SIAM Undergraduate Research Online

Volume 16

SIAM Undergraduate Research Online Volume 12

Symmetry and Free Boundary Points in a Class of Linear Ordinary Differential Equations

Published electronically January 10, 2023
DOI: 10.1137/21S1454110

Authors: Feng Jiang (University of Nottingham Ningbo China), Zhengyang Guo (University of Nottingham Ningbo China), Dong’ang Liu (University of Nottingham Ningbo China), and Yanghao Wang (University of Nottingham Ningbo China)
Project Advisors: Behrouz Emamizadeh (University of Nottingham Ningbo China) and Amin Farjudian (University of Nottingham Ningbo China)

Abstract: This note is concerned with the qualitative properties of the solutions of a class of linear ordinary differential equations. The existence and uniqueness of solutions are addressed, and properties of the graph of the solution when imposing some restrictions are derived. A new notion of derivative, called the force derivative, is introduced and an orthogonality result, between the force derivative of the solution and the force function, is obtained. All the important results are verified by numerical examples using MATLAB. Finally, an inequality result reminiscent of the famous G. Talenti's inequality is proved.

Food Deserts and k-Means Clustering

Published electronically March 9, 2023
DOI: 10.1137/22S1504445

Authors: Garrett Kepler (California State University, East Bay), Maria Palomino (California State University, East Bay)
Project Advisor: Andrea Arauza Rivera (California State University, East Bay)

Abstract: Food deserts are regions where people lack access to healthy foods. In this article we use k-means clustering to cluster the food deserts in two Bay Area counties. The centroids (means) of these clusters are optimal locations for intervention sites (such as food pantries) since they minimize the distance that a person within a food desert cluster would need to travel to reach the resources they require. We present the results of both a standard and a weighted k-means clustering algorithm. The weighted algorithm takes into account the poverty levels in each food desert when determining the placement of a centroid. We find that this weighting can make significant changes to the proposed locations of intervention sites.

Multi-scale Hybridized Topic Modeling: A Pipeline for Analyzing Unstructured Text Datasets via Topic Modeling

Published electronically March 27, 2023
DOI: 10.1137/22S1536832

Authors: Keyi Cheng (University of California, Los Angeles), Stefan Inzer (University of California, Berkeley), Adrian Leung (University of California, Los Angeles), and Xiaoxian Shen (University of California, Los Angeles)
Project advisors: Deanna Needell (University of California, Los Angeles), Todd Presner (University of California, Los Angeles), Michael Perlmutter (University of California, Los Angeles), Michael R. Lindstrom (University of California, Los Angeles), and Joyce Chew (University of California, Los Angeles)

Abstract: We propose a multi-scale hybridized topic modeling method to find hidden topics from transcribed interviews more accurately and more efficiently than traditional topic modeling methods. Our multiscale hybridized topic modeling method (MSHTM) approaches data at different scales and performs topic modeling in a hierarchical way utilizing first a classical method, Nonnegative Matrix Factorization, and then a transformer-based method, BERTopic. It harnesses the strengths of both NMF and BERTopic. Our method can help researchers and the public better extract and interpret the interview information. Additionally, it provides insights for new indexing systems based on the topic level. We then deploy our method on real-world interview transcripts and find promising results.

Numerical Analysis of Crowding Effects in Competing Species

Published electronically March 10, 2023
DOI: 10.1137/22S151042X

Author: Braden Carlson (Southern Utah University)
Project Advisors: Jianlong Han (Southern Utah University), Seth Armstrong (Southern Utah University), and Sarah Duffin (Southern Utah University)

Abstract: In recent decades, scientists have observed that the mortality rate of some competing species increases superlinearly as populations grow to unsustainable levels. This is modeled by terms representing crowding effects in a system of nonlinear differential equations that describes population growth of two species competing for resources under the effects of crowding. After applying nondimensionalization to reduce parameters in the system, the stability of the steady state solutions of the system is examined. A semi-implicit numerical scheme is proposed which guarantees the positivity of the solutions. The long term behavior of the numerical solutions is studied. The error estimate between the numerical solution and the true solution is given.

Quantifying Uncertainty in Ensemble Deep Learning

Published electronically April 17, 2023
DOI: 10.1137/22S1531816

Authors: Emily Diegel (Embry Riddle Aeronautical University), Rhiannon Hicks (Embry Riddle Aeronautical University), Max Prilutsky (San Diego State University), and Rachel Swan (Embry Riddle Aeronautical University)
Project Advisor: Mihhail Berezovski (Embry Riddle Aeronautical University)

Abstract: Neural networks are an emerging topic in the data science industry due to their high versatility and efficiency with large data sets. Past research has utilized machine learning on experimental data in the material sciences and chemistry field to predict properties of metal oxides. Neural networks can determine underlying optical properties in complex images of metal oxides and capture essential features which are unrecognizable by observation. However, neural networks are often referred to as a “black box algorithm” due to the underlying process during the training of the model. This poses a concern on how robust and reliable the prediction model actually is. To solve this ensemble neural networks were created. By utilizing multiple networks instead of one the robustness of the model was increased and points of uncertainty were identified. Overall, ensemble neural networks outperform singular networks and demonstrate areas of uncertainty and robustness in the model.

A Quasi-Optimal Spectral Solver for the Heat and Poisson Equations in a Closed Cylinder

Published electronically April 26, 2023
DOI: 10.1137/22S1502070

Author: David Darrow (Massachusetts Institute of Technology)
Project Advisors: Alex Townsend (Cornell University) and Grady Wright (Boise State University)

Abstract: We develop a spectral method to solve the heat equation in a closed cylinder, achieving a quasi-optimal O(N log N) complexity and high-order, spectral accuracy. The algorithm relies on a Chebyshev–Chebyshev–Fourier (CCF) discretization of the cylinder, which is easily implemented and decouples the heat equation into a collection of smaller, sparse Sylvester equations. In turn, each of these equations is solved using the alternating direction implicit (ADI) method in quasi-optimal time; overall, this represents an improvement in the heat equation solver from O(N4/3) (in previous Chebyshev-based methods) to O(N log N). While Legendre-based methods have recently been developed to achieve similar computation times, our Chebyshev discretization allows for far faster coefficient transforms; this is critical in applications with non-linear forcing, which we discuss in the context of the incompressible Navier–Stokes equations. Lastly, we provide numerical simulations of the heat equation, demonstrating significant speed-ups over traditional spectral collocation methods and finite difference methods.

Opinion Dynamics with Slowly Evolving Zealot Populations

Published electronically May 4, 2023
DOI: 10.1137/22S1515306

Author: Ashlyn DeGroot (Calvin University) and Emma Schmidt (Calvin University)
Project Advisor: Todd Kapitula (Calvin University)

Abstract:We introduce and analyze a model for opinion dynamics comprised of nonlinear ODEs. The variables are the proportion of moderates in the population who hold opinion A, the proportion of zealots who hold opinion A, and the proportion of zealots who hold opinion B (not A). The zealots are willing to change their opinion at a much slower rate than the moderates. Our model takes into account such things as the inherent attractiveness of one opinion over the other, the indoctrination of moderates by the zealots, and deradicalization of the zealots by the moderates. A combination of theoretical and numerical analysis shows there are many different types of asymptotic configurations of the population. Many of these correspond to critical points of the system. The most intriguing finding is that if both A and B are roughly equally attractive, and the rate of indoctrination is roughly equal to the rate of deradicalization, then there will be a stable periodic orbit. The dynamics of this orbit show that a precursor to an opinion being dominant is that the proportion of zealots for the opinion must first grow to some critical value. Moreover, when the periodic orbit exists, there are no other solutions which allow for coexistence between the two opinions.

Predicting Molecular Phenotypes with Single Cell RNA Sequencing Data: An Assessment of Unsupervised Machine Learning Models

Published electronically May 26, 2023
DOI: 10.1137/21S1439985

Author: Anastasia Dunca (Massachusetts Institute of Technology)
Project Advisor: Frederick R. Adler (University of Utah)

Abstract:According to the National Cancer Institute, there were 9.5 million cancer-related deaths in 2018. A challenge in improving treatment is resistance in genetically unstable cells. The purpose of this study is to evaluate unsupervised machine learning on classifying treatment-resistant phenotypes in heterogeneous tumors. This is done with analysis of single cell RNA sequencing (scRNAseq) data using a pipeline and evaluation metrics. scRNAseq quantifies mRNA in cells and characterizes cell phenotypes. One scRNAseq dataset is used (tumor/non-tumor cells of different molecular subtypes and patient identifications). The pipeline consists of data filtering, dimensionality reduction with Principal Component Analysis, projection with Uniform Manifold Approximation and Projection, clustering, and evaluation. Nine approaches for clustering (Ward, BIRCH, Gaussian Mixture Model, DBSCAN, Spectral Clustering, Affinity Propagation, Agglomerative Clustering, Mean Shift, and K-Means) are evaluated. Seven models divided tumor versus non-tumor cells and molecular subtype while six models classified different patient identification (13 of which were presented in the dataset); K-Means, Ward, and BIRCH often ranked highest with ∼ 80% accuracy on the tumor versus non-tumor task and ∼ 60% for molecular subtype and patient ID. An optimized classification pipeline using K-Means, Ward, and BIRCH models was evaluated to be most effective for further analysis. In clinical research where there is currently no standard protocol for scRNAseq analysis, clusters generated from this optimized pipeline can be used to understand cancer cell behavior and malignant growth, directly affecting the success of treatment.