Physical Sciences and Mathematics Commons^™

Analytical And Numerical Analysis Of The Sirs Model, Catherine Nguyen

Student Research Submissions

Mathematical models in epidemiology describe how diseases affect and spread within a population. By understanding the trends of a disease, more effective public health policies can be made. In this paper, the Susceptible-Infected-Recovered-Susceptible (SIRS) Model was examined analytically and numerically to compare with the data for Coronavirus Disease 2019 (COVID-19). Since the SIRS model is a complex model, analytical techniques were used to solve simplified versions of the SIRS model in order to understand general trends that occur. Then by Euler's Method, the Runge-Kutta Method, and the Predictor-Corrector Method, computational approximations were obtained to solve and plot the SIRS model. …

Mathematical Modeling And Examination Into Existing And Emerging Parkinson’S Disease Treatments: Levodopa And Ketamine, Gabrielle Riddlemoser

Undergraduate Honors Theses

Parkinson’s disease (PD) is the second most common neurodegenerative disease across the world, affecting over 6 million people worldwide. This disorder is characterized by the progressive loss of dopaminergic neurons within the substantia nigra pars compacta (SNpc) due to the aggregation of α-synuclein within the brain. Patients with PD develop motor symptoms such as tremors, bradykinesia, and postural instability, as well as a host of non-motor symptoms such as behavioral changes, sleep difficulties, and fatigue. The reduction of dopamine within the brain is the primary cause of these symptoms. The main form of treatment for PD is levodopa, a precursor …

Proof-Of-Concept For Converging Beam Small Animal Irradiator, Benjamin Insley

Dissertations & Theses (Open Access)

The Monte Carlo particle simulator TOPAS, the multiphysics solver COMSOL., and

several analytical radiation transport methods were employed to perform an in-depth proof-ofconcept

for a high dose rate, high precision converging beam small animal irradiation platform.

In the first aim of this work, a novel carbon nanotube-based compact X-ray tube optimized for

high output and high directionality was designed and characterized. In the second aim, an

optimization algorithm was developed to customize a collimator geometry for this unique Xray

source to simultaneously maximize the irradiator’s intensity and precision. Then, a full

converging beam irradiator apparatus was fit with a multitude …

Convergence Estimate Of Minimal Residual Methods And Random Sketching Of Krylov Subspace Methods, Peter Westerbaan

All Dissertations

This study concerns two main issues in numerical linear algebra: convergence estimate of minimal residual methods based on explicit construction of approximate min-max polynomials for in- definite matrices, and development and analysis of Krylov subspace methods using non-orthonormal basis vectors based on random sketching. For a matrix A with spectrum Λ(A), it is well known that the min-max polynomial problem min max |pk (z)| pk ∈Pk, pk (0)=1, z∈Λ(A) is used to bound the relative error of Krylov subspace minimum residual methods or similar methods. For a symmetric positive definite matrix A, the min-max polynomial for the Conjugate Gradient (CG) …

Domain Decomposition Methods For Fluid-Structure Interaction Problems Involving Elastic, Porous, Or Poroelastic Structures, Hemanta Kunwar

All Dissertations

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and Schwarz waveform relaxation (SWR) method using Robin transmission conditions (or the Robin method), for solving fluid-structure interaction systems involving elastic, porous, or poroelastic structure. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent fluid and the structure subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent multiphysics phenomena. For the fluid-porous …

Applications Of Survival Estimation Under Stochastic Order To Cancer: The Three Sample Problem, Sage Vantine

Honors Program Theses and Research Projects

Stochastic ordering of probability distributions holds various practical applications. However, in real-world scenarios, the empirical survival functions extracted from actual data often fail to meet the requirements of stochastic ordering. Consequently, we must devise methods to estimate these distribution curves in order to satisfy the constraint. In practical applications, such as the investigation of the time of death or the progression of diseases like cancer, we frequently observe that patients with one condition are expected to exhibit a higher likelihood of survival at all time points compared to those with a different condition. Nevertheless, when we attempt to fit a …

Modeling And Numerical Analysis Of The Cholesteric Landau-De Gennes Model, Andrew L. Hicks

LSU Doctoral Dissertations

This thesis gives an analysis of modeling and numerical issues in the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs) with cholesteric effects. We derive various time-step restrictions for a (weighted) $L^2$ gradient flow scheme to be energy decreasing. Furthermore, we prove a mesh size restriction, for finite element discretizations, that is critical to avoid spurious numerical artifacts in discrete minimizers that is not well-known in the LC literature, particularly when simulating cholesteric LCs that exhibit ``twist''. Furthermore, we perform a computational exploration of the model and present several numerical simulations in 3-D, on both slab geometries and spherical …

Tools For Biomolecular Modeling And Simulation, Xin Yang

Mathematics Theses and Dissertations

Electrostatic interactions play a pivotal role in understanding biomolecular systems, influencing their structural stability and functional dynamics. The Poisson-Boltzmann (PB) equation, a prevalent implicit solvent model that treats the solvent as a continuum while describes the mobile ions using the Boltzmann distribution, has become a standard tool for detailed investigations into biomolecular electrostatics. There are two primary methodologies: grid-based finite difference or finite element methods and body-fitted boundary element methods. This dissertation focuses on developing fast and accurate PB solvers, leveraging both methodologies, to meet diverse scientific needs and overcome various obstacles in the field.

New Algorithmic Support For The Fundamental Theorem Of Algebra, Vitaly Zaderman

Dissertations, Theses, and Capstone Projects

Univariate polynomial root-finding is a venerated subjects of Mathematics and Computational Mathematics studied for four millenia. In 1924 Herman Weyl published a seminal root-finder and called it an algorithmic proof of the Fundamental Theorem of Algebra. Steve Smale in 1981 and Arnold Schonhage in 1982 proposed to classify such algorithmic proofs in terms of their computational complexity. This prompted extensive research in 1980s and 1990s, culminated in a divide-and-conquer polynomial root-finder by Victor Pan at ACM STOC 1995, which used a near optimal number of bit-operations. The algorithm approximates all roots of a polynomial p almost as fast as one …

Basins Of Attraction And Metaoptimization For Particle Swarm Optimization Methods, David Ma

Honors Projects

Particle swarm optimization (PSO) is a metaheuristic optimization method that finds near- optima by spawning particles which explore within a given search space while exploiting the best candidate solutions of the swarm. PSO algorithms emulate the behavior of, say, a flock of birds or a school of fish, and encapsulate the randomness that is present in natural processes. In this paper, we discuss different initialization schemes and meta-optimizations for PSO, its performances on various multi-minima functions, and the unique intricacies and obstacles that the method faces when attempting to produce images for basins of attraction, which are the sets of …

Penalized Interpolating B-Splines And Their Applications, Kylee L. Hartman-Caballero

Theses and Dissertations

One of the most studied data analysis techniques in Numerical Analysis is interpolation. Interpolation is used in a variety of fields, namely computer graphic design and biomedical research. Among interpolation techniques, cubic splines have been viewed as the standard since at least the 1960s, due to their ease of computation, numerical stability, and the relative smoothness of the interpolating curve. However, cubic splines have notable drawbacks, such as their lack of local control and necessary knowledge of boundary conditions. Arguably a more versatile interpolation technique is the use of B-splines. B-splines, a relative of Bézier curves, allow local control through …

Discontinuous Galerkin Methods For Compressible Miscible Displacements And Applications In Reservoir Simulation, Yue Kang

Dissertations, Master's Theses and Master's Reports

This dissertation contains research on discontinuous Galerkin (DG) methods applied to the system of compressible miscible displacements, which is widely adopted to model surfactant flooding in enhanced oil recovery (EOR) techniques. In most scenarios, DG methods can effectively simulate problems in miscible displacements.
However, if the problem setting is complex, the oscillations in the numerical results can be detrimental, with severe overshoots leading to nonphysical numerical approximations. The first way to address this issue is to apply the bound-preserving
technique. Therefore, we adopt a bound-preserving Discontinuous Galerkin method
with a Second-order Implicit Pressure Explicit Concentration (SIPEC) time marching
method to …

Les-C Turbulence Models And Fluid Flow Modeling: Analysis And Application To Incompressible Turbulence And Fluid-Fluid Interaction, Kyle J. Schwiebert

Dissertations, Master's Theses and Master's Reports

In the first chapter of this dissertation, we give some background on the Navier-Stokes equations and turbulence modeling. The next two chapters in this dissertation focus on two important numerical difficulties arising in fluid flow modeling: poor mass-conservation and nonphysical oscillations. We investigate two different formulations of the Crank-Nicolson method for the Navier-Stokes equations. The most attractive implementation, second order accurate for both velocity and pressure, is shown to introduce non-physical oscillations. We then propose two options which are shown to avoid the poor behavior. Next, we show that grad-div stabilization, previously assumed to have no effect on the target …

Simulation Of Wave Propagation In Granular Particles Using A Discrete Element Model, Syed Tahmid Hussan

Electronic Theses and Dissertations

The understanding of Bender Element mechanism and utilization of Particle Flow Code (PFC) to simulate the seismic wave behavior is important to test the dynamic behavior of soil particles. Both discrete and finite element methods can be used to simulate wave behavior. However, Discrete Element Method (DEM) is mostly suitable, as the micro scaled soil particle cannot be fully considered as continuous specimen like a piece of rod or aluminum. Recently DEM has been widely used to study mechanical properties of soils at particle level considering the particles as balls. This study represents a comparative analysis of Voigt and Best …

Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen

Theses and Dissertations (Comprehensive)

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …

High-Performance Computing In Covariant Loop Quantum Gravity, Pietropaolo Frisoni

Electronic Thesis and Dissertation Repository

This Ph.D. thesis presents a compilation of the scientific papers I published over the last three years during my Ph.D. in loop quantum gravity (LQG). First, we comprehensively introduce spinfoam calculations with a practical pedagogical paper. We highlight LQG's unique features and mathematical formalism and emphasize the computational complexities associated with its calculations. The subsequent articles delve into specific aspects of employing high-performance computing (HPC) in LQG research. We discuss the results obtained by applying numerical methods to studying spinfoams' infrared divergences, or ``bubbles''. This research direction is crucial to define the continuum limit of LQG properly. We investigate the …

New Preconditioned Conjugate Gradient Methods For Some Structured Problems In Physics, Tianqi Zhang

All Dissertations

This dissertation concerns the development and analysis of new preconditioned conjugate gradient (PCG) algorithms for three important classes of large-scale and complex physical problems characterized by special structures. We propose several new iterative methods for solving the eigenvalue problem or energy minimization problem, which leverage the unique structures inherent in these problems while preserving the underlying physical properties. The new algorithms enable more efficient and robust large-scale modeling and simulations in many areas, including condensed matter physics, optical properties of materials, stabilities of dynamical systems arising from control problems, and many more. Some methods are expected to be applicable to …

Controlled Manipulation And Transport By Microswimmers In Stokes Flows, Jake Buzhardt

All Dissertations

Remotely actuated microscale swimming robots have the potential to revolutionize many aspects of biomedicine. However, for the longterm goals of this field of research to be achievable, it is necessary to develop modelling, simulation, and control strategies which effectively and efficiently account for not only the motion of individual swimmers, but also the complex interactions of such swimmers with their environment including other nearby swimmers, boundaries, other cargo and passive particles, and the fluid medium itself. The aim of this thesis is to study these problems in simulation from the perspective of controls and dynamical systems, with a particular focus …

Series Expansions Of Lambert W And Related Functions, Jacob Imre

Electronic Thesis and Dissertation Repository

In the realm of multivalued functions, certain specimens run the risk of being elementary or complex

to a fault. The Lambert $W$ function serves as a middle ground in a way, being non-representable by elementary

functions yet admitting several properties which have allowed for copious research. $W$ utilizes the

inverse of the elementary function $xe^x$, resulting in a multivalued function with non-elementary

connections between its branches. $W_k(z)$, the solution to the equation $z=W_k(z)e^{W_k(z)}$

for a "branch number" $k \in \Z$, has both asymptotic and Taylor series for its various branches.

In recent years, significant effort has been dedicated to exploring …

Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, Ling-Chen Bu

Doctoral Dissertations

In this thesis, we study the mathematical model of one-dimensional microscopic heat conduction of gas particles, applying both both analytical and numerical approaches. The macroscopic law of heat conduction is the renowned Fourier’s law J = −k∇T, where J is the local heat flux density, T(x, t) is the temperature gradient, and k is the thermal conductivity coefficient that characterizes the material’s ability to conduct heat. Though Fouriers’s law has been discovered since 1822, the thorough understanding of its microscopic mechanisms remains challenging [3] (2000). We assume that the microscopic model of heat conduction is a hard ball system. The …

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, Brennan S. Mccann

Doctoral Dissertations and Master's Theses

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive …

Boundary Integral Equation Methods For Superhydrophobic Flow And Integrated Photonics, Kosuke Sugita

Dissertations

This dissertation presents fast integral equation methods (FIEMs) for solving two important problems encountered in practical engineering applications.

The first problem involves the mixed boundary value problem in two-dimensional Stokes flow, which appears commonly in computational fluid mechanics. This problem is particularly relevant to the design of microfluidic devices, especially those involving superhydrophobic (SH) flows over surfaces made of composite solid materials with alternating solid portions, grooves, or air pockets, leading to enhanced slip.

The second problem addresses waveguide devices in two dimensions, governed by the Helmholtz equation with Dirichlet conditions imposed on the boundary. This problem serves as a …

Analysis Of Nonequilibrium Langevin Dynamics For Steady Homogeneous Flows, Abdel Kader A. Geraldo

Doctoral Dissertations

First, we propose using rotating periodic boundary conditions (PBCs) [13] to simulate nonequilibrium molecular dynamics (NEMD) in uniaxial or biaxial stretching flow. These specialized PBCs are required because the simulation box deforms with the flow. The method extends previous models with one or two lattice remappings and is simpler to implement than PBCs proposed by Dobson [10] and Hunt [24]. Then, using automorphism remapping PBC techniques such as Lees-Edwards for shear flow and Kraynik-Reinelt for planar elongational flow, we demonstrate expo-nential convergence to a steady-state limit cycle of incompressible two-dimensional
NELD. To demonstrate convergence [12], we use a technique similar …

Reducing Communication In The Solution Of Linear Systems, Neil S. Lindquist

Doctoral Dissertations

There is a growing performance gap between computation and communication on modern computers, making it crucial to develop algorithms with lower latency and bandwidth requirements. Because systems of linear equations are important for numerous scientific and engineering applications, I have studied several approaches for reducing communication in those problems. First, I developed optimizations to dense LU with partial pivoting, which downstream applications can adopt with little to no effort. Second, I consider two techniques to completely replace pivoting in dense LU, which can provide significantly higher speedups, albeit without the same numerical guarantees as partial pivoting. One technique uses randomized …

Null Space Removal In Finite Element Discretizations, Pengfei Jia

All Theses

Partial differential equations are frequently utilized in the mathematical formulation of physical problems. Boundary conditions need to be applied in order to obtain the unique solution to such problems. However, some types of boundary conditions do not lead to unique solutions because the continuous problem has a null space. In this thesis, we will discuss how to solve such problems effectively. We first review the foundation of all three problems and prove that Laplace problem, linear elasticity problem and Stokes problem can be well posed if we restrict the test and trial space in the continuous and discrete finite element …

Mathematics Behind Machine Learning, Rim Hammoud

Electronic Theses, Projects, and Dissertations

Artificial intelligence (AI) is a broad field of study that involves developing intelligent
machines that can perform tasks that typically require human intelligence. Machine
learning (ML) is often used as a tool to help create AI systems. The goal of ML is
to create models that can learn and improve to make predictions or decisions based on given data. The goal of this thesis is to build a clear and rigorous exposition of the mathematical underpinnings of support vector machines (SVM), a popular platform used in ML. As we will explore later on in the thesis, SVM can be implemented …

Deep Virtual Pion Pair Production, Dilini Lakshani Bulumulla

Physics Theses & Dissertations

This experiment investigates the deep virtual production of both σ− and ρ− mesons, with a particular focus on the microscopic structure of the σ mesons. While the ρ meson is an ordinary qq¯ pair, the σ meson is composed of not only the typical qq¯ pair, making it a topic of controversy for nearly six decades. Although the existence of the σ− meson is now well established, its microscopic structure remains poorly understood. The primary objective of this thesis is to contribute to the understanding of the σ meson by analyzing its deep virtual production. The main focus of this …

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

Master's Theses

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

Hydrodynamic And Physicochemical Interactions Between An Active Janus Particle And An Inactive Particle, Jessica S. Rosenberg

Dissertations, Theses, and Capstone Projects

Active matter is an area of soft matter science in which units consume energy and turn it into autonomous motion. Groups of these units – whether flocks of birds, bacterial colonies, or even collections of synthetically-made active particles – may exhibit complex behavior on large scales. While the large-scale picture is of great importance, so is the microscopic scale. Studying the individual particles that make up active matter will allow us to understand how they move, and whether and under what circumstances their activity can be controlled.

Here we delve into the world of active matter by studying colloidal-sized (100 …

Distributed Control Of Servicing Satellite Fleet Using Horizon Simulation Framework, Scott Plantenga

Master's Theses

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, …