Numerical Analysis and Computation Commons^™

Modelling Spherical Aberration Detection In An Analog Holographic Wavefront Sensor, Emma Branigan, Suzanne Martin, Matthew Sheehan, Kevin Murphy

Conference Papers

The analog holographic wavefront sensor (AHWFS) is a simple and robust solution to wavefront sensing in turbulent environments. Here, the ability of a photopolymer based AHWFS to detect refractively generated spherical aberration is modelled and verified.

(R1511) Numerical Solution Of Differential Difference Equations Having Boundary Layers At Both The Ends, Raghvendra Pratap Singh, Y. N. Reddy

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, numerical solution of differential-difference equation having boundary layers at both ends is discussed. Using Taylor’s series, the given second order differential-difference equation is replaced by an asymptotically equivalent first order differential equation and solved by suitable choice of integrating factor and finite differences. The numerical results for several test examples are presented to demonstrate the applicability of the method.

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 …

Numerical Analysis Of A Model For The Growth Of Microorganisms, Alexander Craig Montgomery, Braden J. Carlson

Rose-Hulman Undergraduate Mathematics Journal

A system of first-order differential equations that arises in a model for the growth of microorganisms in a chemostat with Monod kinetics is studied. A new, semi-implicit numerical scheme is proposed to approximate solutions to the system. It is shown that the scheme is uniquely solvable and unconditionally stable, and further properties of the scheme are analyzed. The convergence rate of the numerical solution to the true solution of the system is given, and it is shown convergence of the numerical solutions to the true solutions is uniform over any interval [0, T ] for T > 0.

Modeling Empirical Stock Market Behavior Using A Hybrid Agent-Based Dynamical Systems Model, Daniel A. Cline, Grant T. Aguinaldo, Christian Lemp

Northeast Journal of Complex Systems (NEJCS)

We describe the development and calibration of a hybrid agent-based dynamical systems model of the stock market that is capable of reproducing empirical market behavior. The model consists of two types of trader agents, fundamentalists and noise traders, as well as an opinion dynamic for the latter (optimistic vs. pessimistic). The trader agents switch types stochastically over time based on simple behavioral rules. A system of ordinary differential equations is used to model the stock price as a function of the states of the trader agents. We show that the model can reproduce key stylized facts (e.g., volatility clustering and …

Statistical Characteristics Of High-Frequency Gravity Waves Observed By An Airglow Imager At Andes Lidar Observatory, Alan Z. Liu, Bing Cao

Publications

The long-term statistical characteristics of high-frequency quasi-monochromatic gravity waves are presented using multi-year airglow images observed at Andes Lidar Observatory (ALO, 30.3° S, 70.7° W) in northern Chile. The distribution of primary gravity wave parameters including horizontal wavelength, vertical wavelength, intrinsic wave speed, and intrinsic wave period are obtained and are in the ranges of 20–30 km, 15–25 km, 50–100 m s−1, and 5–10 min, respectively. The duration of persistent gravity wave events captured by the imager approximately follows an exponential distribution with an average duration of 7–9 min. The waves tend to propagate against the local background winds and …

Optimal Design Of Bacterial Carpets For Fluid Pumping, Minghao W. Rostami, Weifan Liu, Amy Buchmann, Eva Strawbridge, Longhua Zhao

Biology and Medicine Through Mathematics Conference

No abstract provided.

Mathematical Modeling Of Brain Cancer Growth Using A Level-Set Method, Gbocho M. Terasaki

Biology and Medicine Through Mathematics Conference

No abstract provided.

Olfactory Bulb Processing Of Ortho Versus Retronasal Odors, Michelle F. Craft, Andrea Barreiro, Shree Gautam, Woodrow Shew, Cheng Ly

Biology and Medicine Through Mathematics Conference

No abstract provided.

Inferring Dynamics Of Biological Systems, Tracey G. Oellerich

Biology and Medicine Through Mathematics Conference

No abstract provided.

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 …

The Best Linear Approximation To Y= √X On The Interval [0, B] Using The Minimax Error, Hyounkyun Oh

Georgia Journal of Science

This study discusses how to find the best linear approximation y=mx+b to a fundamental function y=sqrt(x) on the interval [0,b], especially using the minimax error in Numerical Analysis. For this aim we employ two mathematical techniques: a) using the MATLAB code, positioning m and n values of the smallest maximum error on a broad range of m, and n value matrix in a rough scale and then repeatedly refining the regions in the smaller scales and b) Finding three-point fitting line to a set of non-colinear three points. We see that both results are successfully obtained and identical …

Data And Algorithmic Modeling Approaches To Count Data, Andraya Hack

Honors College Theses

Various techniques are used to create predictions based on count data. This type of data takes the form of a non-negative integers such as the number of claims an insurance policy holder may make. These predictions can allow people to prepare for likely outcomes. Thus, it is important to know how accurate the predictions are. Traditional statistical approaches for predicting count data include Poisson regression as well as negative binomial regression. Both methods also have a zero-inflated version that can be used when the data has an overabundance of zeros. Another procedure is to use computer algorithms, also known as …

Evaluating The Behaviour Of Centrally Perforated Unreinforced Masonry Walls: Applications Of Numerical Analysis, Machine Learning, And Stochastic Methods, Mohsen Khaleghi, Javid Salimi, Visar Farhangi, Mohammad Javad Moradi, Moses Karakouzian

Civil and Environmental Engineering and Construction Faculty Research

The presence of openings greatly affects the response of unreinforced masonry (URM) walls. This topic greatly attracts the attention of many researchers. Perforated unreinforced masonry (PURM) walls under in-plane loads through the truss discretization method (TDM) along with several machine learning approaches such as Multilayer perceptron (MLP), Group of Method Data Handling (GMDH), and Radial basis function (RBF) are described in this paper. A new method named Multi-pier (MP) that is fast and accurate, is used to determine the behavior of PURM walls. The results of the MP method are expressed as a ratio of lateral load-bearing capacity and initial …

A Conservative Numerical Scheme For The Multilayer Shallow Water Equations, Evan Butterworth

All Theses

An energy-conserving numerical scheme is developed for the multilayer shallow water equations (SWE’s). The scheme is derived through the Hamiltonian formulation of the inviscid shallow water flows related to the vorticity-divergence variables. Through the employment of the skew-symmetric Poisson bracket, the continuous system for the multilayer SWE’s is shown to preserve an infinite number of quantities, most notably the energy and enstrophy. An energy-preserving numerical scheme is then developed through the careful discretization of the Hamiltonian and the Poisson bracket, ensuring the skew-symmetry of the latter. This serves as the groundwork for developing additional schemes that preserve other conservation properties …

Tracking Academic Success And Its Relationship With Student Success Center Usage And Demographics, Samuel D. Johnson

Honors Capstone Projects and Theses

No abstract provided.

On The Consistency Of Alternative Finite Difference Schemes For The Heat Equation, Tran April

Rose-Hulman Undergraduate Mathematics Journal

While the well-researched Finite Difference Method (FDM) discretizes every independent variable into algebraic equations, Method of Lines discretizes all but one dimension, leaving an Ordinary Differential Equation (ODE) in the remaining dimension. That way, ODE's numerical methods can be applied to solve Partial Differential Equations (PDEs). In this project, Linear Multistep Methods and Method of Lines are used to numerically solve the heat equation. Specifically, the explicit Adams-Bashforth method and the implicit Backward Differentiation Formulas are implemented as Alternative Finite Difference Schemes. We also examine the consistency of these schemes.

Mathematical Analysis Of An Sir Disease Model With Non-Constant Transmission Rate, Emma Bollinger, Tayler Valdez, Swarup Ghosh, Sunil Giri

Student Research

• Epidemiology: A branch of medicine that studies causes, transmission, and control methods of diseases at the population level.
• Mathematical epidemiology deals with creating a model for a disease through the study of incidence and distribution of the disease throughout a population.
• Here, we have examined the behavior of a measles-like disease[2] that is characterized by a non-constant transmission rate.

Relative Energy Comparison For Various Water Clusters Using Mp2, Df-Mp2, And Ccsd(T):Mp2 Methods, Qihang Wang

Honors Theses

The study of water clusters is an important area of research in many disciplines, such as biology, physical chemistry, and environmental studies. However, due to the difficulty in studying larger water clusters, such as clathrate hydrates, it is beneficial to obtain accurate descriptions of smaller water clusters to use as models for larger systems via computational methods. By starting with small water clusters, such as (H2O)₆, and moving into larger systems it is possible to build up data on various water structures that can determine the energetics of the various geometries within a certain number of water molecules. …

A Mathematical Model For The Adoption Of Information And Communication Technology In School Libraries In Nigeria, Helen Olubunmi Jaiyeola Akinade, Jeremiah Ademola Balogun, Peter Adebayo Idowu

Library Philosophy and Practice (e-journal)

This study focused on the development of a mathematical model required for estimating the number of adopters of ICT devices among libraries located in Nigeria. Data for this study was collected from 121 respondents selected based on a research survey approach using simple random sampling. 9 ICT devices were identified, namely: PCs, printers/fax machines, search engines, e-library systems, bulk SMS services, library management systems, bar/QR code readers, projectors and video conferencing. The results showed that the earliest ICT devices were adopted for use in 1997, such as: PCs, printers/fax machines and search engines. The remaining ICT devices were adopted in …

Thickness Of Fluvial Deposits Records Climate Oscillations, Xiaoping Yuan, Laure Guerit, Jean Braun, Delphine Rouby, Charles Shobe

Faculty & Staff Scholarship

Fluvial deposits offer Earth’s best-preserved geomorphic record of past climate change over geological timescales. However, quantitatively extracting this information remains challenging in part due to the complexity of erosion, sediment transport and deposition processes and how each of them responds to climate. Furthermore, sedimentary basins have the potential to temporarily store sediments, and rivers subsequently rework those sediments. This may introduce time lags into sedimentary signals and obscure any direct correlation with climate forcing. Here, using a numerical model that combines all three processes—and a new analytical solution—we show that the thickness of fluvial deposits at the outlet of a …

Representing And Analyzing The Dynamics Of An Agent-Based Adaptive Social Network Model With Partial Integro-Differential Equations, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

We formulated and analyzed a set of partial integro-differential equations that capture the dynamics of our adaptive network model of social fragmentation involving behavioral diversity of agents. Previous results showed that, if the agents’ cultural tolerance levels were diversified, the social network could remain connected while maintaining cultural diversity. Here we converted the original agent-based model into a continuous equation-based one so we can gain more theoretical insight into the model dynamics. We restricted the node states to 1-D continuous values and assumed the network size was very large. As a result, we represented the whole system as a set …

Intra-Hour Solar Forecasting Using Cloud Dynamics Features Extracted From Ground-Based Infrared Sky Images, Guillermo Terrén-Serrano

Electrical and Computer Engineering ETDs

Due to the increasing use of photovoltaic systems, power grids are vulnerable to the projection of shadows from moving clouds. An intra-hour solar forecast provides power grids with the capability of automatically controlling the dispatch of energy, reducing the additional cost for a guaranteed, reliable supply of energy (i.e., energy storage). This dissertation introduces a novel sky imager consisting of a long-wave radiometric infrared camera and a visible light camera with a fisheye lens. The imager is mounted on a solar tracker to maintain the Sun in the center of the images throughout the day, reducing the scattering effect produced …

Toward Suicidal Ideation Detection With Lexical Network Features And Machine Learning, Ulya Bayram, William Lee, Daniel Santel, Ali Minai, Peggy Clark, Tracy Glauser, John Pestian

Northeast Journal of Complex Systems (NEJCS)

In this study, we introduce a new network feature for detecting suicidal ideation from clinical texts and conduct various additional experiments to enrich the state of knowledge. We evaluate statistical features with and without stopwords, use lexical networks for feature extraction and classification, and compare the results with standard machine learning methods using a logistic classifier, a neural network, and a deep learning method. We utilize three text collections. The first two contain transcriptions of interviews conducted by experts with suicidal (n=161 patients that experienced severe ideation) and control subjects (n=153). The third collection consists of interviews conducted by experts …

Active Polar Liquid Crystal Channel Flows: Analyzing The Roles Of Nematic Strength And Activation Parameter, Lacey Schenk, Ruhai Zhou

College of Sciences Posters

Suspensions of active polar liquid crystalline polymers (APLC) exhibit complex phenomena such as spontaneous flows, pattern formations and defects. Using the Kinetic Model, which couples the Smoluchowski Equation and the Navier-Stokes Equations, we conduct numerical simulations of APLC in a microfluidic channel to investigate the competitive effect among different material constants, such as the nematic concentration (the strength of the potential for nematic order) and active strength (the individual nano-rods strength of their individual movement) with and without a pressure gradient. Both Dirichlet and Neumann boundary conditions on the mathematical model are employed. Steady states, including isotropic and nematic states, …

A Spatially And Temporally Second Order Method For Solving Parabolic Interface Problems, Kumudu Gamage, Yan Peng

College of Sciences Posters

Parabolic interface problems have many applications in physics and biology, such as hyperthermia treatment of cancer, underground water flow, and food engineering. Here we present an algorithm for solving two-dimensional parabolic interface problems where the coefficient and the forcing term have a discontinuity across the interface. The Crank-Nicolson scheme is used for time discretization, and the direct immersed interface method is used for spatial discretization. The proposed method is second order in both space and time for both solution and gradients in maximum norm.

A New Application Of The Central Limit Theorem, Kenneth Winters

Selected Honors Theses

This paper discusses the Central Limit Theorem (CLT) and its applications. The paper gives an introduction to what the CLT is and how it can be applied to real life. Additionally, the paper gives a conceptual understanding of the theorem through various examples and visuals. The paper discusses the applications of the CLT in fields such as computer science, psychology, and political science. The author then suggests a new mathematical theorem as an application of the CLT and provides a proof of the theorem. The new theorem relates to expected value and probabilities of random variables and provides a link …

A Meshless Approach To Computational Pharmacokinetics, Anthony Matthew Khoury

Doctoral Dissertations and Master's Theses

The meshless method is an incredibly powerful technique for solving a variety of problems with unparalleled accuracy and efficiency. The pharmacokinetic problem of transdermal drug delivery (TDDD) is one such topic and is of significant complexity. The locally collocated meshless method (LCMM) is developed in solution to this topic. First, the meshless method is formulated to model this transport phenomenon and is then validated against an analytical solution of a pharmacokinetic problem set, to demonstrate this accuracy and efficiency. The analytical solution provides a locus by which convergence behavior are evaluated, demonstrating the super convergence of the locally collocated meshless …

Quadratic Neural Network Architecture As Evaluated Relative To Conventional Neural Network Architecture, Reid Taylor

Senior Theses

Current work in the field of deep learning and neural networks revolves around several variations of the same mathematical model for associative learning. These variations, while significant and exceptionally applicable in the real world, fail to push the limits of modern computational prowess. This research does just that: by leveraging high order tensors in place of 2nd order tensors, quadratic neural networks can be developed and can allow for substantially more complex machine learning models which allow for self-interactions of collected and analyzed data. This research shows the theorization and development of mathematical model necessary for such an idea to …