Category: Mathematics

Daniel Chavez and Eli Laird (U): An ODE-Based Model for the Spread of COVID-19 at Southern Methodist University

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Co-authors: Eli Laird

In early 2020, the SARS-CoV-2 outbreak made its way into the United States and began to rapidly breach all existing protocols for dealing with an infectious disease spreading within communities both locally and at large. As a result all academic institutions within the United States disbanded their campus and school communities so as to slow the spread of the novel virus. We modified a standard Susceptible-Exposed-Infected-Recovered (SEIR) compartmental model to examine the intertwined behaviors of both residential and community populations within university and college campuses, with a focus on Dallas' Southern Methodist University. The modified model contains a new quarantined/isolation category and an equilibrium between the susceptible and exposed categories, with a novel exposure function linking the two. Exposure rates for relevant spaces where students frequently flow through were predicted and calculated from official SMU databases and floor plans. These predictions may be used to propose disease-prevention strategies specific for college campuses.

Daniel Chavez
Majors: BS Biochemistry & BBA General Business
Faculty Mentor: Brandilyn Stigler

 

Yuanting Chen: Growth and Decay of Coherent Structures Interacting with Random Waves

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High-amplitude coherent structures have been observed in nonlinear wave systems as diverse as fluids, plasmas and optical waves in matter. We explore the interaction of disordered waves with coherent solitary waves in a nonintegrable version of the nonlinear Schrodinger equation. We show that statistical mechanics explains growth or decay of the coherent structures in detail.

Yuanting Chen
Program: Ph.D. in Mathematics
Faculty mentor: Benno Rumpf

Tyler Evans: Swelling as a stabilizing mechanism in irradiated thin films (the sequel)

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Winner: Mathematics (Graduate)

Co-author: Scott Norris

The fields of nanoscale pattern formation and nanostructural engineering are still in their infancy (relative to many other scientific areas). Much research is still centered around identifying and quantifying the relevant nanoscale mechanisms responsible for experimentally-observed results, since the same physical forces operating at the nanoscale may look very different than at the macroscale. Here, we provide further results on a recently-identified candidate mechanism (swelling, or radiation damage) that could explain the observed angle-independent lack of nanostructuring in thin films amorphized at high energy. We present new analytical and numerical results, characterization of the mechanism in its full parameter space, and an unexpected, mathematically-interesting bifurcation.

Tyler Evans
Program: PhD in Mathematics
Faculty mentor: Scott Norris

Sabrina Hetzel: Modeling the Spread of COVID-19: A University Study

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Understanding, containing, and eradicating COVID-19 is an enormous problem for the scientific community to tackle, and there are myriad facets that need to be solved. Our group in particular wishes to explore the viral spread in a university setting since it is an ideal climate for a COVID-19 outbreak. Up until recently, we aimed to understand how initial mass testing of the student body at the start of the semester, and continual testing throughout the semester reduces the spread of the virus. Looking into the future, we wish to explore the conditions and implications for a nontrivial equilibrium state for the infected population.

Sabrina Hetzel
Program: PhD in Mathematics
Faculty mentor: Alejandro Aceves

Sydney Holder (U): Descriptive Statistics for the Explanatory Factor Analysis of the MMaRS Home Use Survey

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An analysis of the explanatory factor analysis for MMaRS Home Use Survey. The research was rooted in finding whether the Home Use Survey was a reliable way of measuring at home spacial reasoning.

Sydney Holder
Majors: Applied Mathematics, Statistics, Data Science
Faculty mentor: Leanne Ketterlin Geller

Jonathan Lindbloom (U): A Bayesian Gaussian Process Model for COVID-19

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Winner: Undergraduate Top 3
Winner: Dedman III (Undergraduate)

We present the Bayesian approach to parameter inference for SIR ODE models using Markov Chain Monte Carlo (MCMC) methods, along with its computational implementation using the PyMC3 probabilistic programming library. We show how changes in the transmission rate over time can be captured by change-point models. However, these change-point models fail to learn the underlying dynamics of the time-dependent transmission rate. To overcome this pitfall, we demonstrate how using Gaussian processes to place a functional prior over the time-dependent transmission rate does a better job at characterizing uncertainty in forecasts. Our approach removes the need to specify priors over change-points, captures uncertainty in the dynamics of the effective reproduction number, and flexibly fits county or state-level data without modification. To validate our model, we evaluate the accuracy of our model’s forecasts using scoring rules and compare its performance with that of other competing models submitted to the Center for Disease Control (CDC).

Jonathan Lindbloom
Majors: Mathematics, Finance
Faculty Mentor: Alejandro Aceves

Lizuo Liu: Multiscale DNN for Stationary Navier Stokes Equations with Oscillatory Solutions

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Co-authors: Bo Wang, Wei Cai

In this talk, we develop new multi-scale deep neural network to compute oscillatory flows for stationary Navier-Stokes equation in complex domains. The multiscale neural network is the structure that can convert the high frequency components in target to low frequency components thus accelerate the convergence of training of neural network. Navier-Stokes flow with high frequency components in 2-D domain are learned by the multiscale deep neural network. The results show that the new multiscale deep neural network can be trained fast and accurate.

Lizuo Liu
Program: PhD in Mathematics
Faculty mentor: Wei Cai

Molly Robinson: Adapting Weighted Gene Co-Expression Network Analysis for Next Generation Sequencing

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Co-author: Elyssa Sliheet

New technologies such as single-cell RNA-sequencing (scRNA-seq) have become vital to the understanding of cell type heterogeneity of the brain. One limitation of this technique is that the resulting datasets are sparse. That is, genes often have read counts of zero in a given cell. Computational challenges arise when sparse data sets are analyzed with Weighted Gene Co-Expression Network Analysis (WGCNA), a technique that has been used to study the underlying genetic network of bulk data sets. This project aims to study how sparsity degrades the performance of WGCNA. This is done by modifying datasets where the method has been successfully applied. Gene clusters, or modules, of the generated network can then be tracked from the original dataset across varying levels of sparsity. This gives insight into how the network construction is altered when a sparse dataset is used. We will then study imputation and smoothing techniques to recover performance. Finally, we will seek to determine significant statistical features of the data that predict model performance.

Molly Robinson
Program: PhD in Mathematics
Faculty mentor: Andrea Barreiro

Elyssa Sliheet: Genetic Network Analysis

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Co-author: Molly Robinson

There exists the need to develop and study the effects of novel immunotherapies for the treatment of various cancer types. Unlike chemotherapy and radiotherapy which directly target the cite of a tumor, immunotherapies serve as catalysts to activate the body's immune response. Studies have shown the benefits of immunotherapies. However, there is still need to computationally analyze genetic data to identify relevant biomarkers as related to clinical significance for the overall advancement of targeted medicine. In this research, we construct and analyze important properties of a protein-protein interaction network. These properties shed light on potential biomarkers relevant to patient immune response and sensitivity.

Elyssa Sliheet
Program: PhD in Mathematics
Faculty mentor: Andrea Barreiro

Yongjia Xu (U): Computational Mathematics in Calculating Protein pKas

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A common approach to computing protein pKas uses a continuum dielectric model in which the protein is a low dielectric medium with embedded atomic point charges, the solvent is a high dielectric medium with a Boltzmann distribution of ionic charges, and the pKa is related to the electrostatic free energy which is obtained by solving the Poisson-Boltzmann equation. Starting from the model pKa for a titrating residue, the method obtains the intrinsic pKa and then computes the protonation probability for a given pH including site-site interactions. This approach assumes that acid dissociation does not affect protein conformation aside from adding or deleting charges at titratable sites. In this work we demonstrate our treecode-accelerated boundary integral (TABI) solver for the relevant electrostatic calculations. Our next step is to use machine learning to help use find patterns and better predict the pKa. We aim to make our algorithm efficient in that our protein data bank is usually very large. Careful data processing can help us filter irrelevant or less relevant features and focus more on what could potentially affect pKa values.

Yongjia Xu
Major: Mathematics
Faculty mentor: Weihua Geng

Xin Yang: Kernel Independent Treecode Accelerated Kernel Smoothing

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A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. A regular treecode algorithm computes particle-cluster or cluster-cluster interactions instead of particle-particle interactions. The KITC uses barycentric Lagrange interpolation instead for the far-field approximation when particle and cluster are well-separated. It reduces the operation count and therefore it is more efficient.

Xin Yang
Program: PhD in Mathematics
Faculty mentor: Weihua Geng

Wenzhong Zhang: FBSDE based deep neural network methods for solving high-dimensional quasilinear parabolic PDEs

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We propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs). The algorithms relies on a learning process by minimizing the path-wise difference of two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solutions, respectively.

Wenzhong Zhang
Program: PhD in Mathematics
Faculty mentor: Wei Cai