Numerical Analysis and Computation Commons^™

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 …

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 …

The Magnetic Field Of Protostar-Disk-Outflow Systems, Mahmoud Sharkawi

Electronic Thesis and Dissertation Repository

Recent observations of protostellar cores reveal complex magnetic field configurations that are distorted in the innermost disk region. Unlike the prestellar phase, where the magnetic field geometry is simpler with an hourglass configuration, magnetic fields in the protostellar phase are sculpted by the formation of outflows and rapid rotation. This gives rise to a significant azimuthal (or toroidal) component that has not yet been analytically modelled in the literature. Moreover, the onset of outflows, which act as angular momentum transport mechanisms, have received considerable attention in the past few decades. Two mechanisms: 1) the driving by the gradient of a …

Control Of Shear Layers Using Heating Patterns, Shoyon Panday

Electronic Thesis and Dissertation Repository

The presence of spatially modulated flows is universal in nature. Distributed heating and surface roughness are the most common elements to cause non-uniformity in the flows. Spatially distributed heating leads to fundamentally distinct convection, different from the classical Rayleigh-Bénard instability. Interestingly, the onset of convective motion due to horizontal temperature gradients requires no critical conditions – a forced response. At the same time, surface roughness is known to significantly influence flow behaviours and heat transfer characteristics. The current work aims to analyze modulated flows and assess their potential as a mixing technique for low Reynolds number flows. Spanwise modulations (perpendicular …

A Molecular Dynamics Study Of Polymer Chains In Shear Flows And Nanocomposites, Venkat Bala

Electronic Thesis and Dissertation Repository

In this work we study single chain polymers in shear flows and nanocomposite polymer melts extensively through the use of large scale molecular dynamics simulations through LAMMPS. In the single polymer chain shear flow study, we use the Lattice Boltzmann method to simulate fluid dynamics and also include thermal noise as per the \emph{fluctuation-dissipation} theorem in the system. When simulating the nanocomposite polymer melts, we simply use a Langevin thermostat to mimic a heat bath. In the single polymer in shear flow study we investigated the margination of a single chain towards solid surfaces and how strongly the shear flow …

Development Of A Low Field Mri-Based Approach For Observation Of Water Penetration Into Clay: Preliminary Results, Shivam Gupta

Undergraduate Student Research Internships Conference

Magnetic resonance imaging (MRI) are considered one of the most efficient and non-invasive methods of observing water content in permeable substances. MRI can visualize and quantify the movement of water in real time. In this study, MRI was used to observe the water penetration through clay. Furthermore, MRI can acquire three-dimensional data due to its radio-frequency signals from any orientation. The contrast of the images produced by MRI is a display of the fluid concentration. As such, any change in the contrast intensity is interpreted as a regional change in the concentration of fluid. This report summarizes the preliminary results …

Symphas: A Modular Api For Phase-Field Modeling Using Compile-Time Symbolic Algebra, Steven A. Silber

Electronic Thesis and Dissertation Repository

The phase-field method is a common approach to qualitative analysis of phase transitions. It allows visualizing the time evolution of a phase transition, providing valuable insight into the underlying microstructure and the dynamical processes that take place. Although the approach is applied in a diverse range of fields, from metal-forming to cardiac modelling, there are a limited number of software tools available that allow simulating any phase-field problem and that are highly accessible. To address this, a new open source API and software package called SymPhas is developed for simulating phase-field and phase-field crystal in 1-, 2- and 3-dimensions. Phase-field …

Ciculant Matrix And Fft, Thomas S. Devries

Undergraduate Student Research Internships Conference

The goal was to produce all the eigen values for a BOHEMIAN matrices using coefficient set {0, 1, -1, i, -i} of a size 15 vector. There are 5^15 eigen values so it was attempted to be done in parrallel for parts of the algorithm that permitted.

Eigenvalue Problems On Atypical Domains - The Finite Element Method, Toufiic Ayoub

Undergraduate Student Research Internships Conference

Why do we care about eigenvalues and eigenvectors? What's the big deal? For many people enrolled in entry level linear algebra courses, these concepts seem like far fetched abstractions that become pointless exercises in computation. But in reality, these fundamental ideas are vital to how we live our lives every single day. But how?

Mathematical Modelling & Simulation Of Large And Small Scale Structures In Star Formation, Gianfranco Bino

Electronic Thesis and Dissertation Repository

This thesis aims to study the magnetic and evolutionary properties of stellar objects from the prestellar phase up to and including the late protostellar phase. Many of the properties governing star formation are linked to the core’s physical properties and the magnetic field highly dictates much of the core’s stability.

The thesis begins with the implementation of a fully analytic magnetic field model used to study the magnetic properties governing the prestellar core FeSt 1-457. The model is a direct result of Maxwell’s equations and yields a central-to-surface magnetic field ratio in the equatorial plane in cylindrical coordinates. The model …

Longitudinal Partitioning Waveform Relaxation Methods For The Analysis Of Transmission Line Circuits, Tarik Menkad

Electronic Thesis and Dissertation Repository

Three research projects are presented in this manuscript. Projects one and two describe two waveform relaxation algorithms (WR) with longitudinal partitioning for the time-domain analysis of transmission line circuits. Project three presents theoretical results about the convergence of WR for chains of general circuits.

The first WR algorithm uses a assignment-partition procedure that relies on inserting external series combinations of positive and negative resistances into the circuit to control the speed of convergence of the algorithm. The convergence of the subsequent WR method is examined, and fast convergence is cast as a generic optimization problem in the frequency-domain. An automatic …

Hybrid Symbolic-Numeric Computing In Linear And Polynomial Algebra, Leili Rafiee Sevyeri

Electronic Thesis and Dissertation Repository

In this thesis, we introduce hybrid symbolic-numeric methods for solving problems in linear and polynomial algebra. We mainly address the approximate GCD problem for polynomials, and problems related to parametric and polynomial matrices. For symbolic methods, our main concern is their complexity and for the numerical methods we are more concerned about their stability. The thesis consists of 5 articles which are presented in the following order:

Chapter 1, deals with the fundamental notions of conditioning and backward error. Although our results are not novel, this chapter is a novel explication of conditioning and backward error that underpins the rest …

High Strain Dynamic Test On Helical Piles: Analytical And Numerical Investigations, Mohammed Fahad Alwalan

Electronic Thesis and Dissertation Repository

Helical piles are currently considered a preferred foundation option in a wide range of engineering projects to provide high compressive and uplift resistance to static and dynamic loads. In view of the large capacity of large diameter helical piles, there is a need to determine their capacity using accurate and economically feasible testing techniques. The capacity of piles is usually determined by conducting a Static Load Test (SLT). However, the SLT can be costly and time consuming, especially for large capacity piles. The High Strain Dynamic Load Test (HSDT) evaluates the pile capacity using dynamic measurements generated through subjecting the …

Algebraic Companions And Linearizations, Eunice Y. S. Chan

Electronic Thesis and Dissertation Repository

In this thesis, we look at a novel way of finding roots of a scalar polynomial using eigenvalue techniques. We extended this novel method to the polynomial eigenvalue problem (PEP). PEP have been used in many science and engineering applications such vibrations of structures, computer-aided geometric design, robotics, and machine learning. This thesis explains this idea in the order of which we discovered it.

In Chapter 2, a new kind of companion matrix is introduced for scalar polynomials of the form $c(\lambda) = \lambda a(\lambda)b(\lambda)+c_0$, where upper Hessenberg companions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. This construction can …

Some Recent Developments On Pareto-Optimal Reinsurance, Wenjun Jiang

Electronic Thesis and Dissertation Repository

This thesis focuses on developing Pareto-optimal reinsurance policy which considers the interests of both the insurer and the reinsurer. The optimal insurance/reinsurance design has been extensively studied in actuarial science literature, while in early years most studies were concentrated on optimizing the insurer’s interests. However, as early as 1960s, Borch argued that “an agreement which is quite attractive to one party may not be acceptable to its counterparty” and he pioneered the study on “fair” risk sharing between the insurer and the reinsurer. Quite recently, the question of how to strike a balance in risk sharing between an insurer and …

A Survey Of Numerical Quadrature Methods For Highly Oscillatory Integrals, Jeet Trivedi

Electronic Thesis and Dissertation Repository

In this thesis, we examine the main types of numerical quadrature methods for a special subclass of one-dimensional highly oscillatory integrals. Along with a presentation of the methods themselves and the error bounds, the thesis contains implementations of the methods in Maple and Python. The implementations take advantage of the symbolic computational abilities of Maple and allow for a larger class of problems to be solved with greater ease to the user. We also present a new variation on Levin integration which uses differentiation matrices in various interpolation bases.

Validating And Highlighting The Advantages Of The Optimal Estimation Method For Rayleigh Lidar Middle Atmospheric Temperature Retrievals, Ali Jalali

Electronic Thesis and Dissertation Repository

An improved understanding of temperature variations in Earth’s middle atmosphere is important for the improvement of our understanding of climate and weather on the surface. The optimal estimation method (OEM) is an inversion modeling approach, which uses regularized nonlinear regression to retrieve, in this case, the temperature of Earth’s middle atmosphere using Rayleigh-scatter lidar measurements. The OEM regularization term is the a priori knowledge of the atmospheric temperature profile. In this thesis I use lidar temperatures in the altitude range 30–110km to construct a temperature climatology using over 500 nights of measurements obtained by the Purple Crow Lidar in London, …

Investigation Of Chaos In Biological Systems, Navaneeth Mohan

Electronic Thesis and Dissertation Repository

Chaos is the seemingly irregular behavior arising from a deterministic system. Chaos is observed in many real-world systems. Edward Lorenz’s seminal discovery of chaotic behavior in a weather model has prompted researchers to develop tools that distinguish chaos from non-chaotic behavior. In the first chapter of this thesis, I survey the tools for detecting chaos namely, Poincaré maps, Lyapunov exponents, surrogate data analysis, recurrence plots and correlation integral plots. In chapter two, I investigate blood pressure fluctuations for chaotic signatures. Though my analysis reveals interesting evidence in support of chaos, the utility such an analysis lies in a different direction …

Properties And Computation Of The Inverse Of The Gamma Function, Folitse Komla Amenyou

Electronic Thesis and Dissertation Repository

We explore the approximation formulas for the inverse function of Γ. The inverse function of Γ is a multivalued function and must be computed branch by branch. We compare three approximations for the principal branch Γ̌ 0 . Plots and numerical values show that the choice of the approximation depends on the domain of the arguments, specially for small arguments. We also investigate some iterative schemes and find that the Inverse Quadratic Interpolation scheme is better than Newton’s scheme for improving the initial approximation. We introduce the contours technique for extending a real-valued function into the complex plane using two …

Real Solution Of Dae And Pdae System, Zahra Mohammadi, Greg Reid

Western Research Forum

General systems of differential equations don't have restrictions on the number or type of equations. For example, they can be over or under-determined, and also contain algebraic constraints (e.g. algebraic equations such as in Differential-Algebraic equations (DAE) and Partial differential algebraic equations (PDAE). Increasingly such general systems arise from mathematical modeling of engineering and science problems such as in multibody mechanics, electrical circuit design, optimal control, chemical kinetics and chemical control systems. In most applications, only real solutions are of interest, rather than complex-valued solutions. Much progress has been made in exact differential elimination methods, which enable characterization of all …

Numerical Studies Of Electrohydrodynamic Flow Induced By Corona And Dielectric Barrier Discharges, Chaoao Shi

Electronic Thesis and Dissertation Repository

Electrohyrodynamic (EHD) flow produced by gas discharges allows the control of airflow through electrostatic forces. Various promising applications of EHD can be considered, but this requires a deeper understanding of the physical mechanisms involved.

This thesis investigates the EHD flow generated by three forms of gas discharge. First, a multiple pin-plate EHD dryer associated with the positive corona discharge is studied using a stationary model. Second, the dynamics of a dielectric barrier discharge (DBD) plasma actuator is simulated with a time-dependent solver. Third, different configurations of the extended DBD are explored to enhance the EHD flow.

The results of the …

Feasible Computation In Symbolic And Numeric Integration, Robert H.C. Moir

Electronic Thesis and Dissertation Repository

Two central concerns in scientific computing are the reliability and efficiency of algorithms. We introduce the term feasible computation to describe algorithms that are reliable and efficient given the contextual constraints imposed in practice. The main focus of this dissertation then, is to bring greater clarity to the forms of error introduced in computation and modeling, and in the limited context of symbolic and numeric integration, to contribute to integration algorithms that better account for error while providing results efficiently.

Chapter 2 considers the problem of spurious discontinuities in the symbolic integration problem, proposing a new method to restore continuity …

Essays In Market Structure And Liquidity, Adrian J. Walton

Electronic Thesis and Dissertation Repository

Market structure concerns the mechanisms for negotiating trades and the composition of trading participants, and can affect liquidity and price efficiency. More gains from trade can be realized from an asset that is more liquid, and a better allocation of risk and capital can be achieved when an asset’s price is more efficient so it is important to understand market structure. This thesis uses theory and empirical methods to examine the effects of a few specific aspects of market structure.

In Chapter 1, we study a novel market structure on the New York Stock Exchange (NYSE), the Retail Liqudity Program …

A Comparison Of Solution Methods For Mandelbrot-Like Polynomials, Eunice Y. S. Chan

Electronic Thesis and Dissertation Repository

We compare two different root-finding methods, eigenvalue methods and homotopy methods, using three test problems: Mandelbrot polynomials, Fibonacci-Mandelbrot polynomials, and Narayana-Mandelbrot polynomials. For the eigenvalue methods, using both MATLAB and Maple, we computed the eigenvalues of a specialized recursively-constructed, supersparse, upper Hessenberg matrix, inspired by Piers Lawrence's original construction for the Mandelbrot polynomials, for all three families of polynomials. This led us to prove that this construction works in general. Therefore, this construction is genuinely a new kind of companion matrix. For the homotopy methods, we used a special-purpose homotopy, in which we used an equivalent differential equation to solve …

Secondary Electrohydrodynamic Flow Generated By Corona And Dielectric Barrier Discharges, Mohammadreza Ghazanchaei

Electronic Thesis and Dissertation Repository

One of the main goals of applied electrostatics engineering is to discover new perspectives in a wide range of research areas. Controlling the fluid media through electrostatic forces has brought new important scientific and industrial applications. Electric field induced flows, or electrohydrodynamics (EHD), have shown promise in the field of fluid dynamics. Although numerous EHD applications have been explored and extensively studied so far, most of the works are either experimental studies, which are not capable to explain the in depth physics of the phenomena, or detailed analytical studies, which are not time effective. The focus of this study is …

Tropical Cyclone Wind Hazard Assessment For Southeast Part Of Coastal Region Of China, Sihan Li

Electronic Thesis and Dissertation Repository

Tropical cyclone (TC) or typhoon wind hazard and risk are significant for China. The return period value of the maximum typhoon wind speed is used to characterize the typhoon wind hazard and assign wind load in building design code. Since the historical surface observations of typhoon wind speed are often scarce and of short period, the typhoon wind hazard assessment is often carried out using the wind field model and TC track model. For a few major cities in the coastal region of mainland China, simple or approximated wind field models and a circular subregion method (CSM) have been used …

Observational Signatures From Self-Gravitating Protostellar Disks, Alexander L. Desouza

Electronic Thesis and Dissertation Repository

Protostellar disks are the ubiquitous corollary outcome of the angular momentum conserving, gravitational collapse of molecular cloud cores into stars. Disks are an essential component of the star formation process, mediating the accretion of material onto the protostar, and for redistributing excess angular momentum during the collapse. We present a model to explain the observed correlation between mass accretion rates and stellar mass that has been inferred from observations of intermediate to upper mass T Tauri stars. We explain this correlation within the framework of gravitationally driven torques parameterized in terms of Toomre’s Q criterion. Our models reproduce both the …

Options Pricing And Hedging In A Regime-Switching Volatility Model, Melissa A. Mielkie

Electronic Thesis and Dissertation Repository

Both deterministic and stochastic volatility models have been used to price and hedge options. Observation of real market data suggests that volatility, while stochastic, is well modelled as alternating between two states. Under this two-state regime-switching framework, we derive coupled pricing partial differential equations (PDEs) with the inclusion of a state-dependent market price of volatility risk (MPVR) term.

Since there is no closed-form solution for this pricing problem, we apply and compare two approaches to solving the coupled PDEs, assuming constant Poisson intensities. First we solve the problem using numerical solution techniques, through the application of the Crank-Nicolson numerical scheme. …

Study Of Virus Dynamics By Mathematical Models, Xiulan Lai

Electronic Thesis and Dissertation Repository

This thesis studies virus dynamics within host by mathematical models, and topics discussed include viral release strategies, viral spreading mechanism, and interaction of virus with the immune system.

Firstly, we propose a delay differential equation model with distributed delay to investigate the evolutionary competition between budding and lytic viral release strategies. We find that when antibody is not established, the dynamics of competition depends on the respective basic reproduction numbers of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for …

Image Fusion And Axial Labeling Of The Spine, Brandon Miles

Electronic Thesis and Dissertation Repository

In order to improve radiological diagnosis of back pain and spine disease, two new algorithms have been developed to aid the 75% of Canadians who will suffer from back pain in a given year. With the associated medical imaging required for many of these patients, there is a potential for improvement in both patient care and healthcare economics by increasing the accuracy and efficiency of spine diagnosis. A real-time spine image fusion system and an automatic vertebra/disc labeling system have been developed to address this. Both magnetic resonance (MR) images and computed tomography (CT) images are often acquired for patients. …