Applied Mathematics Commons^™

Localized Hermite Method Of Approximate Particular Solutions, Kwesi Acheampong

Dissertations

A localized Hermite method of approximate particular solutions (LHMAPS) is presented in this dissertation. This method is designed to improve the accuracy of the localized method of approximate particular solutions (LMAPS) for solving partial differential equations. LMAPS is a strong-form method that defines local radial basis function approximations to the solution values at a group of collocation points before applying the differential operator to this approximation function. Based on the LMAPS's local scheme, LHMAPS seeks Hermite-type local approximations based on both function values and derivatives. Numerical experiments validate the superior accuracy of the proposed method to the LMAPS by solving …

Effects Of A Protection Zone In A Reaction-Diffusion Model With Shifting Bounded Habitat And Allee Effect., Davis Henderson

Electronic Theses and Dissertations

We consider a reaction-diffusion equation that models a population in a shifting bounded habitat with a protection zone and the Allee effect. We assume that the population growth function exhibits the strong Allee effect and has a positive integral within the protection zone (indicating population persistence) and a negative integral in the surrounding patches (indicating population decay). We prove that the existence of steady-state solutions to this system depends on the length of the protection zone. It is demonstrated that there exists some value H* such that, for a protection zone of size H*, there exists a solution and, for …

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 …

First-Order Algorithms For Convex Smooth Optimization Problems With Homogeneous Linear Constraints, Yidan Guo

All Dissertations

The purpose of this dissertation is to explore the first-order methods that can be used to solve an approximate solution for convex smooth problems with homogeneous linear constraints. It consists of three interconnected research projects.

In the first project, we study the problem of computing the projection of a given vector onto the kernel of a symmetric positive semi-definite matrix. The complexity of an algorithm for computing a numerical solution is evaluated by the total number of matrix-vector multiplications required for computing an approximate solution. Such problems arise commonly in consensus optimization, in which the total number of matrix-vector multiplications …

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 …

Multi-Scale Ocean Modeling Using The Discontinuous Galerkin Method, Yao Gahounzo

Boise State University Theses and Dissertations

Ocean models often employ the hydrostatic assumption for large-scale applications when the horizontal scale is larger than the depth. Nonetheless, the non-hydrostatic effects are of great importance in submesoscale studies, such as modeling the physical processes of ice-ocean interactions. Most non-hydrostatic ocean models used in ice-ocean interaction studies are low-order finite-difference and finite-volume methods, often with significant dispersive errors. Therefore, there is a need for a high-order ocean model for ice-ocean interaction applications.

The first part of this thesis focuses on deriving the boundary conditions at the ice-ocean interface and incorporating them into the high-order discontinuous Galerkin (DG) method-based ocean …

Frameworks For The Techno-Economic Assessment Of Membrane-Based Bioprocessing Platforms, Juan Jose Romero Conde

All Dissertations

This dissertation describes developing and implementing computational frameworks for simulating and optimizing purification processes in the biopharmaceutical industry. The framework performs techno-economic analyses to establish value propositions for new process alternatives, especially membrane technologies. Initially, the focus is developing a framework capable of simulating monoclonal antibody (mAb) capture using membrane and resin media in multi-column chromatography (MCC) platforms for continuous manufacturing. Subsequently, the impact of capture MCC is compared against other intensification strategies in established mAb manufacturing facilities. Finally, the framework application expands to simulate the purification of adeno-associated virus (AAV) vectors for gene therapy.

Chapter 2 details the framework …

Optimization Strategies For Political Redistricting, Blake Splitter

All Dissertations

Political redistricting has remained a hot-button issue in the United States for several decades. Every ten years, most states need to redraw their districts to account for changing populations. Sometimes, these district plans can be drawn with the malevolent intention of aiding one political party over another. This dissertation summarizes four distinct methods of drawing these districts using computer algorithms while keeping several objectives in mind. We test these approaches on the case study state of South Carolina, since it provides a sufficiently challenging problem for us to test various algorithms. We find that many of these approaches improve upon …

Efficient First-Order Methods For Some Smooth Nonlinear Optimization Problems, Yunheng Jiang

All Dissertations

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Delta-Shaped Approximation Based Homotopy Analysis Method For Nonlinear Poisson-Type Partial Differential Equations, Cyril Ocloo

Dissertations

This research aims to solve nonlinear Poisson-type partial differential equations (PDEs) by the approach of the homotopy analysis method (HAM) incorporated with approximate particular solutions (APS) using Delta-shaped basis (DSB) approximations.

With the inclusion of the h auxiliary parameters, we tackle nonlinear problems by studying the mathematical characteristics of the h curve. This is to ensure the numerical convergence of the HAM.

In the solution process, we use the homotopy analysis method to convert a nonlinear PDE into linear inhomogeneous PDEs, which are solved using the method of approximate particular solutions with DSB.

A proper value of the h is …

Visualization Of Species Tree Likelihood Under The Multispecies Coalescent Model, Jaimasan Sutton

Mathematics & Statistics ETDs

A commonly used tool for evolutionary biologists is a phylogenetic tree that represents the ancestry of a set of species and the evolution of traits. Statistical models can be used to predict the probabilities of gene trees which represent ancestral relationships of genes sampled from species. Because of this, we are able to represent the likelihood of a species tree, which represents the evolutionary history of a set of species, as a function of the counts of gene tree topologies, where each gene tree represents the ancestry of a specific genetic locus for multiple species. Because we can represent these …

The Role Of Sensing Modalities In Shaping Collective Motion And Group Behavior, Poorendra P. Ramlall

Doctoral Dissertations and Master's Theses

Collective behavior refers to the coordinated movements that emerge from simple interactions between individuals within a group. Traditionally, researchers have modeled these interactions assuming individuals can sense their surroundings in all directions, like having eyes all around their heads. While this is a useful simplification, it does not capture the diverse ways animals actually sense the world. In this thesis, we take inspiration from the natural world, particularly from animals like bats and dolphins that use a combination of hearing and sight to navigate their environments. We explore how combining these sensory cues in a three-dimensional space affects the way …

Radially Symmetric Positive Solutions Of The Dirichlet Problem For The P-Laplace Equation, Bo Yang

Faculty Articles

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive solutions for the p-Laplace problem are obtained.

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.

Modeling Virus Diffusion On Social Media Networks With The Smirq Model, Justin Browning, Arnav Mazumder, Gowri Nanda

Rose-Hulman Undergraduate Mathematics Journal

As social networking services become more complex and widespread, users become increasingly susceptible to becoming infected with malware and risk their data being compromised. In the United States, it costs the government billions of dollars annually to handle malware attacks. Additionally, computer viruses can be spread through schools, businesses, and individuals’ personal devices and accounts. Malware affecting larger groups of people causes problems with privacy, personal files, and financial security. Thus, we developed the probabilistic SMIRQ (pSMIRQ) model that shows how a virus spreads through a generated network as a way to track and prevent future viruses. Our model is …

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 …

Resonant Solutions Of The Non-Linear Schrödinger Equation With Periodic Potential, Arein Duaibes, Yulia Karpeshina

Mathematics Faculty Publications

The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.

Asymptotic Formula For Scattering Problems Related To Thin Metasurfaces, Zachary Jermain

LSU Doctoral Dissertations

The goal of this work is to develop an asymptotic formula for the behavior of a scattered electromagnetic field in the presence of a thin metamaterial known as a metasurface. By using a carefully chosen Green’s function and the single and double layer potentials we analyze the perturbed scattering problem in the presence of the metamaterial and a background scattering problem. By using Lippman-Schwinger type representation formulas for the two fields we develop the asymptotic formula for the perturbed field. From here we prove the asymptotic formula holds up to a specific error term based on the size of the …

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 …

Employing Interactive Architecture Techniques In Architectural Design Methodologies To Retrieve Ancient Egyptian Identity, Dina Elwy Aly Hamed Hussein Dr. Lobna Abd-Allah Abd-Elfattah Agha , Dr. Azza Sobhi Ragab Elsakka

Journal of Engineering Research

Architectural projects often rely on modern technologies like smart architecture and environmental treatments, neglecting the principles and philosophy that reflect civilization, ideas, culture, and traditions. This leads to short-lived buildings that depend on these technologies without any purpose of preserving them. Technicians must constantly renew technology to improve production and without benefits, as threats of superior artificial intelligence have been warned, that autonomous machines may perform tasks without programmers understanding how to reach this level. Instead of relying solely on environmental or smart technology, we should make technology serve the design idea rather than relying solely on it. Considering this, …