Numerical Analysis and Computation Commons^™

Computation Of Moving Interface Flows In Biophysical Applications, Mohammad Murshed

Masters Theses and Doctoral Dissertations

This dissertation is concerned with the modeling, simulation, and analysis of moving interface problems involving viscous fluids and solid structures. The main computational technique employed in this work is the immersed boundary method, a widely known numerical method for fluid-structure interaction (FSI). In this technique, the fluid equations are solved in an Eulerian grid and the structure is treated as a network of Lagrangian nodes. The communication between the fluid and structure dynamics is established by the use of the Dirac delta function. Utilizing the immersed boundary method, we have studied three biophysical applications. In the first application, we computed …

Scalable Solution Of Time-Dependent Pdes Through Component-Wise Exponential Integrator, Chelsea Drum

Dissertations

Exponential integrators, such as exponential Runge-Kutta or Rosenbrock methods, are designed specifically for the time integration of stiff systems of ordinary differential equations (ODEs) and allow the use of larger time steps than other general-purpose ODE solvers. However, these methods rely on computing matrix function-vector products that are traditionally computed using a Krylov projection, such as Lanczos or Arnoldi iteration, that involves substantial computational expense at high spatial resolution. Krylov Subspace Spectral (KSS) methods' frequency-dependent approach, designed to circumvent stiffness in linear problems, computes these products with greater scalability. We propose the combination of such KSS methods with exponential integrators …

Uniformly Distributing Points On A Sphere, Flavio Arrigoni

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we are going to present and discuss different procedures for distributing points on a sphere's surface. Furthermore, we will assess their quality with three different distribution tests. The MATHEMATICA package that we created for testing and plotting the points is publicly available.

Approximating Equilibria In Restricted Games, Jack W. Doyle

Rose-Hulman Undergraduate Mathematics Journal

We consider optimal play in restricted games with linear constraints, and use ϵ-equilibria to find near-equilibrium states in these games. We present three mathematical optimization formulations -- a mixed-integer linear program (MILP), a quadratic program with linear constraints (QP), and a quadratically constrained program (QCP) -- to both approximate and identify these states. The MILP has a short runtime relative to the QP and QCP for large games (a factor 100 faster for |S|=9) and exhibits linear growth in run time, but provides only relatively weak upper bound. The QP and QCP provide a tight bound and the precise value …

Numerical Issues For A Non-Autonomous Logistic Model, Marina Mancuso, Kaitlyn M. Martinez, Carrie Manore, Fabio Milner

CODEE Journal

The user-friendly aspects of standardized, built-in numerical solvers in
computational software aid in the simulations of many problems solved using
differential equations. The tendency to trust output from built-in numerical
solvers may stem from their ease-of-use or the user’s unfamiliarity with the
inner workings of the numerical methods. Here, we show a case where the
most frequently used and trusted built-in numerical methods in Python’s
SciPy library produce incorrect, inconsistent, and even unstable approxima-
tions for a the non-autonomous logistic equation, which is used to model
biological phenomena across a variety of disciplines. Some of the most com-
monly used …

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

(R2078) Analyzing The Effects Of Fifth And Seventh Order Terms In A Generalized Henon-Heiles Potential, Nandana Madhukara

Applications and Applied Mathematics: An International Journal (AAM)

The Hénon-Heiles system is a 2-dimensional axisymmetric Hamiltonian system that was first formed to determine the third integral of motion in galactic dynamics. After its inception, it has become a paradigm in dynamical systems due to its apparent simplicity but extremely complicated dynamical behavior. In this paper, we perform a series expansion up to the seventh order of a general potential with axial and reflection symmetries. After some transformations, this becomes a generalized Hénon-Heiles (GHH) system where we separate the fifth and seventh-order terms. We qualitatively analyze this system for energies near the threshold between bounded and unbounded motion with …

Pt-Symmetry And Eigenmodes, Tamara Gratcheva

University Honors Theses

Spectra of systems with balanced gain and loss, described by Hamiltonians with parity and time-reversal (PT) symmetry is a rich area of research. This work studies by means of numerical techniques, how eigenvalues and eigenfunctions of a Schrodinger operator change as a gain-loss parameter changes. Two cases on a disk with zero boundary conditions are considered. In the first case, within the enclosing disk, we place a parity (P) symmetric configuration of three smaller disks containing gain and loss media, which does not have PT-symmetry. In the second case, we study a PT-symmetric configuration …

Advances In Computational And Statistical Inverse Problems, Dylan Green

Dartmouth College Ph.D Dissertations

Inverse problems are prevalent in many fields of science and engineering, such as signal processing and medical imaging. In such problems, indirect data are used to recover information regarding some unknown parameters of interest. When these problems fail to be well-posed, the original problems must be modified to include additional constraints or optimization terms, giving rise to so-called regularization techniques. Classical methods for solving inverse problems are often deterministic and focus on finding point estimates for the unknowns. Some newer methods approach the solving of inverse problems by instead casting them in a statistical framework, allowing for novel point estimate …

Identifiability For Pde Models Of Fluorescence Microscopy Experiments, Veronica Ciocanel

Biology and Medicine Through Mathematics Conference

No abstract provided.

Multiscale Modeling Of Microtubule Polarity Mechanisms Following Neuronal Axotomy, Hannah Scanlon

Biology and Medicine Through Mathematics Conference

No abstract provided.

Conjugations Of Unitary Operators, I, Javad Mashreghi, Mraek Ptak, William T. Ross

Department of Math & Statistics Faculty Publications

If U is a unitary operator on a separable complex Hilbert space H, an application of the spectral theorem says there is a conjugation C on H (an antilinear, involutive isometry on H) for which CUC= U*. In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which CUC = U*.

Effect Of Recommending Users And Opinions On The Network Connectivity And Idea Generation Process, Sriniwas Pandey, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

The growing reliance on online services underscores the crucial role of recommendation systems, especially on social media platforms seeking increased user engagement. This study investigates how recommendation systems influence the impact of personal behavioral traits on social network dynamics. It explores the interplay between homophily, users’ openness to novel ideas, and recommendation-driven exposure to new opinions. Additionally, the research examines the impact of recommendation systems on the diversity of newly generated ideas, shedding light on the challenges and opportunities in designing effective systems that balance the exploration of new ideas with the risk of reinforcing biases or filtering valuable, unconventional …

Analytical And Numerical Analysis Of The Sirs Model, Catherine Nguyen

Departmental Honors & Graduate Capstone Projects

Mathematical models in epidemiology describe how diseases affect and spread within a population. By understanding the trends of a disease, more effective public health policies can be made. In this paper, the Susceptible-Infected-Recovered-Susceptible (SIRS) Model was examined analytically and numerically to compare with the data for Coronavirus Disease 2019 (COVID-19). Since the SIRS model is a complex model, analytical techniques were used to solve simplified versions of the SIRS model in order to understand general trends that occur. Then by Euler's Method, the Runge-Kutta Method, and the Predictor-Corrector Method, computational approximations were obtained to solve and plot the SIRS model. …

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 …

Analysis And Construction Of Artificial Neural Networks For The Heat Equations, And Their Associated Parameters, Depths, And Accuracies., Shakil Ahmed Rafi

Graduate Theses and Dissertations

This dissertation seeks to explore a certain calculus for artificial neural networks. Specifically we will be looking at versions of the heat equation, and exploring strategies on how to approximate them.

Our strategy towards the beginning will be to take a technique called Multi-Level Picard (MLP), and present a simplified version of it showing that it converges to a solution of the equation (^∂/_∂t u_d ) (t, x) = (∇² _x u_d)(t, x).

We will then take a small detour exploring the viscosity super-solution properties of solutions to such …

Representation Learning For Generative Models With Applications To Healthcare, Astronautics, And Aviation, Van Minh Nguyen

Theses and Dissertations

This dissertation explores applications of representation learning and generative models to challenges in healthcare, astronautics, and aviation.

The first part investigates the use of Generative Adversarial Networks (GANs) to synthesize realistic electronic health record (EHR) data. An initial attempt at training a GAN on the MIMIC-IV dataset encountered stability and convergence issues, motivating a deeper study of 1-Lipschitz regularization techniques for Auxiliary Classifier GANs (AC-GANs). An extensive ablation study on the CIFAR-10 dataset found that Spectral Normalization is key for AC-GAN stability and performance, while Weight Clipping fails to converge without Spectral Normalization. Analysis of the training dynamics provided further …

Convergence Estimate Of Minimal Residual Methods And Random Sketching Of Krylov Subspace Methods, Peter Westerbaan

All Dissertations

This study concerns two main issues in numerical linear algebra: convergence estimate of minimal residual methods based on explicit construction of approximate min-max polynomials for in- definite matrices, and development and analysis of Krylov subspace methods using non-orthonormal basis vectors based on random sketching. For a matrix A with spectrum Λ(A), it is well known that the min-max polynomial problem min max |pk (z)| pk ∈Pk, pk (0)=1, z∈Λ(A) is used to bound the relative error of Krylov subspace minimum residual methods or similar methods. For a symmetric positive definite matrix A, the min-max polynomial for the Conjugate Gradient (CG) …

Domain Decomposition Methods For Fluid-Structure Interaction Problems Involving Elastic, Porous, Or Poroelastic Structures, Hemanta Kunwar

All Dissertations

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and Schwarz waveform relaxation (SWR) method using Robin transmission conditions (or the Robin method), for solving fluid-structure interaction systems involving elastic, porous, or poroelastic structure. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent fluid and the structure subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent multiphysics phenomena. For the fluid-porous …

Thermogravimetric Analysis And Chemical Kinetics Of Waste Tires, Mahir Mohammed Said

Tanzania Journal of Engineering and Technology (TJET)

The focus of this paper is to analyse the pyrolysis reaction of waste tires. The pyrolysis was studied experimentally using thermal and chemical kinetics. The tires were pyrolyzed using thermogravimetry at a temperature range of 200 to 600°C, with heating rates of 10, 20, and 50°C/min. The activating energy was observed to increase with increased temperature. The highest activating energy was 101.02 kJ/mol at a heating rate of 50°C/min.