Physical Sciences and Mathematics Commons^™

Probabilistic And Data-Driven Methods For Numerical Pdes, Johannes Krotz

Doctoral Dissertations

This dissertation consists of three integral self-contained parts. The first part develops a novel Monte Carlo algorithm, called the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS), to efficiently generate the nodes of a high-quality mesh for the calculation of flow and the associated transport of chemical species in low-permeability fractured rock, such as shale and granite. A good mesh balances accuracy requirements with a reasonable computational cost, i.e., it is generated efficiently, dense where necessary for accuracy, and contains no cells that cause instabilities or blown-up errors. Quality bounds for meshes generated through nMAPS are proven, and its efficiency is demonstrated …

Data-Driven Model Reduction Strategies For Dynamical Systems, Talha Ahmed

Doctoral Dissertations

Many physically occurring phenomena are nonlinear in nature and can be understood through dynamical systems theory which describes how the state of the particular system evolves in time. However, it is generally cumbersome to analyze these processes in depth because of the nonlinearities in the mathematical model or large sets of equations. Model reduction strategies are employed for such nonlinear processes to reduce the model dimensionality and approximate the full model dynamics. In this study, we focus on data driven model reduction strategies for various biological systems where only observable data is available and illustrate their efficacy.

Our first work …

Mathematical Modeling And Numerical Approximations Of Combustion Instability Frequencies And Growth Rates, Harvey B. Ring Iii

Doctoral Dissertations

This dissertation presents a mathematical model and numerical simulations to determine the resonant frequencies and their associated growth rates for longitudinal modes in a combustion system similar to that found in a rocket engine. The mathematical model, which is applicable to a two-duct system with a thin flame between the two ducts, each of which having constant area and properties, considers the case of axial mean velocity and uses a vibrating wall at the inlet to select the frequency so that all modes may be found. The model is applied to the acoustics equations describing pressure and velocity fluctuations, derived …

Multi-Objective Radiological Analysis In Real Environments, David Raji

Doctoral Dissertations

Designing systems to solve problems arising in real-world radiological scenarios is a highly challenging task due to the contextual complexities that arise. Among these are emergency response, environmental exploration, and radiological threat detection. An approach to handling problems for these applications with explicitly multi-objective formulations is advanced. This is brought into focus with investigation of a number of case studies in both natural and urban environments. These include node placement in and path planning through radioactivity-contaminated areas, radiation detection sensor network measurement update sensitivity, control schemes for multi-robot radioactive exploration in unknown environments, and adversarial analysis for an urban nuclear …

Bridging Biological Systems With Social Behavior, Conservation, Decision Making, And Well-Being Through Hybrid Mathematical Modeling, Maggie Renee Sullens

Doctoral Dissertations

Mathematical modeling can achieve otherwise inaccessible insights into bio-logical questions. We use ODE (ordinary differential equations) and Game Theory models to demonstrate the breadth and power of these models by studying three very different biological questions, involving socio-behavioral and socio-economic systems, conservation biology, policy and decision making, and organismal homeostasis.

We adapt techniques from Susceptible-Infected-Recovered (SIR) epidemiological models to examine the mental well-being of a community facing the collapse of the industry on which it’s economically dependent. We consider the case study of a fishing community facing the extinction of its primary harvest species. Using an ODE framework with a …

Generative Adversarial Game With Tailored Quantum Feature Maps For Enhanced Classification, Anais Sandra Nguemto Guiawa

Doctoral Dissertations

In the burgeoning field of quantum machine learning, the fusion of quantum computing and machine learning methodologies has sparked immense interest, particularly with the emergence of noisy intermediate-scale quantum (NISQ) devices. These devices hold the promise of achieving quantum advantage, but they grapple with limitations like constrained qubit counts, limited connectivity, operational noise, and a restricted set of operations. These challenges necessitate a strategic and deliberate approach to crafting effective quantum machine learning algorithms.

This dissertation revolves around an exploration of these challenges, presenting innovative strategies that tailor quantum algorithms and processes to seamlessly integrate with commercial quantum platforms. A …

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 …

Reduced Order Modeling And Analysis Of Cardiac Chaos, Tuhin Subhra Das

Doctoral Dissertations

Numerous physiological processes are functioning at the level of cells, tissues and organs in the human body, out of which some are oscillatory and some are non-oscillatory. Networks of coupled oscillators are widely studied in the modeling of oscillatory or rhythmical physiological processes. Phase-isostable reduction is an emerging model reduction strategy that can be used to accurately replicate nonlinear behaviors in dynamical systems for which standard phase reduction techniques fail. We apply strategies of phase reduction, or isostable reductions in biologically motivated problems like eliminating cardiac alternans to alleviate arrhythmia by applying energy-optimal, non-feedback type control solutions.

Cardiac fibrillation caused …

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 …

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 …

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 …

Birds And Bioenergy: A Modeling Framework For Managed Landscapes At Multiple Spatial Scales, Jasmine Asha Kreig

Doctoral Dissertations

This dissertation examines the design and management of bioenergy landscapes at multiple spatial scales given numerous objectives. Objectives include biodiversity outcomes, biomass feedstock yields, and economic value.

Our study examined biodiversity metrics for 25 avian species in Iowa, including subsets of these species related to ecosystem services. We used our species distribution model (SDM) framework to determine the importance of predictors related to switchgrass production on species richness. We found that distance to water, mean diurnal temperature range, and herbicide application rate were the three most important predictors of biodiversity overall. We found that 76% of species responded positively to …

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 …

Machine Learning With Topological Data Analysis, Ephraim Robert Love

Doctoral Dissertations

Topological Data Analysis (TDA) is a relatively new focus in the fields of statistics and machine learning. Methods of exploiting the geometry of data, such as clustering, have proven theoretically and empirically invaluable. TDA provides a general framework within which to study topological invariants (shapes) of data, which are more robust to noise and can recover information on higher dimensional features than immediately apparent in the data. A common tool for conducting TDA is persistence homology, which measures the significance of these invariants. Persistence homology has prominent realizations in methods of data visualization, statistics and machine learning. Extending ML with …

Spatio-Temporal Modeling Of Crime In Chicago, Illinois, Shelby Scott

Doctoral Dissertations

Gun crime is a major public health concern in the United States. In Chicago, Illinois, gun crime incurs a significant cost of life along with monetary costs and community unrest. Due to past legislation, there is limited research applying quantitative methods to gun crime in Chicago. The overall purpose of this work is to create a cellular automata model to observe and project the epidemic spread of gun crime in Chicago. To create that model, t-test analyses of temporal patterns, a Bayesian point process model, a negative binomial Bayesian subset selection, and a k-selection algorithm are used. The cellular automata …

Mathematical Models In Medicine: The Immune Response Of Celiac Disease And The Environmental Transmission Of Clostridioides Difficile In Healthcare Settings, Cara Jill Sulyok

Doctoral Dissertations

Mathematical modeling is a useful technique to describe dynamics happening within events and allows one to address questions and test hypotheses that may be not be feasible to study in reality. This work uses mathematical models to describe two such phenomena, one relating to immunology and the other to the spread of infectious diseases.

Celiac disease is a hereditary autoimmune disease that affects approximately 1 in 133 Americans. It is caused by a reaction to the protein gluten found in wheat, rye, and barley. After ingesting gluten, a patient with celiac disease may experience a range of unpleasant symptoms while …

Data Driven Models Of Hemlock Woolly Adelgid Impacts And Biological Control, Hannah M. Thompson

Doctoral Dissertations

We present two models of the Adelges tsugae, the hemlock woolly adelgid, an invasive insect pest of Tsuga canadensis, eastern hemlock, in the eastern United States. An A. tsugae infestation often results in the death of T. canadensis within years, and has caused significant changes to hemlock forests. We construct two models composed of systems of ordinary differential equations with time dependent parameters to represent seasonality. The first model captures the coupled cycles in T. canadensis health and A. tsugae density. We use field data from Virginia to develop the model and to perform parameter estimation. The mechanisms …

Nonparametric Bayesian Deep Learning For Scientific Data Analysis, Devanshu Agrawal

Doctoral Dissertations

Deep learning (DL) has emerged as the leading paradigm for predictive modeling in a variety of domains, especially those involving large volumes of high-dimensional spatio-temporal data such as images and text. With the rise of big data in scientific and engineering problems, there is now considerable interest in the research and development of DL for scientific applications. The scientific domain, however, poses unique challenges for DL, including special emphasis on interpretability and robustness. In particular, a priority of the Department of Energy (DOE) is the research and development of probabilistic ML methods that are robust to overfitting and offer reliable …

Root Stage Distributions And Their Importance In Plant-Soil Feedback Models, Tyler Poppenwimer

Doctoral Dissertations

Roots are fundamental to PSFs, being a key mediator of these feedbacks by interacting with and affecting the soil environment and soil microbial communities. However, most PSF models aggregate roots into a homogeneous component or only implicitly simulate roots via functions. Roots are not homogeneous and root traits (nutrient and water uptake, turnover rate, respiration rate, mycorrhizal colonization, etc.) vary with age, branch order, and diameter. Trait differences among a plant’s roots lead to variation in root function and roots can be disaggregated according to their function. The impact on plant growth and resource cycling of changes in the distribution …

Applications With Discrete And Continuous Models: Harvesting And Contact Tracing, Danielle L. Burton

Doctoral Dissertations

Harvest plays an important role in management decisions, from fisheries to pest control. Discrete models enable us to explore the importance of timing of management decisions including the order of events of particular actions. We derive novel mechanistic models featuring explicit within season harvest timing and level. Our models feature explicit discrete density independent birth pulses, continuous density dependent mortality, and density independent harvest level at a within season harvest time. We explore optimization of within-season harvest level and timing through optimal control of these population models. With a fixed harvest level, harvest timing is taken as the control. Then …

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. …

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 …

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 …

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 …

Assessing The Economic Tradeoffs Between Prevention And Suppression Of Forest Fires, Elizabeth Trulia Heines

Doctoral Dissertations

The number of large-scale, high-severity forest fires occurring in the United States is increasing, as is the cost to suppress these fires. These trends have prompted investigations into alternative fuels methods to help prevent these large wildfires. One of the key challenges in studying the costs and benefits of forest fire prevention management is the incorporation of risk and uncertainty surrounding management decisions. We use a technique developed by William Reed to incorporate the stochasticity of the time of a forest fire into our optimal control problems. The goal of these problems is to determine the optimal fire prevention management …

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 …

Analysis Of A Market For Tradable Credits, Policy Uncertainty Effects On Investment Decisions, And The Potential To Supply A Renewable Aviation Fuel Industry With An Experimental Industrial Oilseed, Evan Lawrence Markel

Doctoral Dissertations

This research is aligned with identifying barriers throughout the alternative jet-fuel supply chain. Prices are analyzed in the market for tradable credits known as renewable identification numbers (RINs). The RIN market is a key policy instrument used in the implementation of the renewable fuel standard (RFS). The program is highly complex and drivers of RIN price are not always clear. RIN prices also exhibit multiple regimes where the price of nested RINs converge. Therefore, a smooth transition autoregressive model is employed to examine drivers of RIN price and to identify drivers of price regime change. Through research in the RIN …

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 …