Numerical Analysis and Computation Commons^™

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 …

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 …

Space-Angle Discontinuous Galerkin Finite Element Method For Radiative Transfer Equation, Hang Wang

Doctoral Dissertations

Radiative transfer theory describes the interaction of radiation with scattering and absorbing media. It has applications in neutron transport, atmospheric physics, heat transfer, molecular imaging, and others. In steady state, the radiative transfer equation is an integro-differential equation of five independent variables, which are 3 dimensions in space and 2 dimensions in the angular direction. This high dimensionality and the presence of the integral term present serious challenges when solving the equation numerically. Over the past 50 years, several techniques for solving the radiative transfer equation (RTE) have been introduced. These include, but are certainly not limited to, Monte Carlo …

Data-Driven Computational Methods For Quasi-Stationary Distribution And Sensitivity Analysis, Yaping Yuan

Doctoral Dissertations

The goal of the dissertation is to develop the computational methods for quasi-stationary- distributions(QSDs) and the sensitivity analysis of a QSD against the modification of the boundary conditions and against the diffusion approximation.
Many models in various applications are described by Markov chains with absorbing states. For example, any models with mass-action kinetics, such as ecological models, epidemic models, and chemical reaction models, are subject to the population-level randomness called the demographic stochasticity, which may lead to extinction in finite time. There are also many dynamical systems that have interesting short term dynamics but trivial long term dynamics, such as …

Fourth Order Dispersion In Nonlinear Media, Georgios Tsolias

Doctoral Dissertations

In recent years, there has been an explosion of interest in media bearing quartic
dispersion. After the experimental realization of so-called pure-quartic solitons, a
significant number of studies followed both for bright and for dark solitonic struc-
tures exploring the properties of not only quartic, but also setic, octic, decic etc.
dispersion, but also examining the competition between, e.g., quadratic and quartic
dispersion among others.
In the first chapter of this Thesis, we consider the interaction of solitary waves in
a model involving the well-known φ⁴ Klein-Gordon theory, bearing both Laplacian and biharmonic terms with different prefactors. As a …

Towards Reduced-Order Model Accelerated Optimization For Aerodynamic Design, Andrew L. Kaminsky

Doctoral Dissertations

The adoption of mathematically formal simulation-based optimization approaches within aerodynamic design depends upon a delicate balance of affordability and accessibility. Techniques are needed to accelerate the simulation-based optimization process, but they must remain approachable enough for the implementation time to not eliminate the cost savings or act as a barrier to adoption.

This dissertation introduces a reduced-order model technique for accelerating fixed-point iterative solvers (e.g. such as those employed to solve primal equations, sensitivity equations, design equations, and their combination). The reduced-order model-based acceleration technique collects snapshots of early iteration (pre-convergent) solutions and residuals and then uses them to project …

Hyperspectral Unmixing: A Theoretical Aspect And Applications To Crism Data Processing, Yuki Itoh

Doctoral Dissertations

Hyperspectral imaging has been deployed in earth and planetary remote sensing, and has contributed the development of new methods for monitoring the earth environment and new discoveries in planetary science. It has given scientists and engineers a new way to observe the surface of earth and planetary bodies by measuring the spectroscopic spectrum at a pixel scale. Hyperspectal images require complex processing before practical use. One of the important goals of hyperspectral imaging is to obtain the images of reflectance spectrum. A raw image obtained by hyperspectral remote sensing usually undergoes conversion to a physical quantity representing the intensity of …

Model Based Force Estimation And Stiffness Control For Continuum Robots, Vincent A. Aloi

Doctoral Dissertations

Continuum Robots are bio-inspired structures that mimic the motion of snakes, elephant trunks, octopus tentacles, etc. With good design, these robots can be naturally compliant and miniaturizable, which makes Continuum Robots ideal for traversing narrow complex environments. Their flexible design, however, prevents us from using traditional methods for controlling and estimating loading on rigid link robots.

In the first thrust of this research, we provided a novel stiffness control law that alters the behavior of an end effector during contact. This controller is applicable to any continuum robot where a method for sensing or estimating tip forces and pose exists. …

Numerical Methods For Stochastic Stokes And Navier-Stokes Equations, Liet Vo

Doctoral Dissertations

This dissertation consists of three main parts with each part focusing on numerical approximations of the stochastic Stokes and Navier-Stokes equations.

Part One concerns the mixed finite element methods and Chorin projection methods for solving the stochastic Stokes equations with general multiplicative noise. We propose a modified mixed finite element method for solving the Stokes equations and show that the numerical solutions converge optimally to the PDE solutions. The convergence is under energy norms (strong convergence) for the velocity and in a time-averaged norm (weak convergence) for the pressure. In addition, after establishing the error estimates in second moment, high …

A Weak Fractional Calculus Theory And Numerical Methods For Fractional Differential Equations, Mitchell D. Sutton

Doctoral Dissertations

This dissertation is comprised of four integral parts. The first part comprises a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions.

The second part of this work presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the …

Preconditioned Nesterov’S Accelerated Gradient Descent Method And Its Applications To Nonlinear Pde, Jea Hyun Park

Doctoral Dissertations

We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show …

Examining The Accumulation Statistics Of Index1 Saddle Points On The Potential Energy Surface And Imposing Early Termination On A Rejection Scheme For Off Lattice Kinetic Monte Carlo, Jonathan W. Hicks

Doctoral Dissertations

In the calculation of time evolution of an atomic system where a chemical reaction and/or diffusion occurs, off-lattice kinetic Monte Carlo methods can be used to overcome timescale and lattice based limitations from other methods such as Molecular Dynamics and kinetic Monte Carlo procedures. Off-lattice kinetic Monte Carlo methods rely on a harmonic approximation to Transition State Theory, in which the rate of the rare transitions from one energy minimum to a neighboring minimum require surmounting a minimum energy barrier on the Potential Energy Surface, which is found at an index-1 saddle point commonly referred to as a transition state. …

A Parallel Direct Method For Finite Element Electromagnetic Computations Based On Domain Decomposition, Javad Moshfegh

Doctoral Dissertations

High performance parallel computing and direct (factorization-based) solution methods have been the two main trends in electromagnetic computations in recent years. When time-harmonic (frequency-domain) Maxwell's equation are directly discretized with the Finite Element Method (FEM) or other Partial Differential Equation (PDE) methods, the resulting linear system of equations is sparse and indefinite, thus harder to efficiently factorize serially or in parallel than alternative methods e.g. integral equation solutions, that result in dense linear systems. State-of-the-art sparse matrix direct solvers such as MUMPS and PARDISO don't scale favorably, have low parallel efficiency and high memory footprint. This work introduces a new …

Asymptotic And Numerical Analysis Of Coherent Structures In Nonlinear Schrodinger-Type Equations, Cory Ward

Doctoral Dissertations

This dissertation concerns itself with coherent structures found in nonlinear Schrödinger-type equations and can be roughly split into three parts. In the first part we study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa-Holm NLS (CH-NLS) equation in both one and two space dimensions. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently used to obtain approximate solitary wave solutions for both the 1D and 2D CH-NLS equations. We then use direct numerical simulations to investigate the validity of these approximate solutions, their evolution, and their head-on …

Inexact And Nonlinear Extensions Of The Feast Eigenvalue Algorithm, Brendan E. Gavin

Doctoral Dissertations

Eigenvalue problems are a basic element of linear algebra that have a wide variety of applications. Common examples include determining the stability of dynamical systems, performing dimensionality reduction on large data sets, and predicting the physical properties of nanoscopic objects. Many applications require solving large dimensional eigenvalue problems, which can be very challenging when the required number of eigenvalues and eigenvectors is also large. The FEAST algorithm is a method of solving eigenvalue problems that allows one to calculate large numbers of eigenvalue/eigenvector pairs by using contour integration in the complex plane to divide the large number of desired pairs …

Mathematical Modeling Of Mixtures And Numerical Solution With Applications To Polymer Physics, John Timothy Cummings

Doctoral Dissertations

We consider in this dissertation the mathematical modeling and simulation of a general diffuse interface mixture model based on the principles of energy dissipation. The model developed allows for a thermodynamically consistent description of systems with an arbitrary number of different components, each of which having perhaps differing densities. We also provide a mathematical description of processes which may allow components to source or sink into other components in a mass conserving, energy dissipating way, with the motivation of applying this model to phase transformation. Also included in the modeling is a unique set of thermodynamically consistent boundary conditions which …

Efficient Methods For Multidimensional Global Polynomial Approximation With Applications To Random Pdes, Peter A. Jantsch

Doctoral Dissertations

In this work, we consider several ways to overcome the challenges associated with polynomial approximation and integration of smooth functions depending on a large number of inputs. We are motivated by the problem of forward uncertainty quantification (UQ), whereby inputs to mathematical models are considered as random variables. With limited resources, finding more efficient and accurate ways to approximate the multidimensional solution to the UQ problem is of crucial importance, due to the “curse of dimensionality” and the cost of solving the underlying deterministic problem.

The first way we overcome the complexity issue is by exploiting the structure of the …

Numerical Methods For Non-Divergence Form Second Order Linear Elliptic Partial Differential Equations And Discontinuous Ritz Methods For Problems From The Calculus Of Variations, Stefan Raymond Schnake

Doctoral Dissertations

This dissertation consists of three integral parts. Part one studies discontinuous Galerkin approximations of a class of non-divergence form second order linear elliptic PDEs whose coefficients are only continuous. An interior penalty discontinuous Galerkin (IP-DG) method is developed for this class of PDEs. A complete analysis of the proposed IP-DG method is carried out, which includes proving the stability and error estimate in a discrete W^2;p-norm [W^2,p-norm]. Part one also studies the convergence of the vanishing moment method for this class of PDEs. The vanishing moment method refers to a PDE technique for approximating these PDEs by a …

Information Metrics For Predictive Modeling And Machine Learning, Kostantinos Gourgoulias

Doctoral Dissertations

The ever-increasing complexity of the models used in predictive modeling and data science and their use for prediction and inference has made the development of tools for uncertainty quantification and model selection especially important. In this work, we seek to understand the various trade-offs associated with the simulation of stochastic systems. Some trade-offs are computational, e.g., execution time of an algorithm versus accuracy of simulation. Others are analytical: whether or not we are able to find tractable substitutes for quantities of interest, e.g., distributions, ergodic averages, etc. The first two chapters of this thesis deal with the study of the …

Jet-Hadron Correlations Relative To The Event Plane Pb--Pb Collisions At The Lhc In Alice, Joel Anthony Mazer

Doctoral Dissertations

In relativistic heavy ion collisions at the Large Hadron Collider (LHC), a hot, dense and strongly interacting medium known as the Quark Gluon Plasma (QGP) is produced. Quarks and gluons from incoming nuclei collide to produce partons at high momenta early in the collisions. By fragmenting into collimated sprays of hadrons, these partons form 'jets'. Within the framework of perturbative Quantum Chromodynamics (pQCD), jet production is well understood in pp collisions. We can use jets measured in pp interactions as a baseline reference for comparing to heavy ion collision systems to detect and study jet quenching. The jet quenching mechanism …

Surface Energy In Bond-Counting Models On Bravais And Non-Bravais Lattices, Tim Ryan Krumwiede

Doctoral Dissertations

Continuum models in computational material science require the choice of a surface energy function, based on properties of the material of interest. This work shows how to use atomistic bond-counting models and crystal geometry to inform this choice. We will examine some of the difficulties that arise in the comparison between these models due to differing types of truncation. New crystal geometry methods are required when considering materials with non-Bravais lattice structure, resulting in a multi-valued surface energy. These methods will then be presented in the context of the two-dimensional material graphene in a way that correctly predicts its equilibrium …

Kinetic Monte Carlo Models For Crystal Defects, Kyle Louis Golenbiewski

Doctoral Dissertations

Kinetic Monte Carlo algorithms have become an increasingly popular means to simulate stochastic processes since their inception in the 1960's. One area of particular interest is their use in simulations of crystal growth and evolution in which atoms are deposited on, or hop between, predefined lattice locations with rates depending on a crystal's configuration. Two such applications are heteroepitaxial thin films and grain boundary migration. Heteroepitaxial growth involves depositing one material onto another with a different lattice spacing. This misfit leads to long-range elastic stresses that affect the behavior of the film. Grain boundary migration, on the other hand, describes …

Computational Simulation Of Mass Transport And Energy Transfer In The Microbial Fuel Cell System, Shiqi Ou

Doctoral Dissertations

This doctoral dissertation introduces the research in the computational modeling and simulation for the microbial fuel cell (MFC) system which is a bio-electrochemical system that drives a current by using bacteria and mimicking bacterial interactions found in nature. The numerical methods, research approaches and simulation comparison with the experiments in the microbial fuel cells are described; the analysis and evaluation for the model methods and results that I have achieved are presented in this dissertation.

The development of the renewable energy has been a hot topic, and scientists have been focusing on the microbial fuel cell, which is an environmentally-friendly …

Numerical Methods For Deterministic And Stochastic Phase Field Models Of Phase Transition And Related Geometric Flows, Yukun Li

Doctoral Dissertations

This dissertation consists of three integral parts with each part focusing on numerical approximations of several partial differential equations (PDEs). The goals of each part are to design, to analyze and to implement continuous or discontinuous Galerkin finite element methods for the underlying PDE problem.

Part One studies discontinuous Galerkin (DG) approximations of two phase field models, namely, the Allen-Cahn and Cahn-Hilliard equations, and their related curvature-driven geometric problems, namely, the mean curvature flow and the Hele-Shaw flow. We derive two discrete spectrum estimates, which play an important role in proving the sharper error estimates which only depend on a …

Domain Decomposition Methods For Discontinuous Galerkin Approximations Of Elliptic Problems, Craig Dwain Collins

Doctoral Dissertations

The application of the techniques of domain decomposition to construct effective preconditioners for systems generated by standard methods such as finite difference or finite element methods has been well-researched in the past few decades. However, results concerning the application of these techniques to systems created by the discontinuous Galerkin method (DG) are much more rare.

This dissertation represents the effort to extend the study of two-level nonoverlapping and overlapping additive Schwarz methods for DG discretizations of second- and fourth-order elliptic partial differential equations. In particular, the general Schwarz framework is used to find theoretical bounds for the condition numbers of …

Numerical Analysis Of Convex Splitting Schemes For Cahn-Hilliard And Coupled Cahn-Hilliard-Fluid-Flow Equations, Amanda Emily Diegel

Doctoral Dissertations

This dissertation investigates numerical schemes for the Cahn-Hilliard equation and the Cahn-Hilliard equation coupled with a Darcy-Stokes flow. Considered independently, the Cahn-Hilliard equation is a model for spinodal decomposition and domain coarsening. When coupled with a Darcy-Stokes flow, the resulting system describes the flow of a very viscous block copolymer fluid. Challenges in creating numerical schemes for these equations arise due to the nonlinear nature and high derivative order of the Cahn-Hilliard equation. Further challenges arise during the coupling process as the coupling terms tend to be nonlinear as well. The numerical schemes presented herein preserve the energy dissipative structure …

Impacts Of Climate Change On The Evolution Of The Electrical Grid, Melissa Ree Allen

Doctoral Dissertations

Maintaining interdependent infrastructures exposed to a changing climate requires understanding 1) the local impact on power assets; 2) how the infrastructure will evolve as the demand for infrastructure changes location and volume and; 3) what vulnerabilities are introduced by these changing infrastructure topologies. This dissertation attempts to develop a methodology that will a) downscale the climate direct effect on the infrastructure; b) allow population to redistribute in response to increasing extreme events that will increase under climate impacts; and c) project new distributions of electricity demand in the mid-21^st century.

The research was structured in three parts. The first …

Numerical Methods And Algorithms For High Frequency Wave Scattering Problems In Homogeneous And Random Media, Cody Samuel Lorton

Doctoral Dissertations

This dissertation consists of four integral parts with a unified objective of developing efficient numerical methods for high frequency time-harmonic wave equations defined on both homogeneous and random media. The first part investigates the generalized weak coercivity of the acoustic Helmholtz, elastic Helmholtz, and time-harmonic Maxwell wave operators. We prove that such a weak coercivity holds for these wave operators on a class of more general domains called generalized star-shape domains. As a by-product, solution estimates for the corresponding Helmholtz-type problems are obtained.

The second part of the dissertation develops an absolutely stable (i.e. stable in all mesh regimes) interior …

Finite Difference And Discontinuous Galerkin Finite Element Methods For Fully Nonlinear Second Order Partial Differential Equations, Thomas Lee Lewis

Doctoral Dissertations

The dissertation focuses on numerically approximating viscosity solutions to second order fully nonlinear partial differential equations (PDEs). The primary goals of the dissertation are to develop, analyze, and implement a finite difference (FD) framework, a local discontinuous Galerkin (LDG) framework, and an interior penalty discontinuous Galerkin (IPDG) framework for directly approximating viscosity solutions of fully nonlinear second order elliptic PDE problems with Dirichlet boundary conditions. The developed frameworks are also extended to fully nonlinear second order parabolic PDEs. All of the proposed direct methods are tested using Monge-Ampere problems and Hamilton-Jacobi-Bellman (HJB) problems. Due to the significance of HJB problems …