one or two point scatterers, using the method of stationary phase.

#### Committee:

Robert Burkholder, PhD (Advisor); Prabhakar Pathak, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Electrical Engineering; Electromagnetics; Electromagnetism

#### Keywords:

Electromagnetics; High Frequency Asymptotics; Point Scattering Models; Stationary Phase Approximation

High-Speed Dynamics and Vibration of Planetary Gears, Vibration of Spinning Cantilevered Beams, and An Efficient Computational Method for Gear Dynamics
Doctor of Philosophy, The Ohio State University, 2012, Mechanical Engineering

This study investigates the dynamics and vibration of high-speed planetary gears, spinning cantilevered beams, and gear pairs. High-speed planetary gear dynamics and vibration are analyzed using a lumped-parameter model. A continuous model is used to study spinning cantilever beam vibration. A finite element/contact mechanics model is used for the dynamics of gear pairs.

Chapter 2 investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speed-dependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.

In Chapter 3, the structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears are identified and mathematically proven. Planetary gears have only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds, and whether or not it occurs is determined using a perturbation method. Numerical results verify the critical speed locations and the stability near these critical speeds. Flutter instabilities occur at extremely high speeds, and these are investigated numerically for each mode type.

Chapter 4 demonstrates unusual gyroscopic system eigenvalue behavior observed in a lumped-parameter planetary gear model. While the model has been used for dynamic analyses in industrial applications, the focus is on the eigenvalue phenomena that occur at especially high speeds rather than practical planetary gear behavior. The behaviors include calculation of exact trajectories across critical speeds, uncommon stability features near degenerate critical speeds, and unique stability transitions. These eigenvalue behaviors are not evident in the vast literature on gyroscopic systems.

Chapter 5 investigates eigenvalue sensitivity to model parameters and eigenvalue veering in high-speed planetary gears. The eigenvalue perturbation approach is formulated such that the results apply to discrete, continuous, and hybrid discrete-continuous gyroscopic systems. Third-order perturbation approximations for the eigenvalues are determined. From the second-order perturbation approximation an eigenvalue veering parameter is defined and used to analyze veering in high-speed planetary gears. The sensitivity of the eigenvalues to model parameters are written in terms of modal kinetic and potential energies. Eigenvalue veering is prominent in planetary gears that have disrupted cyclic symmetry.

In Chapter 6, the single-mode vibrations of high-speed planetary gears are investigated in the rotating carrier-fixed and the stationary inertial reference frames. The properties of the structured planetary gear modes result in gear motions with interesting geometry. The frequency content of the motion differs between the rotating carrier-fixed and stationary inertial bases. The results from this work assist the analysis of experimental planetary gear measurements.

A linear model for the bending-bending-torsional-axial vibration of a spinning cantilever beam with a rigid body attached at its free end is derived in Chapter 7 using Hamilton's Principle. The rotation axis is perpendicular to the beam (like a helicopter blade). The equations split into two uncoupled groups: coupled bending in the direction of the rotation axis with torsional motions, and coupled bending in the plane of rotation with axial motions. The practically important first case above is examined in detail. The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system. Using the extended operator structure, the equations are discretized using Galerkin's method, and subsequently the eigenvalues and mode shapes are calculated for varying rotation speeds.

In Chapter 8, the general Euler-Lagrange equations for gyroscopic continuum are derived from Hamilton's Principle using kinetic, potential, and virtual work expressions with specific functional dependencies typical of gyroscopic continua. These equations are useful in problems with multiple variables, where directly taking variations of the Lagrangian is cumbersome. The equations can be used to derive linear and nonlinear governing equations. The formulation is in a form that is suitable for programming in computer algebra software. The resulting Euler-Lagrange equations are applied to axially moving media, rotating shafts, and spinning beams to determine governing equations of motion.

In Chapter 9, a finite element formulation for the dynamic response of gear pairs is proposed. Following an established approach in lumped parameter gear dynamic models, the static solution is used as the excitation in a frequency domain solution of the finite element vibration model. The nonlinear finite element/contact mechanics formulation provides accurate calculation of the static solution and average mesh stiffness that are used in the dynamic simulation. The frequency domain finite element calculation of dynamic response compares well with numerically integrated (time domain) finite element dynamic results and previously published experimental results. Simulation time with the proposed formulation is two orders of magnitude lower than numerically integrated dynamic results. This formulation admits system level dynamic gearbox response, which may include multiple gear meshes, flexible shafts, rolling element bearings, housing structures, and other deformable components.

#### Committee:

Robert Parker (Advisor); Daniel Mendelsohn (Committee Co-Chair); Manoj Srinivasan (Committee Member); Ulrich Gerlach (Committee Member)

#### Subjects:

Applied Mathematics; Mechanical Engineering; Mechanics

#### Keywords:

Planetary gear; gyroscopic system; vibration; spinning beams; finite element analysis

Compressive Sensing for Tomographic Echo Imaging in Two Dimensions
Master of Science, The Ohio State University, 2012, Electrical and Computer Engineering
We present a framework that leverages compressive sensing (CS) for tomographic echo imaging in two dimensions, a specific type of imaging problem that is of interest to both military and civilian researchers. We establish CS guarantees for certain types of far-field tomographic imaging of sparse scenes. Typically, CS guarantees for common tomographic systems can be difficult to establish because of the structure imposed by uniform sampling of echoes. This introduces a high level of coherence between measurements from different nearby reflectors. We overcome these difficulties by introducing randomness in the placement of several monostatic radar sites surrounding the scene and by making simplifying assumptions based on practical engineering constraints. We use a wideband signal to interrogate the scene, allowing for high-resolution imaging. Our main result shows that with high probability, the system model satisfies the restricted isometry property (RIP) under a certain set of assumptions and restrictions. The number of radar sites required to meet RIP is a function of the desired imaging resolution and the RIP parameters. We compare this result to empirical trials and show that there are significant limitations to the practical use of the bounds proven. However, there is value in the novel approach to proving RIP for this type of two-dimensional system. Our results indicate that we can produce a similar image using less sensors with CS compared to more sensors with traditional imaging algorithms that assume no information about the unknown scene.

#### Committee:

Lee Potter, PhD (Advisor); Emre Ertin, PhD (Committee Member); Phillip Schniter, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Electrical Engineering

#### Keywords:

Boundary Integral Techniques in Three Dimensions for Deep Water Waves
Doctor of Philosophy, The Ohio State University, 2011, Mathematics

The motion of deep water waves is a long-existing problem in fluid dynamics. Because it has many applications in academics and industries, numerous scientists have done much research to tackle this problem. The evolution of deep water wave is governed by Euler equations. Because of its nonlinearity, there are few available analytical expressions to fully describe the evolution of the deep water waves. Thus efficient numerical approximation becomes very critical to understand the properties of water wave motion.

An improved boundary integral technique is presented in this thesis to simulate the motion of deep water waves. One difficulty is the singularity in the Green’s function. Two methods to treat this singularity are discussed. One is blob regularization with third-order accuracy, and the other is based on polar coordinates with spectral accuracy. In the blob regularization, we replace the Green’s function by a regularized smooth Green’s function, which provides a good approximation to the original integral. For the other approach, integral identities are applied to reduce the strength of the singularity and then a polar coordinate transformation is applied to obtain a nonsingular integrand. The results from these two methods will be examined.

Another challenge is that the integrands are integrated over an infinite surface. For a doubly periodic water wave, we have to sum the images of Green’s function over the free-surface of the water. Ewald summation technique is used to expedite the calculation. Three-dimensional interpolation technique is suggested to reduce the time spent even further.

The numerical method is tested with several examples and then applied to the motion of a perturbed Stokes wave. The perturbation grows until the resolution fails.

#### Committee:

Gregory Baker, Dr. (Advisor); Chiu-Yen Kao, Dr. (Committee Member); Ed Overman, Dr. (Committee Member)

#### Subjects:

Applied Mathematics; Fluid Dynamics; Mathematics

#### Keywords:

Three Dimensions Deep Water Wave; Singularity; Blob Regularization; Exponetially Accurate Method; Polar Coordinate Transformation; Third-Order Accurate Method; Ewald Summation; Green's function; Simulation of Perturbed Stokes Wave

Prior Information Guided Image Processing and Compressive Sensing
Doctor of Philosophy, Case Western Reserve University, 2013, Applied Mathematics
Signal/image processing and reconstruction based on mathematical modeling and computational techniques have been well developed and still attract much attention due to their broad applications. It becomes challenging to build mathematical models if the given data lacks some certainties. Prior information, including geometric priors, high frequency priors, spatially variant intensity variations and image regularities, assists to establish mathematical models by providing a more accurate description of the underlying signal/image. We have been exploring applications of the extracted prior information in two directions: integrating prior information into the image denoising explained in nonlocal means (NL-means) denoising framework; enhancing the compressive sensing signal/image reconstruction with the guidance of prior information. The first topic is geometric information based image denoising, where we develop a segmentation boosted image denoising scheme, balancing the removal of excessive noise and preservation of fine features. By virtue of segmentation algorithms and more general geometry extraction schemes, we are able to obtain the phase or geometric prior information. Based on the NL-means method, we introduce a mutual position function to ensure that averaging is only taken over pixels in the same image phase. To further improve the performance, we provide the respective selection scheme for the convolution kernel and the weight function. To address the unreliable segmentation due to the presence of excessive noise, the phase prior is relaxed to a more general geometric prior. The second topic is prior information guided compressive sensing signal/image reconstruction. Concerning the 1D signal reconstruction, we extract high frequency subbands as prior to boost the subsequent reconstruction. In 2D image reconstruction realm, we propose a novel two-stage intensity variation prior guided image reconstruction method using pixel-to-pixel varying weights associated to the total variation. By incorporating high order image regularity prior, we develop one total generalized variation (TGV) based image reconstruction model. Unlike the traditional wavelet which is only able to detect locations of singularities, shearlet transform can efficiently provide more geometric information of singularities in images, e.g. direction. Therefore we adopt the shearlet transform to boost the sparsity in image reconstruction algorithms. In addition, our work in signal/image denoising and reconstruction can be easily generalized to deal with other kinds of noise or measurements.

#### Committee:

Weihong Guo (Advisor); Daniela Calvetti (Committee Member); Erkki Somersalo (Committee Member); David Wilson (Committee Member)

#### Subjects:

Applied Mathematics

#### Keywords:

image denoising; compressive sensing; prior information

Lagrange-Chebyshev Based Single Step Methods for Solving Differential Equations
Master of Science, University of Akron, 2012, Applied Mathematics
Many numerical methods are the result of replacing a function by its interpolating polynomial; quadrature formulas are one such method. In this research a special class of quadrature formulas are used that incorporate equally spaced points and zeros of Chebyshev polynomials simultaneously. Some properties of these quadrature formulas are investigated, and they will be used to develop single step methods for solving ordinary differential equations. Examples are presented to compare the approximated solutions with exact solutions.

#### Committee:

Ali Hajjafar, Dr. (Advisor); John Heminger, Dr. (Other)

#### Subjects:

Applied Mathematics

#### Keywords:

Fast Sweeping Methods for Steady State Hyperbolic Conservation Problems and Numerical Applications for Shape Optimization and Computational Cell Biology
Doctor of Philosophy, The Ohio State University, 2013, Mathematics
This thesis consists of three parts. In the first part, we develop a numerical solver for steady states of hyperbolic conservation problems with high order of accuracy and the capability to resolve shocks. Fast sweeping methods are efficient iterative numerical schemes originally designed for solving stationary Hamilton-Jacobi equations. Their efficiency relies on Gauss-Seidel type nonlinear iterations, and a finite number of sweeping directions. We generalize the fast sweeping methods to hyperbolic conservation laws with source terms. The algorithm is obtained through finite difference discretization, with the numerical fluxes evaluated in WENO (Weighted Essentially Non-oscillatory) fashion, coupled with Gauss-Seidel iterations. In particular, we consider mainly the Lax-Friedrich numerical fluxes. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving shocks of the proposed methods. In the second part, an inverse problem, arising from the design of some optical materials to localize waves at a specific wavelength, is solved by the steepest descent method. The method of steepest descent is a classical approach to find the minimum/maximum of an objective function or functional based on a first order approximation. The method works in spaces of any number of dimensions, even in infinite-dimensional spaces. This method can converge more efficiently than methods which do not require derivative information; however, in certain circumstances the "cost function space" may become discontinuous and as a result, the derivatives may be difficult or impossible to determine. Here, we discuss eigenfunction optimization for representing the topography of a dielectric environment and efficient techniques to solve different material design problems. Numerous results are shown to demonstrate the robustness of the gradient-based approach. In the last part, we model and simulate an important biological process called cell polarization. Yeast cell mating is a well-studied biological system which involves external chemical stimuli, reactions and surface diffusion of multiple proteins on the cell membrane. In this part, we use the level set method to do the simulation and couple it with the extended diffusion equations. The level set method efficiently captures the motion of a closed curve by embedding the curve of interest as the zero contour of a level set function defined on a Cartesian grid. It allows us to update the level set function on a fixed grid mesh instead of moving grid points on the curve. This strategy not only works well for the complex system of a single cell, but also shows great advantage in multi-cell environment since for different cells different level set functions with the same kind of systems of equations will be assigned. In this part, we apply the level set method to reproduce the results obtained from using Lagrangian framework for a single cell with artificial pheromone gradient in extracellular space, and use it to establish a more refined model with extracellular diffusion based on previous work. Moreover, it allows us to investigate simulations involving multiple cells and reveal more biological mechanisms behind that.

#### Committee:

Ching-Shan Chou (Advisor); Ed Overman (Committee Member); Chuan Xue (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics

#### Keywords:

steady states; hyperbolic conservation laws; fast sweeping; Lax-Friedrichs; WENO; shape optimization; optical device; eigenvalue problems; steepest descent; computational cell biology; yeast mating; level set method

Error Analysis of RKDG Methods for 1-D Hyperbolic Conservation Laws
Doctor of Philosophy (Ph.D.), Bowling Green State University, 2012, Mathematics and Statistics
Obtaining reasonable error estimates for computed solutions to hyperbolic conservation laws using the Runge-Kutta / Discontinuous Galerkin Method (RKDG) has been an issue for many years. Zhang and Shu have written papers dealing with an a priori estimate for a third order Runge-Kutta scheme. In this dissertation a new way to compute an a posteriori error estimate for numerical solutions computed using RKDG methods is provided. These techniques can be applied to any order Runge-Kutta scheme as well as any order polynomial. The idea of smoothness indicators is discussed in the papers by Sun, Sun and Fillippova, and Rumsey and Sun. These smoothness indicators are used to compute an a posteriori error estimate. In providing the framework for this new technique, only the regions where the solution is smooth are considered. Two examples, Burgers' equation and the traffic law, are provided to give the full details of the error analysis and to show that the errors computed are reasonable. In the final remarks some insight into how to adapt this method for regions where the solution has a fully developed shock or a partially developed shock are discussed.

#### Committee:

Tong Sun, PhD (Committee Chair); Kit Chan, PhD (Committee Member); So-Hsiang Chou, PhD (Committee Member); John Laird, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics

#### Keywords:

RKDG;conservation law;error analysis;hyperbolic

Responses of Rumen Microbes to Excess Carbohydrate
Doctor of Philosophy, The Ohio State University, 2013, Nutrition Program, The Ohio State University
Some species of rumen microbes respond to excess carbohydrate by synthesizing reserve carbohydrate, but others respond by spilling energy (producing heat alone). The response of mixed cultures of species, however, has not been directly studied, and elucidating the response was the focus of the present research. The first aim was to evaluate methods for detecting reserve carbohydrate, in an attempt to find one that was quantitative. Methods compared were those based on the (i) anthrone reaction and (ii) hydrolysis with amyloglucosidase. Compared to the amyloglucosidase hydrolysis method, the anthrone method detected a larger increase (P = 0.017) in cell carbohydrate when glucose (20 mM) was dosed in cultures. Additionally, it detected a larger decrease (P = 0.049) in cell carbohydrate after glucose was exhausted. This result indicated that the anthrone method detected more carbohydrate that functions as a reserve material. For the anthrone method, recoveries for energy (97.5%), carbon (100.2%), and cell components (99.8%) were high, indicating carbohydrate was completely detected. For the amyloglucosidase hydrolysis method, recoveries were lower. The anthrone method appeared to accurately quantify changes in reserve carbohydrate and shows merit for quantitative studies. The second aim was to determine if a mixed rumen microbes would respond to excess carbohydrate by accumulating reserve carbohydrate, spilling energy, or both. Mixed microbes from the rumen were washed with N-free buffer and dosed with glucose. Total heat production was measured by calorimetry. Energy spilling was calculated as heat production not accounted by (i) endogenous metabolism and (ii) synthesis of reserve carbohydrate. For cells dosed with 5 mM glucose, synthesis of reserve carbohydrate and endogenous metabolism explained nearly all heat production (93.7%); no spilling was detected (P = 0.226). For cells dosed with 20 mM glucose, energy spilling was not detected immediately after dosing, but it became significant (P < 0.05) approximately 30 min after dosing glucose. Energy spilling accounted for as much as 38.7% of heat production in one incubation. As documented for some pure cultures, mixed microbial communities from the rumen can respond to large excesses of carbohydrate by spilling energy. The third aim was to determine how Gibbs energy (ΔG)&#x2014;a thermodynamic property of reactions—was impacted by the specified physical state of gases. The second experiment (above) required calculation of thermodynamic properties, but it was found different authors specify different states for gases for these calculations. Our analysis indicated the aqueous, not gaseous, state is that used by microbes and should be the one specified in calculations. Compilation of literature values showed that microbial reactions create disequilibrium between aqueous and gaseous concentrations of gases, and the pattern of disequilibrium suggests the aqueous state is used. The greater the disequilibrium, the greater ΔG was impacted (up to 60.50 kJ/mol) when changing from the gaseous to the aqueous state. Because ΔG was so impacted, our results suggest that aqueous gas concentrations must be measured to accurately estimate ΔG. By establishing appropriate calculations for ΔG, our results should advance understanding of microbial processes where energetics play a key role.

#### Committee:

Jeffrey Firkins, Ph.D. (Advisor); Daniel Bond, Ph.D. (Committee Member); Thaddeus Ezeji, Ph.D. (Committee Member); Gonul Kaletunc, Ph.D. (Committee Member); William Weiss, Ph.D. (Committee Member)

#### Subjects:

Agriculture; Animal Sciences; Applied Mathematics; Biochemistry; Biostatistics; Energy; Experiments; Livestock; Microbiology; Nutrition

#### Keywords:

rumen; microbiology; animal nutrition; energy spilling; reserve carbohydrate; glycogen; cow; glucose

One Dimensional Approach to Modeling Damage Evolution of Galvanic Corrosion in Cylindrical Systems
Master of Science, University of Akron, 2013, Applied Mathematics
A one-dimensional mathematical model is developed to describe the damage evolution caused by galvanic corrosion in an aluminum/copper system consisting of two wires, tubes, or pipes (or other similar cylindrical objects) end-to-end. Laplace’s equation is solved in the electrolyte film surrounding the object to simulate the effects of corrosion over time. This solution is obtained through use of an asymptotic procedure that takes advantage of the disparity in length scales within the model. This model is then able to track the damage caused by the galvanic action through use of numerical techniques (a finite difference method) solved by MATLAB. Solutions are obtained for various area ratios, electrolyte thicknesses, and object shapes. Current density and corrosion potential are determined as a function of experimental polarization curves. The model is restricted to thin-film domains, assumes that the electrolyte solution is well-mixed, assumes there is no mass transport, and is restricted by the Wagner relationship. A short investigation is also conducted into the behavior of the model when the radius of the wire becomes very large. The iterative model shows a damage profile that is directly related to IR drop and current density in the electrolyte.

#### Committee:

Gerald Young, Dr. (Advisor); Curtis Clemons, Dr. (Committee Member); Kevin Kreider, Dr. (Committee Member)

#### Subjects:

Applied Mathematics; Engineering

#### Keywords:

galvanic corrosion; cylindrical; wire; pipe; mathematical modeling

Mathematical Models and Genetic Algorithm Approaches to Simultaneously Perform Workforce Overtime Capacity Planning and Schedule Cells
Master of Science (MS), Ohio University, 2012, Industrial and Systems Engineering (Engineering and Technology)
The problem studied in this thesis was observed in an actual textile company. The problem is more complex than usual scheduling problems in that we compute overtime requirements and make scheduling decisions simultaneously. Since having tardy jobs is not desirable, overtime work is allowed to minimize the number tardy jobs or total tardiness. Two different problems are considered; Problem1, to maximize the total profits by delivering jobs on or before time. The tardy jobs in this case are considered as lost sales. Problem2, to minimize the total tardiness and overtime costs. In this case tardy jobs are delivered with associated tardiness penalty costs. In problem1, various mathematical models are presented reflecting different overtime workforce hiring practices. To solve the same problem for one particular hiring policy, a Genetic Algorithm (GA) approach is also discussed. GA includes some newly proposed mutation operators, dynamic and twin. The proposed twin mutation strategy produced the best results in all problem sizes. Mathematical Model 2 was the best mathematical model with respect to both profit and execution time. This model considered partial overtime periods and also allowed different overtime periods on cells. In problem2, a mathematical model is presented to solve this complex problem. Experimentation has been carried out using three different problem types with five instances each based on the data collected from the company. For most problems, the mathematical model gave results in seconds.

#### Committee:

Gursel Suer, PhD (Advisor); Dusan Sormaz, PhD (Committee Member); Tao Yuan, PhD (Committee Member); Faizul Huq, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Engineering; Industrial Engineering; Information Science; Information Systems; Information Technology

#### Keywords:

Scheduling; Genetic Algorithm; Mathematical Model; decision making

Beltrami Flows
BS, Kent State University, 2018, College of Arts and Sciences / Department of Mathematical Science
Our goal will be to find a weak solution to the Beltrami flow. A Beltrami flow in three-dimensional space is an incompressible (divergence free) vector field that is everywhere parallel to its curl. That is, curl(B) = λ B for some function. These flows arise naturally in many physical problems. In astrophysics and in plasma fusion Beltrami fields are known as force-free fields. They describe the equilibrium of perfectly conducting pressure-less plasma in the presence of a strong magnetic field. In fluid mechanics, Beltrami flows arise as steady states of the 3D Euler equations. Numerical evidence suggests that in certain regimes turbulent flows organize into a coherent hierarchy of weakly interacting superimposed approximate Beltrami flows. Given the importance of Beltrami fields, there are several approaches to proving existence of solutions, for instance use the calculus of variations, and use fixed point arguments. In this thesis we instead use a Hilbert space approach through the Lax-Milgram lemma.

#### Committee:

Benjamin Jaye (Advisor); Andrew Tonge (Committee Member); Dexheimer Veronica (Committee Member); Jeremy Williams (Committee Member)

#### Subjects:

Applied Mathematics; Astrophysics

#### Keywords:

Beltrami Flows; Pressureless Fields; Plasma Fusion; Magnetic Field; Sun Corona; Finite Elements Method; Dirichlet; Neumann;

Modelling The Financial Market Using Copula
Master of Science, University of Akron, 2017, Applied Mathematics
This project is to track the differences and the movements between the Actual and theoretical future prices using Copula. Standard & Poor's 500 Index (SPX) and 10-year treasury bond yield rate was downloaded from Yahoo! website and SPX future prices were downloaded from Moore Research Centre website and their observations from January 2, 2001 to May 27, 2016 were used for this analysis. Log-returns of the future prices were taken to model and analyse the direct movements of the future prices. The distributions of the marginals and the best family of copula was selected and simulated. We compared the copula method to the classical method after 2000 simulation. A high level of mis-pricing in the future price which corresponds to the period 2008-2009 was observed. This observed mis-pricing could be as a result of relative over-reaction of the Financial market compared to future market. Inverse relationship between the performance of SPX and the volatility of future prices was observed. Standardised Student's t-distribution was concluded to be the marginal distribution using the maximum likelihood method to estimate their distribution parameters. Student t-Copula was concluded to be the best family of copula to measure the dependence. In further studies, modelling the risk associated with futures stock price and pricing with copula based simulation will be a major red flag to be addressed.

#### Committee:

Nao Mimoto, Dr (Advisor); Patrick Wilber, Dr (Other); Kevin Kreider, Dr (Other)

#### Subjects:

Applied Mathematics

#### Keywords:

Copula, Theoretical Future Price and Actual Future Price

The Wildlife-Livestock Interface of Infectious Disease Dynamics: A One Health Approach
Doctor of Philosophy, The Ohio State University, 2016, Comparative and Veterinary Medicine
Surveillance for wildlife diseases is critical to our understanding of the emergence, transmission, persistence and control of infectious diseases at the interface of humans, domestic animals, and wildlife populations. Neospora caninum is a protozoan parasite capable of infecting a wide range of canid and ungulate species. The importance of the disease relates to economic losses, mainly derived from endemic or epidemic abortions in cattle. In the United States, coyotes and dogs are believed to be the main definitive hosts and white-tailed deer and cows are the main intermediate hosts. Our overall aim was to better understand the wildlife-livestock interface of N. caninum in natural settings. First, we estimated the true prevalence of N. caninum in three ruminant species by using Bayesian inference. We identified and discussed differences between apparent and true prevalence (TP). Differences in TP for some species suggest differences in the epidemiology of N. caninum for these co-located populations. Second, we evaluated the environmental phase of N. caninum shed in wild canid scats. Results suggested that the role of this environmental phase in the transmission to ruminants is likely minor. Finally, we evaluated the role of host species heterogeneity in the epidemiology of N. caninum circulating in a community. We identified differences in the patterns of immunity, age structure, and maternal and/or fetal antibody duration in three intermediate (ruminant) host species. Also, we estimated the species-specific contributions to the persistence of this pathogen in a community. This research was approached from the One Health perspective and provided a better understanding of N. caninum dynamics at the wildlife-livestock interface in an ecosystem.

#### Committee:

Rebecca Garabed (Advisor); Mark Moritz (Committee Member); Barbara Wolfe (Committee Member); William Saville (Committee Member)

#### Subjects:

Animal Diseases; Applied Mathematics; Biology; Biostatistics; Computer Science; Conservation; Cultural Anthropology; Ecology; Environmental Health; Epidemiology; Geographic Information Science; Health Sciences; Livestock; Parasitology; Veterinary Services; Wildlife Conservation; Zoology

#### Keywords:

multi-host parasites; Neospora caninum; wildlife-livestock interface; infectious disease modeling; disease ecology; epidemiology; One-Health; community; human dimensions; prevalence; wildlife conservation; multidisciplinary; complexity; parasitology

Segregation Dynamics Motivated by Territorial Markings: The Transition from a Particle to a Continuum Model
Doctor of Philosophy, Case Western Reserve University, 2016, Applied Mathematics
We present an agent-based model to simulate gang territorial development motivated by graffiti marking on a two-dimensional discrete lattice. Moreover, we study and analyse the dynamics and steady-state solutions of gang agents and their graffiti markings. For simplicity, we assume that there are only two rival gangs present, and they compete for territory. In this model, agents represent gang members who move according to a biased random walk. All agent interactions are indirect, with the interactions occurring through the graffiti field. Using numerical simulations, we show that gang segregation and territory formation may happen for different system parameters. We also show that a phase transition occurs between a well-mixed state and a segregated state. The numerical results show that the inverse temperature, decay rate and graffiti rate effect the phase transition. From the discrete model, we derive a continuum system for territorial development. Using the continuum equations, we perform a linear stability analysis to determine the stability of the equilibrium solutions. From the results of the linear stability analysis, we show that the critical values of both the discrete model and the continuum system have the same behaviour.

#### Committee:

Alethea Barbaro (Advisor); Erkki Somersalo (Committee Member); Wanda Strychalski (Committee Member); Wojbor Woyczynski (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics