SIURO | Volume 15 | SIAM


SIAM Undergraduate Research Online

Volume 15

SIAM Undergraduate Research Online Volume 12

Clustering COVID-19 Lung Scans

Published electronically January 24, 2022
DOI: 10.1137/20S1365053

Authors: Andrew Householder, Jacob Householder, and John Paul Gomez-Reed (Whittier College)
Project Advisor: Fredrick Park (Whittier College)

Abstract: With the ongoing COVID-19 pandemic, understanding the characteristics of the virus has become an important and challenging task in the scientific community. While tests do exist for COVID-19, the goal of our research is to explore other methods of identifying infected individuals. Our group applied supervised clustering techniques to explore a dataset of lung scans of COVID-19 infected, Viral Pneumonia infected, and healthy individuals. This is an important area to explore as COVID-19 is a novel disease that is currently being studied in detail. Our methodology explores the potential that unsupervised clustering algorithms have to reveal important hidden differences between COVID-19 and other respiratory illnesses. Our experiments use: Principal Component Analysis (PCA), K-Means++ (KM++) and the recently developed Robust Continuous Clustering algorithm (RCC).We evaluate the performance of KM++ and RCC in clustering COVID-19 lung scans using the Adjusted Mutual.

Finding the Optimum Times to Flip a Beefsteak Using a Genetic Algorithm

Published electronically February 2, 2022
DOI: 10.1137/21S1414917

Authors: Rebekah Yu-En Chin (Hong Kong Baptist University)
Project Advisor: Leevan Ling (Hong Kong Baptist University)

Abstract:This paper attempts to find the best times at which to flip a beefsteak so that the steak is cooked to medium-rare, subject to a fixed minimum temperature. The steak, pan, and a layer of oil is modeled with a partial differential equation. The physical parameters of the model are approximated and their effects on the model are discussed. Appropriate boundary conditions are selected to allow for heat convection with the air, heat to enter from a stove, and heat diffusion between the steak, oil, and pan. The model is compared to experimental results and evaluated. The model is then converted into an optimization problem and minimized with a genetic algorithm (GA). The solution obtained with GA lists the optimum times to flip a steak to minimize its mean temperature and performed better than a single flip procedure.

Spectral Touching Points in Two-Dimensional Materials

Published electronically February 17, 2022
DOI: 10.1137/21S143889X

Authors: Andrea Wynn (Rose-Hulman Institute of Technology)
Project Advisor: Dr. Tracy Weyand (Rose-Hulman Institute of Technology)

Abstract: A two-dimensional (2D) material is a crystalline material consisting of a single layer of atoms. These materials are used in many applications including photovoltaics, semiconductors, electrodes, and water purification. These materials’ atomic structures can be represented as a discrete infinite periodic graph. Using Floquet-Bloch theory, the spectrum of the Schrodinger operator can be calculated on these infinite graphical representations by computing the eigenvalues of the magnetic flux Schrodinger operator on a fundamental domain for every possible value of magnetic flux. Previous researchers have conjectured a relationship between the special physical properties of one 2D material, graphene, and the Dirac conical points which appear in the spectrum of its Schrodinger operator. However, graphene was the only material studied with respect to these Dirac conical points. The existence of spectral touching points in different two-dimensional materials is proved, including muscovite, quartz, and transition metal oxides, under certain conditions on electric potential. The spectral touching points found in transition metal oxides are not the Dirac conical points found in graphene, but rather a previously unknown type of spectral touching point, named the mesa touching point, which appears in the Schrodinger operator for transition metal oxides under certain conditions.

A Numerical Investigation of Rayleigh-Benard Convection with an Obstruction

Published electronically April 1, 2022
DOI: 10.1137/21S1454535

Authors: Harieth Mhina and Samira Souley Hassane (Trinity College)
Project Advisor: Matthew McCurdy (Trinity College)

Abstract: The phenomenon of convection is found in a wide variety of settings on different scales– from applications in the cooling technology of laptops to heating water on a stove, and from the movement of ocean currents to describing astrophysical events with the convective zones of stars. Given its importance in these diverse areas, the process of convection has been the focus of many research studies over the past two centuries. However, much less research has been conducted on how the presence of an obstruction in the flow can impact convection. In this work, we find that the presence of an obstruction can greatly affect convection. We note occurrences where the presence of an obstruction yields similar behavior to flow without an obstruction. Additionally, we find cases with markedly different features in comparison to their counterpart without an obstruction– notably, exhibiting long-term periodic behavior instead of achieving a constant steady-state, or the formation of convection cells versus an absence of them.

Alternating Minimization for Computed Tomography with Unknown Geometry Parameters

Published electronically April 1, 2022
DOI: 10.1137/21S1441638

Authors: Mai Phuong Pham Huynh (Emory University) and Manuel Santana (Utah State University)
Project Advisor: James Nagy (Emory University)

Abstract: Due to the COVID-19 pandemic, there is an increasing demand for portable CT machines worldwide in order to diagnose patients in a variety of settings. This has led to a need for CT image reconstruction algorithms that can produce high quality images in the case when multiple types of geometry parameters have been perturbed. In this paper we present an alternating minimization algorithm to address this issue, where one step minimizes a regularized linear least squares problem, and the other step minimizes a bounded non-linear least squares problem. Additionally, we survey existing methods to accelerate convergence of the algorithm and discuss implementation details. Finally, numerical experiments are conducted to illustrate the effectiveness of the algorithm.

Accelerated Alternating Minimization for X-ray Tomographic Reconstruction

Published electronically April 27, 2022
DOI: 10.1137/21S1437470

Author: Peijian Ding (Emory University)
Project Advisor: James Nagy (Emory University)

Abstract: While Computerized Tomography (CT) images can help detect disease such as Covid-19, regular CT machines are large and expensive. Cheaper and more portable machines suffer from errors in geometry acquisition that downgrades CT image quality. The errors in geometry can be represented with parameters in the mathematical model for image reconstruction. To obtain a good image, we formulate a nonlinear least squares problem that simultaneously reconstructs the image and corrects for errors in the geometry parameters. We develop an accelerated alternating minimization scheme to reconstruct the image and geometry parameters.

On Application of Block Kaczmarz Methods in Low-Rank Matrix Factorization

Published electronically April 28, 2022
DOI: 10.1137/20S1376467

Author: Edwin Chau (UCLA)
Project Advisor: James Haddock (Harvey Mudd College)

Abstract: Matrix factorization techniques compute low-rank product approximations of high dimensional data matrices and as a result, are often employed in recommender systems and collaborative filtering applications. However, many algorithms for this task utilize an exact least-squares solver whosecomputation is time consuming and memory-expensive. In this paper we discuss and test a block Kaczmarz solver that replaces the least-squares subroutine in the common alternating scheme for lowrank matrix factorization. This variant trades a small increase in factorization error for significantlyfaster algorithmic performance. In doing so we find block sizes that produce a solution comparable to that of the least-squares solver for only a fraction of the runtime and working memory requirement.

Comparison of Atlas-Based and Neural-Network-Based Semantic Segmentation for DENSE MRI Images

Published electronically May 26, 2022
DOI: 10.1137/21S1448392
Supplementary Materials

Authors: Emma Hart (corresponding author – Colgate University), Elle Buser (Wyoming University), and Ben Huenemann (University of Utah)
Project Advisor: Lars Ruthotto (Emory University)

Abstract: Two segmentation methods, one atlas-based and one neural-network-based, were compared to see how well they can each automatically segment the brain stem and cerebellum in Displacement Encoding with Stimulated Echoes Magnetic Resonance Imaging (DENSE-MRI) data. The segmentation is a pre-requisite for estimating the average displacements in these regions, which have recently been proposed as biomarkers in the diagnosis of Chiari Malformation type I (CMI). In numerical experiments, the segmentations of both methods were similar to manual segmentations provided by trained experts. It was found that, overall, the neural-network-based method alone produced more accurate segmentations than the atlas-based method did alone, but that a combination of the two methods - in which the atlas-based method is used for the segmentation of the brain stem and the neural-network is used for the segmentation of the cerebellum - may be the most successful.

Remote Work: Fad or Future

Published electronically June 16, 2022
DOI: 10.1137/22S1493136
M3 Introduction

Authors: Eric Wan (corresponding author – Homestead High School, Mequon, WI) ORCIDiD_iconbw16x16.png, Adam Garsha  (Homestead High School), Jacob Schmidman (Homestead High School), and Ethan Wang (Homestead High School)
Project Advisor: Weizhong Wang (Homestead High School, Mequon, WI and University of Wisconsin – Milwaukee, WI)

Abstract: Since the onset of the coronavirus pandemic, the share of American and British workers working remotely has dramatically increased. Employees and business owners have rapidly adapted to the significant shift towards online work, dramatically revolutionizing the labor landscape.

Implementation of Statewide Election Data to Examine Fairness of South Carolina District Maps: A Comparative Analysis of Approaches for Approximating Results in Uncontested Races

Published electronically July 11, 2022
DOI: 10.1137/21S1437342

Author: Alfie-Louise Brownless (Wofford College)
Project Advisor: Anne J. Catlla (Wofford College)

Abstract: After each census, researchers analyze election data to provide information relevant to the redistricting process. South Carolina is among a collection of states which face certain issues regarding election analysis of fairness due to the presence of a large percentage of uncontested races. Although uncontested results are known to create analysis challenges, there is not a universal consensus on how to best handle these situations. Here we explore quantification of partisan fairness and the impact of using statewide election county-level data as a proxy for estimating uncontested results. We develop a district approximation method using statewide elections at the county scale and use known metrics to qualitatively and quantitatively evaluate resulting election characteristics in historical and simulated election contexts. The same metrics were then used to perform a thorough comparative analysis of other common approximation methods. We find county-level election data to be an effective tool in approximating uncontested elections by providing evidence to support the notion that county-level data is effective under multiple election conditions. Furthermore, analysis of different approximation methods show how measures of partisan fairness for a particular election can change based upon a particular approximation method, potentially affecting future interpretations of uncontested election results.

Statistical Learning for Best Practices in Tattoo Removal

Published electronically July 18, 2022
DOI: 10.1137/21S1421325

Author: Richard P. Yim (UCLA)
Project Advisor: Jamie Haddock (UCLA) and Deanna Needell (UCLA)

Abstract: The causes behind complications in laser-assisted tattoo removal are currently not well understood, and in the literature relating to tattoo removal the emphasis on removal treatment is on removal technologies and tools, not best parameters involved in the treatment process. Additionally, the very challenge of determining best practices is difficult given the complexity of interactions between fac- tors that may correlate to these complications. In this paper we apply a battery of classical statistical methods and techniques to identify features that may be closely correlated to causes of complication during the tattoo removal process, and report quantitative evidence for potential best practices. We develop elementary statistical descriptions of tattoo data collected by the largest gang rehabilitation and reentry organization in the world, Homeboy Industries; perform parametric and nonparametric tests of significance; and finally, produce a statistical model explaining treatment parameter inter- actions, as well as develop a ranking system for treatment parameters utilizing bootstrapping and gradient boosting.

Neural Network Approach to NFL Position Classification

Published electronically July 28, 2022
DOI: 10.1137/21S1444485

Author: Sithija Manage (Texas A&M University)
Project Advisor: Sai-Mang Pun (Texas A&M University)

Abstract: With an ever-increasing captivation of the United States sports-viewing audience, the National Football League continues to produce some of the world’s most capable, physical athletes. In this work, athletes’ positions C, OG, OT, DE, and DT were categorized as on the line, while the remaining positions were categorized as not on the line. In this work, a predictive neural network is applied to classify 2,022 National Football League players into the two classifications using scouting combine data of height, weight, and 40-Yard dash time, outperforming the current standard logistic regression. The two measures utilized to compare the strength of the methods were total accuracy and area under ROC curve, with the neural network yielding a slightly higher average in both. In terms of total accuracy, the neural network had an accuracy of 0.9134 to the logistic model’s 0.9065, and in terms of area under ROC curve, the neural network had an area of 0.9578 compared to the logistic model’s 0.9567. As a head-to-head iteration-wise comparison, the neural network had a winning Win-Loss-Tie ratio of 7-2-1 and 5-5-0 in the two measures respectively.

Using Artificial Neural Networks to Classify Optimal Microswimmers Based on Their Shapes

Published electronically July 28, 2022
DOI: 10.1137/22S1479816

Authors: Niyizhen Jin (corresponding author – University of Michigan), Xinyue Qie (University of Michigan), Nicole Surgent (University of Michigan), Wanting Huang (University of Michigan)
Sponsor: Hanliang Guo (University of Michigan)

Abstract: Studies of microswimmers have received increasing attention since the 2000s fueled by the advancements in micro-manufacturing and their potential for exciting biomedical applications. One of the popular mathematical research directions is the optimization of the flagella- or cilia-kinematics to maximize the swimming efficiency, usually for isolated microswimmer. The collective behaviors, on the other hand, are affected by the types of microswimmers (e.g., pusher, puller, or neutral). Understanding the connections between the optimal activation of a given shape and its swimming-type can have important implications on designing artificial microswimmers. In this work, we build an artificial neural network (ANN) that can predict the types of optimal microswimmers based solely on their shapes. More interestingly, we show that the tangent vector information along the microswimmer surface is important for the ANN to successfully classify the microswimmers.

Forecasting COVID-19 Vaccine Distribution in the United States, Japan, Taiwan, and China Using the Auto-Regressive Integrated Moving Average (ARIMA) Model

Published electronically August 2, 2022
DOI: 10.1137/21S145584X

Author: Kenneth (Hsuan An) Chen (University of California, Los Angeles)
Project Advisor: Michael Tsiang (University of California, Los Angeles)

Abstract: Developed at unprecedented speeds, vaccines have thus far played a crucial role in slowing down the COVID-19 pandemic around the world. Therefore, it is an absolute necessity for countries to be able to accurately forecast the distribution of vaccines. This paper uses an Auto-Regressive Integrated Moving Average (ARIMA) model to analyze and forecast 30 days of COVID-19 vaccine distribution for the United States, Japan, Taiwan, and China. Specifically, for the United States and Japan, the predicted variable was the percent of the population that was fully vaccinated while the predicted variable for Taiwan and China was the total number of doses administered. The data used to fit our model was pulled from a publicly available dataset compiled from various sources around the world. For each country, the training data consisted of that country’s vaccination data from whenever they first administered vaccines until July 19, 2021. After fitting the model on the training data, the model was then tested against 30 days of data from July 20, 2021 to August 18, 2021. The paper found that the univariate ARIMA model was able to, on average, forecast the distribution of COVID-19 vaccines within 5% of the actual value for each country.

A Numerical Study on Sparse Learning of Interaction Laws in Homogeneous Multiparticle Systems

Published electronically August 5, 2022
DOI: 10.1137/22S1469341

Author: Ritwik Trehan (corresponding author -- University of California, Santa Barbara), Hao-Tien Chuang (University of California, Los Angeles), Dongyang Li (University of California, Santa Barbara), Shelby Malowney (University of California, Santa Barbara)
Sponsor: Sui Tang (University of California, Santa Barbara)

Abstract:Multi-agent systems have found wide applications in science and engineering ranging from opinion dynamics to predator-prey systems. A grand challenge encountered in these areas is to reveal the interaction laws between individual agents leading to collective behaviors. In this article, we consider a system of ODEs that is often used in modeling opinion dynamics, where the laws of the interaction are dependent on pairwise distances. We leverage recent advancements in sparsity-promoted algorithms and propose a new approach to learning the interaction laws from a small amount of data. Numerical experiments demonstrate the effectiveness and robustness of the proposed approach in a small, noisy data regime and show the superiority of the proposed approach.

A Tensor SVD-based Classification Algorithm Applied to fMRI Data

Published electronically August 29, 2022
DOI: 10.1137/21S1456522

Authors: Katherine Keegan (corresponding author – Mary Baldwin University) ORCIDiD_iconbw16x16.png, Tanvi Vishwanath (Texas A&M University), and Yihua Xu (Georgia Institute of Technology)
Project Advisor: Elizabeth Newman (Emory University)

Abstract: To analyze the abundance of multidimensional data, tensor-based frameworks have been developed. Traditionally, the matrix singular value decomposition (SVD) is used to extract the most dominant features from a matrix containing the vectorized data. While the SVD is highly useful for data that can be appropriately represented as a matrix, this step of vectorization causes us to lose the high-dimensional relationships intrinsic to the data. To facilitate efficient multidimensional feature extraction, we utilize a projection-based classification algorithm using the t-SVDM, a tensor analog of the matrix SVD. Our work extends the t-SVDM framework and the classification algorithm, both initially proposed for tensors of order 3, to any number of dimensions. We then apply this algorithm to a classification task using the StarPlus fMRI dataset. Our numerical experiments demonstrate that for this fMRI classification task, the t-SVDM-based algorithm obtains noticeably superior performance when compared to the best possible equivalent matrix-based approach. Our results illustrate the advantages of our chosen tensor framework, provide insight into beneficial choices of parameters, and could be further developed for classification of more complex imaging data. We provide our Python implementation at

A Mathematical Analysis of Reconstruction Artifacts in Radar Limited Data Tomography

Published electronically September 7, 2022
DOI: 10.1137/21S1468759

Authors: Elena Martinez (corresponding author – Loyola Marymount University)
Project Advisor: Eric Todd Quinto (Tufts University)

Abstract: In the study of tomography, there are often missing data values. This leads artifacts to present themselves in data reconstructions. We investigate this problem in a bistatic radar system that has a radio transmitter in a fixed location and a receiver flying around the transmitter in a circular path. Our data is collected by integrating over all ellipses in a given space that have the transmitter and receiver as foci. We reconstruct this numerical data and analyze the artifacts that present themselves when we place objects within and outside of the receiver's path. Our research demonstrates how objects outside the receiver's path can create artifacts inside the receiver's path and vice versa. This shows an intrinsic limitation to a method that works well when the scanned region outside the receiver's path is clear.

Discretized Migration Flow: A Vector Field Based Tool for Avian Mobility in Patchy Mechanistic Models of Early Pathogenic Spread

Published electronically September 20, 2022
DOI: 10.1137/21S1461381

Authors: Raphael Chiemezie Kelly (corresponding author – Archbishop MacDonald High School)
Project Advisor: Peter D. Harrington (University of Alberta) and Mark A. Lewis (University of Alberta)

Abstract: Various pathogens are spread through avian hosts. The spread of these pathogens can have massiveeconomic and health consequences. Spatially explicit models of spread are needed in order to anticipatewhere and when diseases will spread. However, making predictions from models for suchdiseases has traditionally been challenging due to the complexity of bird movements, and a lack ofcomprehensive data on them. This paper proposes a model for the directional movement of birdsbetween patches within epidemiological models. This model considers bird mobility in two ways:directed migration and random diffusion. Migration is incorporated through a vector field that representsaverage movements each migratory season, generated based on continental flyways. Diffusionis incorporated between neighbouring patches and segmented between each of the major flyways.Migration and diffusion are combined into a large, temporally varying mobility matrix that representsthe movement of each bird in one patch to another. The mobility matrix is then used with a systemof susceptible-infected (SI) differential equations to determine the spread of disease. The system wassolved and results verified against infection data on the West Nile virus (WNv) outbreak in the US in1999 and Turdus migratorius distributions, demonstrating the model’s ability to accurately predict boththe major spatio-temporal phases of WNv spread as well as the phases of American robin migration.This approach, here called discretized migration flow (DMF), can be further developed and exploredfor application in early stage emerging disease models.

Analysis of Legal Documents via Non-negative Matrix Factorization Methods

Published electronically September 27, 2022
DOI: 10.1137/21S1414486

Authors: Pengyu Li (corresponding author – University of California, Los Angeles), Ryan Budahazy (author -- Towson University, Maryland), Lu Cheng (author -- University of California, Los Angeles), Yihuan Huang (author -- University of California, Los Angeles), Andrew Johnson (author -- Baruch College, New York), Joshua Vendrow (author -- University of California, Los Angeles), and Zhoutong Wu (author -- University of California, Los Angeles)
Project Advisors: Denali Molitor (University of California, Los Angeles), Elizaveta Rebrova (Princeton University), and Deanna Needell (University of California, Los Angeles)

Abstract: The California Innocence Project (CIP), a clinical law school program aiming to free wrongfully convicted prisoners, evaluates thousands of mails containing new requests for assistance and corresponding case files. Processing and interpreting this large amount of information presents a significant challenge for CIP ocials, which can be successfully aided by topic modeling techniques. In this paper, we apply Non-negative Matrix Factorization (NMF) method and implement various offshoots of it to the important and previously unstudied data set compiled by CIP. We identify underlying topics of existing case files and classify request files by crime type and case status (decision type). The results uncover the semantic structure of current case files and can provide CIP ocials with a general understanding of newly received case files before further examinations. We also provide an exposition of popular variants of NMF with their experimental results and discuss the benefits and drawbacks of each variant through the real-world application.

An Agent-Based Model of COVID-19 on the Diamond Princess Cruise Ship

Published electronically September 29, 2022
DOI: 10.1137/21S1462520

Authors: Naomi Rankin (corresponding author – Howard University)

Project Advisor: Katharine Gurski (Howard University)

Abstract: We model the COVID-19 outbreak and shipboard quarantine with a 3-D agent-based simulation of a SEIR model which preserves the ratios of crew, passengers, and shipboard space. The stochastic model captures the movement patterns of passengers and crew members on-board the ship, as well as how this movement changed once quarantine is established. The study includes the derivation of the basic reproduction number based on contact numbers and transmission rates. We capture the number of contacts between two people when they remain within the model equivalent of a 3-foot radius for 60 minutes and the transmission probability per contact. We show that, based on the measured reproduction number, an outbreak is bound to occur in the majority of simulations even with quarantine imposed on the ship. We also show that most infection on board occurs by others of the same group (passenger or crew), with passengers causing the majority of infections.

Rapid Testing in COVID and Modified SIR Model

Published electronically October 4, 2022
DOI: 10.1137/21S1460399

Authors: Jiawei Chen (corresponding author – University of Toronto), Ran Li (University of Toronto), and Junru Lin (University of Toronto)

Project Advisor: Lisa Jeffrey (University of Toronto)

Abstract: The COVID pandemic has swept the globe since 2019, posing a grave threat to human life. There are multiple ways for the government to control the pandemic, including promoting the vaccination, limiting the number of people in public places, requiring people to wear masks in public places, and suggesting infected people isolate themselves. In this paper, we used a compartmental model to analyze the spread of COVID-19 under the promotion of rapid tests. The result shows that popularization of rapid tests may have a significant impact on controlling the pandemic. With an estimated minimum requirement for the use of rapid tests, we are able to put forward suggestions on reasonable ways to curtail the pandemic.

Accelerating Parameter Inference in Diffusion-Reaction Models of Glioblastoma Using Physics-Informed Neural Networks

Published electronically October 11, 2022
DOI: 10.1137/22S1472814

Authors: Andy Zhu (corresponding author – Northwood High School, Irvine, CA)

Project Advisors: Jonathan Vo (University of California, Irvine) and John Lowengrub (University of California, Irvine)

Abstract: Glioblastoma is an aggressive brain tumor with cells that infiltrate and proliferate rapidly into surrounding brain tissue. Current mathematical models of glioblastoma growth capture this behavior using partial differential equations (PDEs) that are simulated via numerical solvers—a highly-efficient implementation can take about 80 seconds to complete a single forward evaluation. However, clinical applications of tumor modeling are often framed as inverse problems that require sophisticated numerical methods and, if implemented naively, can lead to prohibitively long runtimes that render them inadequate for clinical settings. Recently, physics-informed neural networks (PINNs) have emerged as a novel method in scientific machine learning for solving nonlinear PDEs. Compared to traditional solvers, PINNs leverage unsupervised deep learning methods to minimize residuals across mesh-free domains, enabling greater flexibility while avoiding the need for complex grid constructions. Here, we describe and implement a general method for solving time-dependent diffusion-reaction PDE models of glioblastoma and inferring biophysical parameters from numerical data via PINNs. We evaluate the PINNs over patient-specific geometries, accounting for individual variations with diffusion mobilities derived from pre-operative MRI scans. Using synthetic data, we demonstrate the performance of our algorithm in patient-specific geometries. We show that PINNs are capable of solving parameter inference inverse problems in approximately one hour, expediting previous approaches by 20–40 times owing to the robust interpolation capabilities of machine learning algorithms. We anticipate this method may be sufficiently accurate and efficient for clinical usage, potentially rendering personalized treatments more accessible in standard-of-care medical protocols.

Quantum Evolution in One Dimensional Disordered Systems: Localisation and Oscillations

Published electronically October 13, 2022
DOI: 10.1137/22S1516373

Author: Edward Sharp (University of Bristol)

Project Advisor: Francisco Gonzalez Montoya (University of Bristol)

Abstract: This work is a simple example of the quantum dynamics of a particle in a disordered system in one dimension. In particular, we illustrate numerically the phenomenon of Anderson localization of a wave packet using a basic model constructed with small random rectangular potential barriers. Also, we study the dynamics of a quantum particle in a disordered potential formed by an harmonic oscillator perturbed by random rectangular barriers. To show the effects of disorder on the dynamics of the system, we compare the time evolution of the wave function of the unperturbed harmonic oscillator with the wave function of the disordered system. We do this by taking the scalar product between the unperturbed and perturbed wave functions at each timestep for different values of the perturbation parameters affecting the disordered wave packet.

A Probabilistic Analysis of Shotgun Sequencing for Metagenomics

Published electronically October 14, 2022
DOI: 10.1137/22S1472437

Author: Marlee Herring (University of North Carolina at Charlotte)
Project Advisor: Kevin McGoff (University of North Carolina at Charlotte)

Abstract: Genome sequencing is the basis for many modern biological and medicinal studies. With recent technological advances, metagenomics has become a problem of interest. This problem entails the analysis and reconstruction of multiple DNA sequences from different sources. Shotgun genome sequencing works by breaking up long DNA sequences into shorter segments called reads. Given this collection of reads, one would like to reconstruct the original collection of DNA sequences. For experimental design in metagenomics, it is important to understand how the minimal read length necessary for reliable reconstruction depends on the number and characteristics of the genomes involved. Utilizing simple probabilistic models for each DNA sequence, we analyze the identifiability of collections of M genomes of length N in an asymptotic regime in which N tends to infinity and M may grow with N. Our first main result provides a threshold in terms of M and N so that if the read length exceeds the threshold, then a simple greedy algorithm successfully reconstructs the full collection of genomes with probability tending to one. Our second main result establishes a lower threshold in terms of M and N such that if the read length is shorter than the threshold, then reconstruction of the full collection of genomes is impossible with probability tending to one.

Characterising Dark Matter Substructure in Gravitational Lens Galaxies with Deep Learning

Published electronically October 28, 2022
DOI: 10.1137/22S1478033

Author: Owen Scutt (University of Nottingham)

Project Advisor: Simon Dye (University of Nottingham)

Abstract: We investigate the novel application of two sequential convolutional neural networks (CNNs) for the characterization of dark matter substructure in lensing galaxies from galaxy-galaxy strong gravitational lensing images. In our configuration, an initial CNN predicts the number of substructures from a gravitationally lensed image and then this number, along with the same image, is input to a second CNN which predicts the power-law slope of the substructure mass distribution function. We have trained and tested the CNNs on simulated images created by lensing a galaxy-like light distribution with a foreground galaxy mass. We find that training and testing the CNNs on images created with a fixed lens geometry allows the number of substructures and the mass function power-law slope to be retrieved well. We then explore the effect of reducing the resolution of images such that the image pixel scale is halved finding that the accuracy of the number of predicted substructures decreases by only 7% while the accuracy of the predicted mass function slope decreases by 25%. When we allow variation in lens geometry between images in the test set, to mimic more physically motivated lens samples, we observe a decrease in accuracy of the number of predicted substructures and the mass function slope of 57% and 81% respectively. We attribute this significant degradation in predicting the mass function power-law slope to the degradation in the performance of the number-predicting CNN by comparing with predictions of the slope that are made when the CNN is given the true number of substructures. We discuss future possible improvements and the impact of the computing hardware available for this work.

The Effect of Academic Performance on Athletic Success in Collegiate Athletic Programs

Published electronically November 28, 2022
DOI: 10.1137/22S1491216

Author: Derek Brickley (Lawrence University)

Project Advisors: Andrew J. Sage (Lawrence University) and Jonathan Lhost (Lawrence University)

Abstract: Since the National Collegiate Athletics Association (NCAA) was formed, student-athletes have been representing their school through athletic programs. This leads us to ask how student-athletes' academic performance impacts their teams' athletic success. Utilizing multiple mixed-effects models, we explore the relationship between academics and athletics measured using Academic Progress Rate (APR) and win percentage respectively. In doing so we find that athletic programs with a greater percentage of athletes remaining eligible and higher rates of athlete retention have higher win percentages on average

COVID-19, Crowdedness, and CMC Dining: An Agent-Based Model Approach to Reducing the Spread of COVID-19

Published electronically December 7, 2022
DOI: 10.1137/22S1483633

Authors: Reia Li (Corresponding Author – Pomona College), Ruth Efe (Claremont McKenna College), and Zintan Mwinila-Yuori (Pomona College)

Project Advisor: Christina J. Edholm (Scripps College)

Abstract: To make a COVID-safe return to in-person learning in the fall 2021 semester, Claremont McKenna College (CMC) created a new outdoor dining option to decrease the number of students inside the dining hall at one time: food trucks. However, crowding often occurs at both the inside and outside dining options. And so, we constructed an Agent-Based Model (ABM) to simulate the flow of students to the dining options over the lunch time hours. We use our ABM to investigate when over the lunch period crowding occurs and how often both or either option is crowded. Our analysis examines three different student behaviors namely, the ability to stick to an initial preference, the ability to sense crowding, and the ability to be influenced by other students. We find that the behavior that influences the level of crowding in the dining areas the most is having a strong preference for one of the dining areas. We also explore two different control measures that CMC could take to reduce crowding: adding another outdoor food option or increasing the amount of grab-and-go options. We find that adding another dining area is more effective in reducing crowding.

The Effects of Seasonality on Competition for a Limiting Resource

Published electronically December 19, 2022
DOI: 10.1137/21S1458132

Author: Lluc Briganti Wiprachtiger (University of Dundee)

Project Advisor: Lukas Eigentler (University of Dundee)

Abstract: Theoretical studies of PDE/ODE models describing ecosystem dynamics usually ignore seasonality in environmental conditions. In this paper I study a model of two generic consumer species that compete for a single limiting resource. I first consider constant resource input and then compare it to the case when resource input is dependent on time with a seasonal (periodic) pattern. The model with constant resource input is analysed analytically, by looking at the linear stability of every equilibrium. The model with seasonal resource input is analysed through numerical simulations. Results of the analysis show that seasonality has a significant effect on the outcome of the system, as when resource input is dependent on time, there could be stable coexistence, which is not possible under constant resource input. Moreover, metastable coexistence states exist for both resource input regimes if the average fitness difference between species is small. Finally, times until extinction become longer if resource input is not constant.

Numerical Computation of Fractional Derivatives of Complex-Valued Analytic Functions

Published electronically December 27, 2022
DOI: 10.1137/22S1520566

Author: Austin Higgins (Michigan Technological University)ORCIDiD_iconbw16x16.png

Project Advisors: Cecile Piret (Michigan Technological University) and Bengt Fornberg (University of Colorado)

Abstract: Highly accurate numerical approximations of analytic Caputo fractional derivatives are difficult to compute due to the upper bound singularity in its integral definition. However, it has been recently demonstrated that Caputo fractional derivatives of analytic functions can be numerically evaluated to double-precision accuracy by utilizing only function values in a grid. This is done by considering a modified Trapezoidal Rule (TR) and placing equispaced finite difference (FD) correction stencils at both endpoints. In terms of complex-valued analytic functions f(z), these fractional derivatives are multi-valued. In this paper, we provide several test functions for this numerical method of evaluating Caputo fractional derivatives. We produce figures of the principal branch of the functions’ approximated fractional derivatives, and include error plots of these approximations.

hp Gauss-Legendre Quadrature for Layer Functions

Published electronically December 27, 2022
DOI: 10.1137/22S1514866

Author: Kleio Liotati (University of Cyprus)
Project Advisor: Christos Xenophontos (University of Cyprus)

Abstract: We consider the numerical approximation of integrals involving layer functions, which appear as components in the solution of singularly perturbed boundary value problems. The hp version of the Gauss-Legendre composite quadrature, from [1], is utilized in conjunction with the Spectral Boundary Layer mesh from [2]. We show that the error goes to zero exponentially fast, as the number of Gauss points increases, independently of the singular perturbation parameter. Numerical examples illustrating the theory are also presented.