Applied Mathematics Commons^™

Analysis Of Multi Grade Deep Learning, Ronglong Fang

Mathematics & Statistics Theses & Dissertations

Multi-Grade Deep Learning (MGDL) is a training framework that incrementally builds deep neural networks. It does this by dividing the training process into multiple “grades,” where each grade sequentially trains a shallow neural network to learn the residue from the previous one, using the outputs of prior grades as input. This approach progresses from shallow to deep architectures. This dissertation offers a comprehensive theoretical and numerical analysis of the MGDL methodology.

We first demonstrate that MGDL can effectively learn target functions within the sum-composition learning format. In this context, MGDL approximates high-frequency components by composing multiple low-frequency functions. This unique …

Two Analytic Issues Arising From Models In Probability And Materials Science, Giang Vu Thanh Nguyen

Mathematics & Statistics Theses & Dissertations

This dissertation explores two distinct topics centered on mathematical models in probability theory and materials science. The first part investigates a series of functions derived from an adaptive algorithm designed to address the score-based secretary problem, a classic challenge in probability theory. This problem involves making immediate decisions to select the best candidate from a sequence of interviews. The algorithm aims to maximize the probability of selecting the optimal candidate based on observed scores. We prove two fundamental analytic properties of this sequence of functions as a theoretic support of the algorithm: first, the functions in the sequence each possess …

A Strict Physicality-Preserving Scheme For A 2d Q-Tensor Flow With A Singular Potential, Md Mashud Parvez

Mathematics & Statistics Theses & Dissertations

Nematic liquid crystals are a state of matter that exhibit properties between those of conventional liquids and solid crystals. Their unique ability to align molecules in specific directions makes them essential in various applications, including display technologies and advanced materials. To model their complex behavior, mathematical frameworks such as the Q-tensor model are used to describe the orientation and degree of molecular order. In this work, we introduce a numerical scheme for a two-dimensional (2D) dynamic Q-tensor model, which is formulated as an L²-gradient flow driven by the liquid crystal free energy and incorporates a singular potential to …

Stability Analysis In The Twist-Bend Nematic Liquid Crystal Model, Zhenqiang Li

Mathematics & Statistics Theses & Dissertations

The recently discovered twist-bend nematic liquid crystal (LC) phase is characterized by a nanoscale helical modulation of the nematic director n, forming a conical helix along the z-axis at an oblique angle θ. While many models assume a constant cone angle and equal elastic constants K11 = K22 = K33, this dissertation removes both assumptions by considering a fully anisotropic elastic energy with K11 ≠ K22 ≠ K33, and allowing θ to vary spatially. We analyze the stability of this system under frustrated and free boundary conditions using variational methods. …

Parallel-In-Time Implicit Schemes For Nonlinear Pdes, Subhash Paudel

Mathematics & Statistics Theses & Dissertations

An accurate prediction of unsteady physical phenomena arising in various applications (e.g., rotorcraft and turbomachinery flows, fluid-structure interaction, maneuvering flight conditions, etc.) requires a very large number of time steps, thus considerably increasing the total computational time, because conventional time integrators are inherently sequential. Parallel-in-time methods offer a promising direction for drastically reducing the computational time and achieving such scalability that is required for solving these unsteady problems on modern supercomputers with hundreds of thousands of computing cores. The parallel performance of existing parallel-in-time algorithms for nonlinear equations especially of the hyperbolic or mixed type is far from being satisfactory. …

Deep Learning In Reproducing Kernel Banach Spaces, Mingsong Yan

Mathematics & Statistics Theses & Dissertations

Deep learning has achieved immense success in the past decade. The goal of this dissertation is to understand deep learning through the framework of reproducing kernel Banach spaces (RKBSs), which were originally proposed for promoting sparse solutions. We begin by considering learning problems in a general functional setting, and establishing explicit and data-dependent representer theorems for both minimal norm interpolation (MNI) problems and regularization problems. These theorems provide a crucial foundation for the subsequent results derived for both sparse learning and deep learning. Next, we investigate the essential properties of RKBSs capable of encouraging sparsity in learning solutions. With the …

On Weighted Sequence Spaces, Gilbert D. Acheampong

Mathematics & Statistics Theses & Dissertations

The space ℓ^p,α of complex sequences a = (a₀,a₁,a₂, . . .) for which

∞

∥a∥p,α = ( Σ|ak|^p(k+1)^α)^1/p < ∞

k=0

is studied. Each such sequence can be identified with the analytic function with power series

∞

f (z) = ∑ akz^k.

k=0

In this setting, the point evaluation and the difference quotient mappings are shown to be bounded; the cases are identified in which ℓ^p,α is boundedly contained in ℓ^r,β . …

Accelerating The Efficiency Of Multiscale Hybridizable Discontinuos Galerkin Methods For Flows In Heterogeneous Media, Tony Charles Haines

Mathematics & Statistics Theses & Dissertations

A plethora of scientific and engineering problems encountered are multiscale in nature. This multiscale feature often influences simulation efforts wherever large disparities in spatial scales are experienced. Notable examples include composite materials, fluid flow through porous media and turbulent transport in high Reynolds number flow. Although there are promising results from the advancement of modern supercomputer, obtaining direct numerical solution of multiscale problems is very laborious. This difficulty stems from the tremendous amount of computer memory and CPU time required. Parallel computing may be one obvious choice in remedying this issue. However, the complexity and size of the discrete problem …

Contributions To Nonparametric Testing In Clustered Data, Hasika Kalani Wickrama Senevirathne

Mathematics & Statistics Theses & Dissertations

Clustered data refers to a specific kind of correlated data where units within the same cluster are correlated while units from different clusters are independent. The number of units in each cluster, known as the cluster size, can be associated with the cluster’s outcome. This is known as the informative cluster size (ICS) and affects the inference drawn from clustered data. Recently, a hypothesis testing method has been developed to detect the presence of ICS. However, considering ICS alone may not be sufficient when comparing outcomes across multiple groups of units within clustered data. The size of a group within …

Scaled And Graduated Learning In Deep Relu Networks And Reconstructing Depp Inelastic Scattering Kinematics, Abdullah Ayar Farhat

Mathematics & Statistics Theses & Dissertations

To address computational challenges in learning deep neural networks, properties of deep RELU networks were studied to develop a multi-scale learning model. The multi-scale model was compared to the multi-grade learning models. Unlike the deep neural network learned from the standard single-scale, single-grade model, the multi-scale neural networks use low scale information from all hidden layers, and thusly provide a robust approximation method that requires fewer parameters, lower computational time, and is resistant to noise. It is shown that the multiscale method is not subject to issues arising from the vanishing gradient problem. This allows very deep multi-scale networks to …

Copula Based Models For Bivariate Zero-Inflated Count Time Series Data, Dimuthu Fernando

Mathematics & Statistics Theses & Dissertations

Count time series data have multiple applications. The applications can be found in areas of finance, climate, public health and crime data analyses. In most scenarios, time is an important part of the data. Time series counts then come as multivariate vectors that exhibit not only serial dependence within each time series but also with cross-correlation among the series. When considering these observed counts, and when a value, say zero, occurs more often than usual, analysis presents crucial challenges. There is presence of zeroinflation in the data. The literature on bivariate or multivariate count time series, as well as zero-inflated …

Inference For Multiple Utility In Time-Dependent Choice Pairs Under Copula-Based Models, Sasanka Adikari

Mathematics & Statistics Theses & Dissertations

Models for discrete choice experiments (DCE) are frequently used to analyze consumer choices about products and services. A family of DCE, best-worst scaling experiments, offers more in-depth insights into consumer preferences by eliciting a best and worst choice from a set of options, rather than just a single preference. Traditional approaches often assume that choices are mutually exclusive over time, which may not always be the case. This dissertation proposes a novel model for DCE that takes into account the changing nature of consumer choices over time and the priority constraint of transition probabilities. The model introduces a copula combination …

Statistical Methods For Meta-Analysis In Large-Scale Genomic Experiments, Wimarsha Thathsarani Jayanetti

Mathematics & Statistics Theses & Dissertations

Recent developments in high throughput genomic assays have opened up the possibility of testing hundreds and thousands of genes simultaneously. With the availability of vast amounts of public databases, researchers tend to combine genomic analysis results from multiple studies in the form of a meta-analysis. Meta-analysis methods can be broadly classified into two main categories. The first approach is to combine the statistical significance (pvalues) of the genes from each individual study, and the second approach is to combine the statistical estimates (effect sizes) from the individual studies. In this dissertation, we will discuss how adherence to the standard null …

Kinetic Simulations Of Active Nematic Polymers In Channel Flow, Lacey Savoie Schenk

Mathematics & Statistics Theses & Dissertations

Suspensions of active nematic liquid crystalline polymers exhibit complex phenomena such as spontaneous flows, pattern formations, and defects. They have many applications in industry, commercial settings, and our daily lives. We employ the Kinetic Model for our research, an extensive model that couples the Smoluchowski Equation and the incompressible Navier-Stokes Equations to solve for the active nanorod number density function–a function dependent on the polymer’s physical orientation and space at a given time. Using this function, we can derive the polymer’s polarity and nematic orientations as well as other rheological properties. In this research, we conduct numerical simulations of active, …

Inexact Fixed-Point Proximity Algorithms For Nonsmooth Convex Optimization, Jin Ren

Mathematics & Statistics Theses & Dissertations

The aim of this dissertation is to develop efficient inexact fixed-point proximity algorithms with convergence guaranteed for nonsmooth convex optimization problems encountered in data science. Nonsmooth convex optimization is one of the core methodologies in data science to acquire knowledge from real-world data and has wide applications in various fields, including signal/image processing, machine learning and distributed computing. In particular, in the context of image reconstruction, compressed sensing and sparse machine learning, either the objective functions or the constraints of the modeling optimization problems are nondifferentiable. Hence, traditional methods such as the gradient descent method and the Newton method are …

A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Janani Gamage

Mathematics & Statistics Theses & Dissertations

Interface problems have many applications in physics. In this dissertation, we develop a direct method for solving three-dimensional elliptic interface problems and study their application in solving parabolic interface problems. As many of the physical applications of interface problems can be approximated with partial differential equations (PDE) with piecewise constant coefficients, our derivation of the model is focused on interface problems with piecewise constant coefficients but have a finite jump across the interface. The critical characteristic of the method is that our computational framework is based on a finite difference scheme on a uniform Cartesian grid system and does not …

On The P-Inner Functions Of ℓPA, James G. Dragas

Mathematics & Statistics Theses & Dissertations

Define ℓ^p_A as the space of all functions holomorphic over the unit disk whose Taylor coefficients are p-summable. Despite their classical origins and simple definition, these spaces are not as well understood as one might expect. This is particularly true when compared with the Hardy spaces, which provide a useful road map for the types of questions we might consider reasonable. In this work we examine the zero sets of ℓ^p_A, p ∈ (1;∞), as well as a notion of inner function that is consistent with the approach taken on numerous other function spaces. …

High-Order Positivity-Preserving L2-Stable Spectral Collocation Schemes For The 3-D Compressible Navier-Stokes Equations, Johnathon Keith Upperman

Mathematics & Statistics Theses & Dissertations

High-order entropy stable schemes are a popular method used in simulations with the compressible Euler and Navier-Stokes equations. The strength of these methods is that they formally satisfy a discrete entropy inequality which can be used to guarantee L₂ stability of the numerical solution. However, a fundamental assumption that is explicitly or implicitly used in all entropy stability proofs available in the literature for the compressible Euler and Navier-Stokes equations is that the thermodynamic variables (e.g., density and temperature) are strictly positive in the entire space{time domain considered. Without this assumption, any entropy stability proof for a numerical scheme …

Finite Difference Schemes For Integral Equations With Minimal Regularity Requirements, Wesley Cameron Davis

Mathematics & Statistics Theses & Dissertations

Volterra integral equations arise in a variety of applications in modern physics and engineering, namely in interactions that contain a memory term. Classical formulations of these problems are largely inflexible when considering non-homogeneous media, which can be problematic when considering long term interactions of real-world applications. The use of fractional derivative and integral terms naturally relax these restrictions in a natural way to consider these problems in a more general setting. One major drawback to the use of fractional derivatives and integrals in modeling is the regularity requirement for functions, where we can no longer assume that functions are as …

Electrohydrodynamic Simulations Of Capsule Deformation Using A Dual Time-Stepping Lattice Boltzmann Scheme, Charles Leland Armstrong

Mathematics & Statistics Theses & Dissertations

Capsules are fluid-filled, elastic membranes that serve as a useful model for synthetic and biological membranes. One prominent application of capsules is their use in modeling the response of red blood cells to external forces. These models can be used to study the cell’s material properties and can also assist in the development of diagnostic equipment. In this work we develop a three dimensional model for numerical simulations of red blood cells under the combined influence of hydrodynamic and electrical forces. The red blood cell is modeled as a biconcave-shaped capsule suspended in an ambient fluid domain. Cell deformation occurs …

Inference And Estimation In Change Point Models For Censored Data, Kristine Gierz

Mathematics & Statistics Theses & Dissertations

In general, the change point problem considers inference of a change in distribution for a set of time-ordered observations. This has applications in a large variety of fields and can also apply to survival data. With improvements to medical diagnoses and treatments, incidences and mortality rates have changed. However, the most commonly used analysis methods do not account for such distributional changes. In survival analysis, change point problems can concern a shift in a distribution for a set of time-ordered observations, potentially under censoring or truncation.

In this dissertation, we first propose a sequential testing approach for detecting multiple change …

Investigating The Feasibility And Stability For Modeling Acoustic Wave Scattering Using A Time-Domain Boundary Integral Equation With Impedance Boundary Condition, Michelle E. Rodio

Mathematics & Statistics Theses & Dissertations

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet and environmentally friendly aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction and are characterized by a frequency-dependent impedance. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners

This work considers using either an impedance or an admittance (inverse …

Electrohydrodynamic Simulations Of The Deformation Of Liquid-Filled Capsules, Pai Song

Mathematics & Statistics Theses & Dissertations

A comprehensive two- and three-dimensional framework for the electrohydrodynamic simulation of deformable capsules is provided. The role of a direct current (DC) electric field on the deformation and orientation of a liquid-filled capsule is thoroughly considered numerically. This framework is based on lattice Boltzmann method for the fluid, finite element method for the membrane structure of the capsule, fast immersed interface method for the electric field and immersed boundary method being used to consider the fluid-structure-electric interaction. Under the effect of electric field, two different types of equilibrium states, prolate or oblate are obtained. The numerical algorithm is also applied …

Extended Poisson Models For Count Data With Inflated Frequencies, Monika Arora

Mathematics & Statistics Theses & Dissertations

Count data often exhibits inflated counts for zero. There are numerous papers in the literature that show how to fit Poisson regression models that account for the zero inflation. However, in many situations the frequencies of zero and of some other value k tends to be higher than the Poisson model can fit appropriately. Recently, Sheth-Chandra (2011), Lin and Tsai (2012) introduced a mixture model to account for the inflated frequencies of zero and k. In this dissertation, we study basic properties of this mixture model and parameter estimation for grouped and ungrouped data. Using stochastic representation we show …

A Partitioned Approach For Computing Fluid-Structure Interaction, With Application To Tumor Modeling And Simulation, Asim Timalsina

Mathematics & Statistics Theses & Dissertations

Modeling and Simulation plays a critical role in understanding complex physical and biological phenomena as it provides an efficient and controlled test environment, without the risk of costly experiments and clinical trials. In this dissertation, we present an extensive study of two such systems with integrated application: Fluid structure interaction (FSI) and virotherapy on tumor. Moreover, we substantiate a few preliminary results of FSI application on tumor.

The FSI problem comprises of fluid forces exerted on the solid body and the motion of the structure affecting the fluid flow. FSI problems are of great interest to applied industries, however they …

Modeling And Simulation Of Molecular Couette Flows And Related Flows, Wei Li

Mathematics & Statistics Theses & Dissertations

In this thesis, molecular Couette flow is clearly defined and the modeling and simulation of this kind of flow is systematically investigated. First, the integral equations for the velocity of gaseous Couette flow and related flows are derived from linearized Boltzmann BGK equation with Maxwell boundary condition and solved with high precision by using Chebyshev collocation and chunk-based collocation methods. The velocity profiles of gaseous Couette flows and related flows with a wide range of Knudsen number and the Maxwell boundary condition of various accommodation ratios are obtained. Moreover, the order of convergence of the numerical methods is also discussed …

Uniform L1 Behavior Of A Time Discretization Method For A Volterra Integrodifferential Equation With Convex Kernel; Duality Of The Weak Parallelogram Laws On Banach Spaces, Charles Benjamin Harris

Mathematics & Statistics Theses & Dissertations

The first chapter of this thesis concerns the stability and convergence of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation

u'(t) + ∫^t₀ β (t--s) Au(s) ds = 0, t ≥ 0, u( 0) = u_0,

which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space and β(t) is locally integrable, nonnegative, nonincreasing, convex, with -- β'(t) and β''(t) convex. We establish …

Analyzing Cholera Dynamics In Homogeneous And Heterogeneous Environments, Drew Posny

Mathematics & Statistics Theses & Dissertations

Cholera continues to be a serious public health concern in developing countries and the global increase in the number of reported outbreaks suggests that activities to control the diseases and surveillance programs to identify or predict the occurrence of the next outbreaks are not adequate. Mathematical models play a critical role in predicting and understanding disease mechanisms, and have long provided basic insights in the possible ways to control infectious diseases. This dissertation is concerned with mathematical modeling and analysis of cholera dynamics. First, we study an autonomous model in a homogeneous environment with added controls that involves both direct …

Modeling And Simulation Of Shape Changes Of Red Blood Cells In Shear Flow, John Gounley

Mathematics & Statistics Theses & Dissertations

A description of the biomechanical character of red blood cells is given, along with an introduction to current computational schemes which use deformable capsules to simulate red blood cell shape change. A comprehensive two- and three-dimensional framework for the fluid-structure interaction between a deformable capsule and an ambient flow is provided. This framework is based on the immersed boundary method, using lattice Boltzmann and finite element methods for the fluid and structure, respectively. The characteristic response and recovery times of viscoelastic circular and spherical capsules are compared, and their dependence on simulation parameters is shown. The shape recovery of biconcave …

Topics In Electromagnetic, Acoustic, And Potential Scattering Theory, Umaporn Nuntaplook

Mathematics & Statistics Theses & Dissertations

With recent renewed interest in the classical topics of both acoustic and electromagnetic aspects for nano-technology, transformation optics, fiber optics, metamaterials with negative refractive indices, cloaking and invisibility, the topic of time-independent scattering theory in quantum mechanics is becoming a useful field to re-examine in the above contexts. One of the key areas of electromagnetic theory scattering of plane electromagnetic waves — is based on the properties of the refractive indices in the various media. It transpires that the refractive index of a medium and the potential in quantum scattering theory are intimately related. In many cases, understanding such scattering …