Numerical Analysis and Scientific Computing Commons^™

A Simulation Assessment Of The 'Law Of One Price', Caleb Wilkins

Computational and Data Sciences (MS) Theses

The ‘law of one price’ is an appealing notion regarding pricing of tradeable commodities that are priced in different currencies. It states that the prices of the same good in different markets should be equal after adjustment for exchange rates and that equality should persist through exchange rate fluctuations.

My research simulates the market conditions that should precipitate the ‘law of one price.’ Data was obtained from the simulated trade between algorithmic artificial intelligence agents that operated under induced boundedly rational market behaviors. Trade took place in two initially separate markets, a high-price market with a higher equilibrium price and …

Information Theory Analysis Of Water Vapor Stable Isotopes From The Sail Campaign, Matthew John Rybecky

Earth and Planetary Sciences ETDs

Understanding the processes that control water vapor isotopic composition in mountain environ- ments is essential for interpreting isotope records and predicting water resource responses to cli- mate change. This thesis applies information theory to continuous, high-resolution water vapor stable isotope measurements from the Surface Atmosphere Integrated Field Laboratory (SAIL) campaign in the East River watershed of Colorado’s Upper Gunnison Basin, spanning the winter- to-spring transition of 2022–2023. The analysis employs Shannon entropy, mutual information, transfer entropy, and joint transfer en- tropy (JTE) to quantify how environmental variables, including surface meteorology, radiation, tur- bulent fluxes, and ERA5 reanalysis products, transfer information …

Low-Complexity Structured Neural Networks And Their Usage In Image And Signal Processing, Adam Kuzmicki

Doctoral Dissertations and Master's Theses

Conventional neural networks face significant challenges due to high computational costs, large parameter counts, and reliance on backpropagation, which restricts their application in resource-constrained and real-time settings. To address these challenges, this thesis proposes three structured neural network (NN) architectures grounded in the theories of sparse and self-contained factorizations of transforms, with applications to image compression, reconstruction, classification, encryption, and also adaptive wideband multi-beam beamforming. The first neural network architecture, named DCTrix-Net, replaces conventional spatial con- volution with highly sparse factorization of the discrete Cosine transform (DCT) complemented by Toeplitz-structured weight initialization, achieving at least 97% FLOP reduction over CNNs, …

Detecting And Repairing Conflicting Constraints In Co-Trained Physics-Informed Neural Networks For Composite Curing Processes, Cooper J. Evans

Dissertations, Master's Theses and Master's Reports

Composite materials have become a critical component of modern manufacturing, especially in the automotive and aerospace industries. The curing process for these composites has been modeled using a variety of partial differential equations representing the heat transfer and composite curing kinetics. Optimizing the applied temperature profile is critical for maximizing the efficiency and capacity of composite part manufacturers. Constraints must be placed on the inputs and outputs of the model, including but not limited to, the applied temperature profile, part temperature, and final degree of cure. Conflicting sets of constraints are easy to unknowingly impose due to the highly coupled …

A 1d Symmetric Interior Penalty Discontinuous Galerkin Solver In Rust, William Aey

Williams Honors College, Honors Research Projects

This honors project will build a 1D Symmetric Interior Discontinuous Galerkin (SIPDG) solver in Rust for Stum-Liouville type problems such as the Poisson equation, with Robin, Dirichlet, and Neumann boundary conditions. The work will cover the full pipeline: starting from the strong form of the PDE, deriving the DG weak form, implementing element and interface operators, and assembling or apply the discrete operator. Rust's safety and concurrency (e.g, via Rayon) will be used to explore serial and parallel performance. A test-driven development approach will be used to maintain a strong suite of tests. The project will result in a documented …

Performance Of Numerical Methods Applied To The Black–Scholes Model, Scott Cameron Williams

UNF Graduate Theses and Dissertations

We compare five numerical approaches for approximating solutions to the Black–Scholes partial differential equation for pricing European call options: FTCS, BTCS, Crank– Nicolson, Monte Carlo simulation, and a physics–informed neural network (PINN). These methods span finite difference techniques, probabilistic simulation, and machine learning. Performance is evaluated based on computational efficiency and accuracy relative to the analytical Black–Scholes solution.

Among the methods, Crank–Nicolson and the PINN demonstrated the strongest overall performance. Crank–Nicolson achieved the highest accuracy but exhibited increased runtime as the number of underlying stock price grid points grew. In contrast, the PINN produced slightly less accurate results but with …

Stability Analysis Of Thermohaline Convection With A Time-Varying Shear Flow Using The Lyapunov Method, Kalin Kochnev

Honors Scholar Theses

This work applies the Lyapunov method to identify instabilities and compute the growth rate of a linear time-varying system. The linear system studied describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. A time-dependent weighting matrix is employed to construct a Lyapunov function candidate. The resulting linear matrix inequalities are discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted by the Lyapunov method or Floquet theory, used for comparison, will converge to the same value obtained from numerical simulations. Furthermore, the …

Universal Systems Simulation Via Constraint Hypergraphs With Applications To Digital Twins, John Morris

All Dissertations

The characterization of systems encompasses a variety of modeling frameworks designed to capture specific behaviors and components of various system domains. Whatever the framework, the core elements of a system representation are the information of the system and a description of how that information is related. The relations in deterministic systems are functions, which, when composed to form executable processes, can be used to simulate system data. A declarative modeling framework is one that encodes mechanisms for preparing these simulations within the model structure, allowing an external agent to form the execution processes required for a given context. To date, …

Modeling Synaptic Dysfunction As Neural Contagion: A Graph-Based Sedr Framework For Simulating Signal Spread, Michelle Marfo, Dr. Padmanabhan Seshaiyer, Alonso Ogueda-Oliva

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Cancer Cell Growth Predictions Based On Classical Mathematical Models With Physics-Informed Neural Network, Widodo Samyono

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Quantification Of Parameters To Predict The Rupture Of Intracranial Saccular Aneurysms Using Physics Informed Neural Networks, Alonso Gabriel Ogueda, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Meshless Discrete Velocity Boltzmann Model For Porous Media Flow, Amandine Maidenberg

Doctoral Dissertations and Master's Theses

This dissertation explores the combination of two sophisticated techniques for addressing computational fluid dynamics: the discrete velocity Boltzmann equation (DVBE) and the localized collocation meshless model with upwinding (U-LCMM). The DVBE is a high-level model that describes the foundations of transport phenomena by addressing the microscale motions of particles themselves and the effect of their aggregate behaviors on continuum principles. This equation integrates multiple scales of phenomena; while it can be used for fluid flow at Navier-Stokes scales, it can also resolve fine features that can only be described at the molecular level. This type of model is necessary for …

Algebraic Multigrid Methods For Nonsymmetric And Indefinite Problems: Theory And Applications, Ahsan Ali

Mathematics & Statistics ETDs

Algebraic multigrid (AMG) is a well-established and highly efficient solver for symmetric positive definite (SPD) systems arising from elliptic and parabolic PDEs, while nonsymmetric systems from hyperbolic PDEs remain a significant challenge. This dissertation develops AMG methods and theory for nonsymmetric problems. First, we develop a novel approach combining mode constraints from energy-minimization AMG with local approximations of ideal restriction in $\ell$AIR, resulting in constrained $\ell$AIR (C$\ell$AIR), which demonstrates scalable convergence across advective and diffusive problems. Second, we extend optimal AMG theory by deriving spectral radius estimates for the two-grid error transfer operator using matrix-induced orthogonality, enabling convergence predictions for …

Filling Gaps In Scientific Data Sets Using Physics Informed Neural Networks: A Case Study In Velocity Fields, Ellen Saunders

Master's Theses

Gaps in scientific data sets are a persistent issue for researchers in a variety of fields, and while nothing makes up for missing out on real data, well-simulated synthetic data can be a useful tool. In the world of image processing, machine learning techniques have become quite sophisticated at taking an image with a missing component and filling in that space with something believable. The aim of this thesis is to take machine learning techniques similar to what gets used in image processing and repurpose them to infill gaps in scientific data sets in a realistic manner. This thesis compares …

Using Gaussian Process Regression To Learn Thermodynamic Equations Of State With Uncertainty Quantification, Austen T. Lee

Chemical Engineering Undergraduate Honors Theses

This study investigates the use of derivative-informed Gaussian Process (GP) models to estimate thermodynamic behavior across temperature and density by building a Helmholtz-based equation of state. Argon, a stable monatomic gas, was chosen as a case study within the vapor region. The GP model was trained using values of experimentally measurable properties found by taking first and second derivatives of the original potential function. Results show that while the GP model offered uncertainty quantification and informed thermodynamic behavior, it predicted values that deviated from the ground truth depending on the property. The model exhibited high confidence in regions with substantial …

Physics Embedded Neural Network: A Novel Data-Free Numerical Method For Solving Computational Physics Problems, Pawan Gaire

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

A novel approach for solving partial differential equations (PDEs) using neural networks for scientific computing is introduced. The proposed approach, referred to as physics-embedded neural network (PENN), features a unique architecture that incorporates the PDE and boundary conditions information directly within the final fully-connected layer of the feed-forward neural network (NN). The key aspect of PENN is the parallel numerical embedding of a differential equation associated with physical problems within the activation function of the network’s final layer. This integration leads to a new class of computational solvers competitive with classical methods like the Finite Element Method (FEM) and capable …

Towards Advancing Streamflow And Peak Flow Prediction With Machine Learning: Identifying Infrastructure At Risk, Sudan Pokharel

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Due to climate change and its impact, the need for adaptive strategies for natural disaster mitigation and resource management has never been more urgent. Central to this is water resource management, which is essential for sustainable human activities, ecological balance, and the mitigation of natural hazards like floods. Streamflow is a crucial element of water resource management and plays a vital role in planning and building water infrastructure, implementing emergency response plans, supporting flood mitigation initiatives, and regulating agricultural and industrial use. However, accurate prediction of streamflow still remains a challenge due to the complex non-linear and non-stationary interaction between …

Novel Generative And Language Model Architectures With Applications, Edison Mucllari

Theses and Dissertations--Mathematics

This dissertation investigates novel architectures to address fundamental challenges in machine learning, particularly focusing on transformer models, recurrent neural networks, GAN and continual learning and their applications in natural language processing and computer vision. We propose the Neumann-Cayley Gated Recurrent Unit (NC-GRU), which leverages a Neumann series-based Scaled Cayley transformation to maintain orthogonal weight matrices, effectively mitigating exploding gradients problems while improving long-term memory retention across prediction tasks. We demonstrate the practical applications of NC-GRU by implementing our proposed architecture into an autoencoder to derive neural molecular fingerprints. Building upon these advancements, we turn our attention to the transformer architecture, …

Scalable Parallel-In-Time Integration For Equations Of Motion, Nathan W. Chapman

All Master's Theses

Physical simulations always need to balance accuracy and run-time. This work implements the Parareal Algorithm using graphics processing units across a distributed system to accurately simulate time-dependent physics while attempting to minimize runtime. Data-transfer latency is identified as the primary bottleneck, for which mitigation methods are provided. Benchmarks comparing single-GPU and distributed implementations on a spectrum of coarse and fine discretizations are analyzed.

The Presence Of Outer Giant Planets And Their Role In Inner Planet Formation With And Without Their Influence, Mateo E. Guerra Toro

Graduate Theses/Dissertations

We performed dynamical simulations of the giant impact phase of planet formation to investigate the formation of inner terrestrial planets under the influence of 4 solar system-like outer giant planets. We developed a new code using the N-body simulation suite REBOUND and REBOUNDx (Rein et al. (2019) and Tamayo et al. (2019)) to simulate 2 stages of planetary formation: a residual gaseous protoplanetary disk phase and subsequent dynamical evolution after the disk photoevaporates. The initial conditions for the inner planetary embryos were taken by Morrison et al. (2020) based on a range of solid surface densities that produced Super-Earth terrestrial …

Improving Infectious Disease Predictions Through The Use Of Metapopulation Sir Modeling And Graph Convolutional Neural Networks, Petr Kisselev, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

An Introduction To The Time-Independent Schrödinger Equation And Methods To Solve It, Vu Giang, Alex Gnech

OUR Journal: ODU Undergraduate Research Journal

The Time-Independent Schrödinger Equation is a linear elliptic PDE that describes quantum-mechanical systems. Its significance in the science of submicroscopic phenomena, particularly quantum mechanics, is as central as Newton’s laws of motion are to classical mechanics. This study uses various methods, including novel neural networks and finite difference schemes, to solve the one-dimensional two-body equation.

Certifying Stability In Runge-Kutta Schemes: Algebraic Conditions And Semidefinite Programming, Austin Juhl

Dissertations

Numerical stability is a critical property for a time-integration scheme. In the context of Runge-Kutta methods applied to stiff differential equations, A-stability is one of the most basic and practically important notions of stability. Dating back to the work of Dahlquist, it has been known that A-stability is equivalent to the Runge-Kutta stability function satisfying a particular convex feasibility problem. Specifically, up to a transformation, the stability function lies in the convex cone of positive functions. In recent years, sum-of-squares optimization and semidefinite programming have become valuable tools in developing rigorous certificates of stability in dynamical systems. Therefore, it is …

Parallel Multigrid In Time For Chaotic Dynamical Systems, David Alan Vargas

Mathematics & Statistics ETDs

Despite the fact that Parallel-in-Time (PinT) methods are predicted to become necessary to fully utilize next-generation exa- and zettascale machines, there are currently no known practical methods which scale well with the length of the time-domain for chaotic problems, due to exponential dependence of the condition number on the fastest chaotic timescale. I present modifications to the coarse-grid equations along with a novel rediscretization approach which together greatly improve convergence of the multigrid reduction in time (MGRIT) algorithm and allow the first known PinT speedup for a chaotic PDE. The novel Local Shadowing Relaxation (LSR) is presented as an alternative …

Analysis And Computation Of Constrained Sparse Coding On Emerging Non-Von Neumann Devices, Kyle Henke

Mathematics & Statistics ETDs

This dissertation seeks to understand how different formulations of the neurally inspired Locally Competitive Algorithm (LCA) represent and solve optimization problems. By studying these networks mathematically through the lens of dynamical and gradient systems, the goal is to discern how neural computations converge and link this knowledge to theoretical neuroscience and artificial intelligence (AI). Both classical computers and advanced emerging hardware are employed in this study. The contributions of this work include:

1. Theoretical Work: A comprehensive convergence analysis for networks using both generic Rectified Linear Unit (ReLU) and Rectified Sigmoid activation functions. Exploration of techniques to address the binary …

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

Dissertations and Theses (Open Access)

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

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

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

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

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

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

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

converging beam irradiator apparatus was fit with a multitude …

Mathematically Rigorous Deep Learning Paradigms For Data-Driven Scientific Modeling, Owen Nicholas Davis

Mathematics & Statistics ETDs

This dissertation explores the crucial role of data-driven modeling in science and engineering, with a focus on developing surrogate models to accelerate large-scale computational tasks, aiding in both outer-loop functions like uncertainty quantification and expensive inner-loop tasks within broader computational frameworks. Challenges arise with increased problem dimension and sparse, noisy training data, particularly significant when constructing surrogates for very expensive computational models where acquiring sufficient high-fidelity training data is unfeasible. In such scenarios, training surrogates from an ensemble of multifidelity information sources of varying accuracy and cost becomes essential. We emphasize neural network-based modeling paradigms, which are flexible in integrating …

Quantitative Verification For Massive Linear Systems, Qing Liu

School of Computing: Dissertations, Theses, and Student Research

The verification of linear systems has been an active area of research for decades. Reachability analysis is a key component in verification problems. It involves computing the system’s reachable set, the set of reachable states in the state space from a given set of initial states. Most verification methods primarily focus on qualitative verification, which answers whether or not a system may violate specified safety conditions. This paper extends this qualitative verification to quantitative verification by introducing a novel approach, employing probabilistic stars (Probstars) to compute reachable sets, which augment traditional star sets by integrating Gaussian-distributed random variables with …

The Computational Search For Unidentified Central Configurations Of The Newtonian N-Body Problem, Hannah G. Havel

CURE Proceedings

The N-body problem is a field of study in mathematics and physics that involves predicting the motion of particles moving under their mutual gravitational attraction. It is vital in celestial mechanics, such as planning collision-free satellite orbit trajectories. When beginning to understand the N-body problem, we can start by looking at equal masses of these particles or celestial bodies. As particles move, their position and velocity change, both energy and angular momentum are conserved. Sets of constant energy and angular momentum, known as integral manifolds, are higher-dimensional figures that represent constraints of movement to a system. Integral manifolds are described …

Adaptive Neh With Constrained Nearest Neighbor Subtours For The Electric Vehicle Routing Problem With Time Windows, Andrew Struthers

All Master's Theses

The development of electric vehicles is currently considered one of the most innovative areas in manufacturing. Largely driven by the desire to reduce greenhouse emissions, electric vehicles are seen as a viable alternative to internal combustion engine cars. Starting from consumer cars, a dedicated effort is being made to translate this into commercial vehicles for freight and delivery. This research introduces a novel adaptive Nawaz, Enscore, Ham (NEH) algorithm with constrained nearest neighbor subtour (NEH-NN). This algorithm is tested on the standard benchmark problems in literature and used as a seed solution for the Genetic Algorithm (GA). The performance and …