Numerical Analysis and Computation Commons^™

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, …

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 …

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 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.

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 …

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 …

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 …

Physics-Informed Neural Networks For Agent-Based Epidemiological Model Calibration, Alvan C. Arulandu, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Domain Decomposition Methods For Linear And Non-Linear Elliptic Problems, Tadanaga Takahashi

Dissertations

The primary purpose of this dissertation is to expand upon the circle of domain decomposition methods (DDM) which are algorithms that reformulate a boundary value problem in terms of multiple localized problems on subdomains. The first project involves expanding upon DDMs in a relatively mature field: the Helmholtz equation for wave scattering applications. The proposed method is an adaptation of a continuous cross-point Finite Element Non-overlapping DDM algorithm. The usual unbounded computational domain is truncated and then the near-field wave pattern is solved with a parallelized finite element method. Several improvements over the standard transmission operator are discussed in this …

Covid-19 In Casinos: Analysis Of Covid-19 Contamination And Spread With Economic Impact Assessment, Anastasia (Stasi) D. Baran, Jason D. Fiege

International Conference on Gambling & Risk Taking

Abstract:

The COVID-19 pandemic caused tremendous disruption for casinos, with the virus causing various lengths of shutdowns, capacity restrictions, and social distancing strategies such as machine removals or section closures. Although most of the world has now eased off these measures, it is important to review lessons learned to understand, and better prepare for similar circumstances in the future. We present Monte Carlo slot floor simulation software customized to simulate players spreading COVID-19 on the slot floor. We simulate the amount of touch surface contamination; the number of potential surface contact exposure events per day, and a proximity exposures statistic …

Head And Neck Tumor Histopathological Image Representation With Pre- Trained Convolutional Neural Network And Vision Transformer, Ranny Rahaningrum Herdiantoputri, Daisuke Komura, Tohru Ikeda, Shumpei Ishikawa

Journal of Dentistry Indonesia

Image representation via machine learning is an approach to quantitatively represent histopathological images of head and neck tumors for future applications of artificial intelligence-assisted pathological diagnosis systems. Objective: This study compares image representations produced by a pre-trained convolutional neural network (VGG16) to those produced by a vision transformer (ViT-L/14) in terms of the classification performance of head and neck tumors. Methods: W hole-slide images of five oral t umor categories (n = 319 cases) were analyzed. Image patches were created from manually annotated regions at 4096, 2048, and 1024 pixels and rescaled to 256 pixels. Image representations were …

A Graphical User Interface Using Spatiotemporal Interpolation To Determine Fine Particulate Matter Values In The United States, Kelly M. Entrekin

Honors College Theses

Fine particulate matter or PM2.5 can be described as a pollution particle that has a diameter of 2.5 micrometers or smaller. These pollution particle values are measured by monitoring sites installed across the United States throughout the year. While these values are helpful, a lot of areas are not accounted for as scientists are not able to measure all of the United States. Some of these unmeasured regions could be reaching high PM2.5 values over time without being aware of it. These high values can be dangerous by causing or worsening health conditions, such as cardiovascular and lung diseases. Within …

Practical Implementation Of The Immersed Interface Method With Triangular Meshes For 3d Rigid Solids In A Fluid Flow, Norah Hakami

Mathematics Theses and Dissertations

When employing the immersed interface method (IIM) to simulate a fluid flow around a moving rigid object, the immersed object can be replaced by a virtual fluid enclosed by singular forces on the interface between the real and virtual fluids. These forces represent the impact of the rigid motion on the fluid flow and cause jump discontinuities across the interface in the whole flow field. Then, the IIM resolves the fluid flow on a fixed computational domain by directly incorporating the jump conditions across the interface into numerical schemes. Previous development of the method is limited to simple smooth boundaries. …

Chatgpt As Metamorphosis Designer For The Future Of Artificial Intelligence (Ai): A Conceptual Investigation, Amarjit Kumar Singh (Library Assistant), Dr. Pankaj Mathur (Deputy Librarian)

Library Philosophy and Practice (e-journal)

Abstract

Purpose: The purpose of this research paper is to explore ChatGPT’s potential as an innovative designer tool for the future development of artificial intelligence. Specifically, this conceptual investigation aims to analyze ChatGPT’s capabilities as a tool for designing and developing near about human intelligent systems for futuristic used and developed in the field of Artificial Intelligence (AI). Also with the helps of this paper, researchers are analyzed the strengths and weaknesses of ChatGPT as a tool, and identify possible areas for improvement in its development and implementation. This investigation focused on the various features and functions of ChatGPT that …

Dynamic Function Learning Through Control Of Ensemble Systems, Wei Zhang, Vignesh Narayanan, Jr-Shin Li

Publications

Learning tasks involving function approximation are preva- lent in numerous domains of science and engineering. The underlying idea is to design a learning algorithm that gener- ates a sequence of functions converging to the desired target function with arbitrary accuracy by using the available data samples. In this paper, we present a novel interpretation of iterative function learning through the lens of ensemble dy- namical systems, with an emphasis on establishing the equiv- alence between convergence of function learning algorithms and asymptotic behavior of ensemble systems. In particular, given a set of observation data in a function learning task, we …

Physics-Informed Neural Networks For Informed Vaccine Distribution In Heterogeneously Mixed Populations, Alvan Arulandu, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova

Journal of Humanistic Mathematics

This article describes the use of computer software to optimize the design of an academic hat and an ice cream cone!

Computational Models To Detect Radiation In Urban Environments: An Application Of Signal Processing Techniques And Neural Networks To Radiation Data Analysis, Jose Nicolas Gachancipa

Beyond: Undergraduate Research Journal

Radioactive sources, such as uranium-235, are nuclides that emit ionizing radiation, and which can be used to build nuclear weapons. In public areas, the presence of a radioactive nuclide can present a risk to the population, and therefore, it is imperative that threats are identified by radiological search and response teams in a timely and effective manner. In urban environments, such as densely populated cities, radioactive sources may be more difficult to detect, since background radiation produced by surrounding objects and structures (e.g., buildings, cars) can hinder the effective detection of unnatural radioactive material. This article presents a computational model …

A Bidirectional Formulation For Walk On Spheres, Yang Qi

Dartmouth College Master’s Theses

Poisson’s equations and Laplace’s equations are important linear partial differential equations (PDEs)
widely used in many applications. Conventional methods for solving PDEs numerically often need to
discretize the space first, making them less efficient for complex shapes. The random walk on spheres
method (WoS) is a grid-free Monte-Carlo method for solving PDEs that does not need to discrete the
space. We draw analogies between WoS and classical rendering algorithms, and find that the WoS
algorithm is conceptually identical to forward path tracing.
We show that solving the Poisson’s equation is equivalent to solving the Green’s function for every
pair of …

Understanding The Influence Of Perceptual Noise On Visual Flanker Effects Through Bayesian Model Fitting, Jordan Deakin, Dietmar Heinke

MODVIS Workshop

No abstract provided.

A Novel Method For Sensitivity Analysis Of Time-Averaged Chaotic System Solutions, Christian A. Spencer-Coker

Theses and Dissertations

The direct and adjoint methods are to linearize the time-averaged solution of bounded dynamical systems about one or more design parameters. Hence, such methods are one way to obtain the gradient necessary in locally optimizing a dynamical system’s time-averaged behavior over those design parameters. However, when analyzing nonlinear systems whose solutions exhibit chaos, standard direct and adjoint sensitivity methods yield meaningless results due to time-local instability of the system. The present work proposes a new method of solving the direct and adjoint linear systems in time, then tests that method’s ability to solve instances of the Lorenz system that exhibit …