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.
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.
Published electronically February 17, 2022DOI: 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.
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.
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.
Published electronically April 27, 2022 DOI: 10.1137/21S1437470
Authors: 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.
Published electronically April 28, 2022 DOI: 10.1137/20S1376467
Authors: 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.