Volume 14

DOI: 10.1137/20S1369142

Authors

Huandong Chang (Grinnell College), Chuming Chen (Grinnell College)

Project Advisors

Nicole Eikmeier (Grinnell College)

Abstract

In the field of network analysis, incorporating higher-order features into network models has become increasingly routine. In this paper we introduce the HyperType network model: an extension of a simple typing model with better clustering due to the focus on triangles instead of single edges. In addition to more realistic clustering, we empirically show HyperType retains many features from the original typing model. We empirically fit HyperType to real data, and show an interesting relationship to a recursive Kronecker product.

DOI: 10.1137/21S1423531

Authors

Collin Tully (Purdue University)

Project Advisors

Jyotirmoy Sarkar (Purdue University)

Abstract

We study symmetric random walks on the vertices of a wheel-and-spokes graph. We consider the following questions. How long does it take for the walk to go from one vertex to another? Starting from one vertex, how long does it take to visit all vertices? Having visited all vertices, how much additional time does it take to return to the starting vertex? The answers to these questions are random variables for which we desire the exact probability distributions, if possible; otherwise, we seek at least their means and standard deviations. We compare our results to those of symmetric random walks on the vertices of polygons.

DOI: 10.1137/20S133467X

Authors

Amy Creel (University of Mary Washington)

Project Advisors

Leo Lee (University of Mary Washington)

Abstract

This paper investigates a model for cellular and viral interactions following Human Immunodeficiency Virus Type 1 (HIV-1) infection. A simplified version of this model, which considers interactions between the populations of susceptible CD4+ T cells, infected CD4+ T cells, infectious virus, and non-infectious virus, under the effects of reverse transcriptase and protease inhibitors, is presented and solved analytically. The solution is obtained through an iterative method after isolating one dependent variable and performing various substitutions. Although an analytical solution is more difficult to obtain than numerical approximations, it produces ex-act results to the system of equations. As such, the analytical solution can be used to study the behavior of HIV-1 and its interactions with various treatment methods in an infected patient.

DOI: 10.1137/21S1420551

Authors

Vincent Nguyen(Simon Fraser University), Sohrab Ganjian (Simon Fraser University)

Project Advisors

Razvan Fetecau (Simon Fraser University)

Abstract

This article develops a generalizable agent-based simulation model of crime. This model is a direct application of a diffusion model of residential burglary and offender mobility by Short et al. [31] to real world crime. Offender mobility is modeled through biased random movement and crime attractiveness of a location is derived from past crimes, diffusion of information and memory decay of past events. Simulations use realistic spatial and crime data for Vancouver, Canada with an error analysis performed at the macro, meso, and micro level. The experimental results are convincing, achieving Pearson Correlation values of 0.988, 0.909, and 0.732 for each level respectively using a Truncated Levy Flight mobility framework. Without a mobility framework, we were able to achieve 0.997, 0.954, and 0.645 for these same levels.

DOI: 10.1137/20S1361584

Authors

William Li (Delbarton School, Morristown, NJ)

Project Advisors

Yingbin Liang (The Ohio State University)

Abstract

In mathematics, a dynamical system is a system in which a point moves in a geometrical space as a function of time. This paper considers one type of dynamical systems where the time dependence function is given by high dimensional ordinary differential equations. This mathematical model can be used to describe a wide variety of real world phenomena as simple as a clock pendulum or as complex as a chaotic Lorenz system. Stability is an important topic in the studies of the dynamical system. A major challenge is that the analytical solution of a time-varying nonlinear dynamical system is in general not known. Lyapunov’s direct method is a classical approach used for many decades to study stability without explicitly solving the dynamical system, and has been successfully employed in numerous applications ranging from aerospace guidance systems, chaos theory, to traffic assignment. Roughly speaking, an equilibrium is stable if an energy function monotonically decreases along the trajectory of the dynamical system. This paper extends Lyapunov’s direct method by allowing the energy function to follow a rich set of dynamics. More precisely, the paper proves two theorems, one on globally uniformly asymptotic stability and the other on stability in the sense of Lyapunov, where stability is guaranteed provided that the evolution of the energy function satisfies an inequality of a non-negative Hurwitz polynomial differential operator, which uses not only the first-order but also high-order time derivatives of the energy function. The classical Lyapunov theorems are special cases of the extended theorems. The paper provides an example in which the new theorem successfully determines stability while the classical Lyapunov’s direct method fails.

DOI: 10.1137/21S1416813

Authors

Nicole Stock (James Madison University)

Project Advisors

Minah Oh (James Madison University)

Abstract

In this paper, we construct a new family of higher order Fourier finite element spaces to discretize the axisymmetric Hodge Laplacian problems. We demonstrate that these new higher order Fourier finite element methods provide improved computational efficiency as well as increased accuracy.

DOI: 10.1137/20S1378144

Authors

Ethan Lu (Carnegie Mellon University), Jonathan Jenkins (Carnegie Mellon University), Carolyn Lee (Carnegie Mellon University), Yuxuan Liu (Carnegie Mellon University), Desmond Reed (Carnegie Mellon University)

Project Advisors

Ian Tice (Carnegie Mellon University

Abstract

In 1966, V. Arnold established an important connection between the incompressible Euler equations and a particular set of geodesic flows, using variational techniques to characterize the latter as solutions to the former. Motivated by his results, we investigate a series of similar PDEs characterizing constrained critical points of action functionals, paying particular interest to those associated with surfactant dynamics. Starting with the Arnold functional, we introduce various complications, adding terms associated to potential energies, surface tension, and surfactant momentum to derive different PDEs.

DOI: 10.1137/21S1411664

Authors

Jennifer Zheng (Emory University), Katherine Keegan (Mary Baldwin University), David Melendez (University of Central Florida)

Project Advisors

Dr. Minah Oh (James Madison University)

Abstract

We introduce the singular value decomposition (SVD) along with some of its key properties, illustrate its utility in image processing and audio processing, and show how this relates to watermarking and digital ownership protection. After establishing experimental results regarding image processing, we propose a modified version of a watermarking scheme introduced in Jain, Arora, and Panigrahi, A reliable SVD based watermarking scheme, CoRR abs/0808.0309 (2008) which offers improved robustness and imperceptibility properties.

DOI: 10.1137/21S1422847

Authors

Vivek Sreejithkumar (Florida Atlantic University)

Project Advisors

Necibe Tuncer (Florida Atlantic University)

Abstract

It is important to accurately forecast a new infection such as COVID-19 in order to effectively implement control measures. For this purpose, we study whether theepidemiological parameters such as the rate of infection, incubation period, and rate of recovery for the COVID-19 disease can be identified from daily incidences and death data. The data are obtained from the Florida Department of Health, which reports the numbers of daily COVID-19 cases and disease-induced casualties. Two mathematical models that consist of a system of ordinary differential equations are used to simulate the spread of the coronavirus in the Florida population. Structural identifiability analysis is conducted on the models to determine whether the models are well-structured to forecast the outbreak. Analysis revealed that the SEIR model is structurally identifiable, while the social distancing model is not structurally identifiable. If the model is structurally unidentifiable, it may not accurately forecast the pandemic, and in turn, may lead to inaccurate control measures. Then, the practical identifiability of parameter estimates that provide the best _t was investigated using Monte Carlo simulations. The practical identifiability analysis revealed that all of the parameters in the SEIR model are practically identifiable, but the parameters _; _E; and _ were found to be unidentifiable in the social distancing model. By comparing two models in this project, we were able to determine the effectiveness of social distancing in preventing incidences and saving lives from the disease in Florida. Furthermore, we consider how people's behavior changes over time, and how this may affect the rate of disease spread in the population. To represent this, we develop a recipe to determine the time-dependent transmission rate, _(t), from the data and introduce a methodology of how to accomplish this.

DOI: 10.1137/20S1365284

Authors

Darja Strahlberg (Hamburgh University of Technology)

Project Advisors

Dr. Michael Kolbe (Hamburg University of Technology)

Abstract

The rise of antibiotic resistance is a growing challenge for global health. Antibiotics are used for disease treatment, as well as for medical procedures, for instance, operations and transplants. The aim of this work is to compare auto-regressive integrated moving average (ARIMA) and recurrent neural net-works (RNN) to forecast the spread of drug-resistant bacterial infections at the community level. The comparison of two algorithms is performed for a multi-step time series univariate dataset. Five distinct time series were modelled, each one representing the number of episodes per single ESKAPE infecting pathogen, that has occurred quarterly between 2008 and 2018 calendar years in Germany. The forecast quality is evaluated by the root mean squared error between the forecasted values and the test data set. The experimental results show that multi-neural network forecasting RNN is significantly poorer than ARIMA for multi-step forecasting on univariate datasets. Finally, the paper provides a conclusion, that machine learning complexity is not always adding skill to the forecast. The forthcoming challenges are setting conditions when machine learning models can perform well for the real-world applications. The code used to evaluate the concept is available.

DOI: 10.1137/20S1380612

Authors

Tanav Choudhary (Singapore International School)

Project Advisors

Guillermo Goldsztein (Georgia Institute of Technology)

Abstract

In this paper, we consider the deterministic paths traversed by a particle in a randomly configured infinite cylinder. We derive a formula for the expected value of the number of loops that a particle takes around the cylinder in one period of the path given that the path is periodic.

DOI: 10.1137/21S1403242

Authors

Derek Lilienthal (California State University, Monterey Bay)

Project Advisors

Dr. Elizabeth Gooch (Naval Postgraduate School)

Abstract

In this research, we quantify foreign actors media events in the South Pacific involving an environmental theme. We use the GDELT (Global Database of Events, Language, and Tone) database to compare the tones of articles produced by Western, Chinese, and South Pacific (Local) media sources that involve an environmental theme and when a great power (United States, China, Australia, New Zealand, Japan, and Russia) is involved as an actor. We found when comparing Western, Chinese, and Local news sources, the average sentimental analysis of Western tones is negative, the average of Local tones is slightly positive, and the average of Chinese tones are very positive. When comparing the difference in means by each set of news sources, we used the Welch’s two sample t-test because the distribution of Western, Chinese, and Local tones followed a normal distribution but had unequal variances among the groups. After conducting our statistical analysis, we found there is strong evidence to conclude the difference in means of tones between the three media sources are statistically significant between each pairwise comparison.

DOI: 10.1137/21S1406192

Authors

Tianzhu Liu (Bucknell University)

Project Advisors

Lucas Waddell (Bucknell University)

Abstract

The quadratic assignment problem (QAP) is perhaps the most widely studied discrete nonlinear optimization problem, due both to its many practical applications as well as the difficulty associated with solving it. One popular approach to efficiently find good solutions to large instances of the QAP is to use genetic algorithms (GAs), which are metaheuristics based on Charles Darwin’s theory of evolution and his idea of survival of the fittest. One of the most important features of a genetic algorithm is the initial population. Some practitioners prefer to use completely random initial populations, while others prefer to seed the initial population with good heuristic solutions. While the choice of initial population method can typically have a significant impact on the performance of a GA, previous work has not clearly suggest whether GA has significant effect when solving the QAP. In this study, we provide evidence that the choice of method for creating the initial population does not affect (in a statistically significant way) the solution obtained by a GA when solving the QAP.

DOI: 10.1137/20S1383161

Authors

Benjamin Lee (University of Minnesota, Twin Cities)

Project Advisors

Fadil Santosa (Johns Hopkins)

Abstract

The Lotka-Volterra model is a system of differential equations often used to predict changes in populations of organisms in an ecosystem over time. The Lotka-Volterra model is dependent on parameters such as growth rate and species-species interactions. These parameters in turn can determine certain population dynamics such as cyclical behavior over time or even extinction of one or more species. This means that determining parameters from population measurements can provide useful predictions on how populations will change in the future. In this work, we study models that exhibit (i) stable equilibrium, (ii) limit cycles, (iii) extinction, (iv) chaos. Our goal is to understand how well the parameters in the models can be determined from population data. The approach we take is to reduce the problem to that of linear regression by estimating the time rate-of-change of the populations from data. Since the regression problem can be ill-conditioned, we consider regularization strategies to ensure stability in the parameter estimation even when there is noise in the data. Numerical experiments are conducted to gain further insights into the parameter estimation problem in the four types of behavior.

DOI: 10.1137/20S1379149

Authors

Miles Roberts (Michigan State University), Helena Seymour (Washington State University, Vancouver)

Project Advisors

Alex Dimitrov (Washington State University)

Abstract

SARS-CoV-2, the virus responsible for COVID-19, has killed hundreds of thousands of Americans. Physical distancing measures and record-setting vaccine roll out played a key role in slowing COVID-19 spread, but the advent of new SARS-CoV-2 variants remains a real threat. Implementing strategies to minimize COVID-19 hospitalizations will be key to controlling the toll of COVID-19 variants and future novel pathogens, but non-physical distancing strategies receive relatively little attention. We present a novel system of differential equations designed to predict the relative effects of hospitalizing fewer COVID-19 patients and increasing ICU bed availability on delaying ICU bed shortages. This model, which we call SEAQIRD, was calibrated to mortality data on two US states with different peak infection times from mid-March { mid-May 2020. It found that when the probability of hospitalization is already low, decreasing it further can have a large effect on delaying an ICU bed shortage in both states. Meanwhile, altering the proportion of ICU beds available to COVID-19 patients had markedly different effects on when a bed shortage was reached in the two states. This trend remained consistent when the model's most sensitive parameters were altered.

DOI: 10.1137/20S1368859

Authors

Georgia Lazaridou (corresponding, University of Cyprus), Amalia Chatzigeorgiou (University of Cyprus), Konstantina Kyriakou (University of Cyprus)

Project Advisors

Dr. Angelos Georghiou (University of Cyprus)

Abstract

Disneyland is a magical place for young and old alike. Yet the size of the park, the age restrictions of the attractions, the preferences amongst different age groups and the time a family could spend in the park, makes it difficult to decide which attractions to visit. By collecting data from the Disneyland Paris website and other reliable sources which include ratings of the attractions for each age group, we formulate an Orienteering Problem that aims to find the route achieving the maximum total rating. Our case study tries to answer two questions: (i) how should each family member individually navigate the park maximizing his/hers total rating, and (ii) how should a family as a single group navigate the park such that all family members achieve their goals. Our results show that if the family stay as a single group, the loss in total rating is negligible compared to each family member navigating the park individually.

DOI: 10.1137/20S1364047

Authors

Lauren Schmiedeler (St Louis University), Ellen Stonner (St Louis University), Nolan Murphy (St Louis University)

Project Advisors

Benjamin Hutz (St Louis University)

Abstract

Consider a "forest" of infinitely thin trees arranged on the lattice Z x Z. If you are standing at the origin, (0; 0), not all trees are visible despite the fact that they are infinitely thin. In particular, of the trees all lying on a line through (0; 0), only one such point is visible. In this article we conclusively classify all closest occurring invisible rectangular n x m blocks of points for 1 _ n;m _ 4. This (partially) resolves a question posed by Goins-Harris-Kubik-Mbirika. Furthermore, we compile statistics for all occurring arrangements up to size 4_4 and discuss interesting patterns that appear in that data.

DOI: 10.1137/21S1417922

Authors

Edward Wang (Livingston High School, Livingston, NJ) , Charles Yu (Livingston High School, Livingston, NJ) , Aditya Desai (Livingston High School, Livingston, NJ) , Sidhant Srivastava (Livingston High School, Livingston, NJ) , Leo Stepanewk (Livingston High School, Livingston, NJ)

Project Advisors

Cheryl Coursen (Livingston High School, Livingston, NJ)

Abstract

As the world becomes increasingly reliant on the internet, from online schooling to working from home, broader and higher quality access has never been more important. However, expanding internet infrastructure presents a unique challenge in terms of cost, economic efficiency, and capacity requirements. Our team aims to optimize the process of improving connectivity by predicting the price of bandwidth over the next 10 years, calculating band-width needs for a variety of household scenarios, and determining the best distribution of cellular nodes over a given region.

M3 Challenge Introduction

DOI: 10.1137/20S1368227

Authors

Keegan Kresge (Rochester Institute of Technology), Natalie Petruzelli (St. John Fisher College)

Project Advisors

Dr. Eben Kenah (The Ohio State University))

Abstract

COVID-19 epidemics in many parts of the United States and the world have shown unexpected shifts from exponential to linear growth in the number of daily new cases. Epidemics on configuration model networks typically produce exponential growth, while epidemics on lattices produce linear growth. We explore a network-based epidemic model that interpolates between lattice-like and configuration model networks while keeping the degree distribution and basic reproduction number (R0) constant. This model starts with nodes assigned random locations in a unit square and connected to their nearest neighbors. A proportion p of the edges are disconnected and reconnected in a configuration model subnetwork. As p increases, we observe a shift from linear to exponential growth. Realistic human contact networks involve many local interactions and fewer long-distance interactions, so social distancing affects both the effective reproduction number Rt and the proportion of long-distance connections in the network. While the impact of changes in Rt is well-understood, far less is understood about the effect of more subtle changes in network structure. Our analysis indicates that the threshold between linear and exponential growth may occur even with a small percentage of reconfigured edges. Additionally, the number of total infected individuals in an epidemic substantially increases around this threshold even when R0 remains constant. This study reveals that implementing and relaxing social distancing restrictions can have more complex and dramatic effects on epidemic dynamics than previously thought.

DOI: 10.1137/20S1357342

Authors

Jennifer Lew (Palos Verdes Peninsula High School)

Project Advisors

Derek Fong (California Public Utilities Commission)

Abstract

The analytical solution for the large deformation of a cantilever beam under a point load, typically applied to the tip of the cantilever and perpendicularly to its axis, has been widely studied and published. However, the more complex case of two angled point loads applied to the cantilever has not been published. The current research delved into the following scenario: an upright cantilever, e.g. a pole, has point loads applied at two locations on the cantilever, where each point load is angled, i.e. the point load has both a horizontal component (which may result from wind loading) and downward vertical component of force (such as from weights). The aim of the research is to develop a methodology for finding, At the two locations where the point loads are applied, the angle of deflection, the horizontal deflection, and the vertically deflected height. Ultimately, the research yielded a methodology based on the Complete and Incomplete Elliptic Integrals of the First Kind and Second Kind. The analytical solution developed in this research - specifically the method for calculating the angles of deflections - was compared against Finite Element Analysis and was found to produce nearly identical results. We conclude that the methodology shown can be extended to any number of point loads and will be a contribution to the field of non-linear mechanics.

DOI: 10.1137/20S1369841

Authors

Henry Stewart (Emory University), Megan Johnston (Emory University), Jesse Sun (Emory University), David Zhang (Emory University)

Project Advisors

Alessandro Veneziani (Emory University)

Abstract

Since the end of 2019, COVID-19 has threatened human life around the globe. As the death toll continues to rise, development of vaccines and antiviral treatments have progressed at unprecedented speeds. This paper uses an SIR-type model, extended to include asymptomatic carrier and deceased populations as a basis for expansion to the effects of a time-dependent drug or vaccine. In our model, a drug is administered to symptomatically infected individuals, decreasing recovery time and death rate. Alternatively, a vaccine is administered to susceptible individuals and, if effective, will move them into the recovered population. We observe final mortality outcomes of these countermeasures by running simulations across different release times with differing effectivenesses. As expected, the earlier the drug or vaccine is released into the population, the smaller the death toll. We find that for earlier release dates, difference in the quality of either treatment has a large effect on total deaths. However as their release is delayed, these differences become smaller. Finally, we find that a vaccine is much more effective than a drug when released early in an epidemic. However, when released after the peak of infections, a drug is marginally more effective in total lives

DOI: 10.1137/20S1362930

Authors

Shane Doyle (National University of Ireland, Galway)

Project Advisors

Dr. Petri Piiroinen (Chalmers University of Technology)

Abstract

The spread of the novel coronavirus SARS-CoV-19 throughout a population can be modelled through the use of compartment models. Here we will use age-cohort separation to design a system of ordinary differential equations, which will be solved with numerical methods in order to model the spread of the virus in Ireland by age-cohort. From here we analyse policy decisions made by the Irish Government throughout the COVID-19 pandemic in early 2020 in terms of their effect on differently aged people within the population. Simulations are generated of alternative policies that could be enacted in the future, with the aim of analysing the effectiveness of policies such as lockdown and cocooning. The results of this analysis indicate that a reduction in social interaction is a major driving force in the suppression of new infections and that reducing the contacts of vulnerable members of the population leads to a slower rate of increase in infections for the population at large. The testing for the model is done by varying the level of social interaction within the population over a 160 day interval from February 29th, 2020 until August 7th, 2020, with all projections past this date based on assumptions made relating to future levels of social interaction resulting from future policies.

DOI: 10.1137/20S1367155

Authors

Manuel Santana (Utah State University), Abby Brauer (Lewis and Clark University), Megan Krawick (Youngstown State University)

Project Advisors

Jun Kitagawa (Michigan State University)

Abstract

Numerical methods for the optimal transport problem is an active area of research. Recent work of Kitagawa and Abedin shows that the solution of a time-dependent equation converges exponentially fast as time goes to infinity to the solution of the optimal transport problem. This suggests a fast numerical algorithm for computing optimal maps; we investigate such an algorithm here in the 1-dimensional case. Specifically, we use a finite-difference scheme to solve the time-dependent optimal transport problem and carry out an error analysis of the scheme. A collection of numerical examples is also presented and discussed.

DOI: 10.1137/20S1361870

Authors

Srinath Mahankali (Stuyvesant High School)

Project Advisors

Yunan Yang (Courant Institute of Mathematical Sciences)

Abstract

Full-waveform inversion (FWI) is a method used to determine properties of the Earth from information on the surface. We use the squared Wasserstein distance (squared W2 distance) as an objective function to invert for the velocity of seismic waves as a function of position in the Earth, and we discuss its convexity with respect to the velocity parameter. In one dimension, we consider constant, piecewise increasing, and linearly increasing velocity models as a function of position, and we show the convexity of the squared W2 distance with respect to the velocity parameter on the interval from zero to the true value of the velocity parameter when the source function is a probability measure. Furthermore, we consider a two-dimensional model where velocity is linearly increasing as a function of depth and prove the convexity of the squared W2 distance in the velocity parameter on large regions containing the true value. We discuss the convexity of the squared W2 distance compared with the convexity of the squared L2 norm, and we discuss the relationship between frequency and convexity of these respective distances. We also discuss multiple approaches to optimal transport for non-probability measures by first converting the wave data into probability measures.

DOI: 10.1137/20S137208110.1137/20S1372081

Authors

Timothy Getscher (Woods Hole Oceanographic Institution)

Project Advisors

Kevin McIlhany (United States Naval Academy)

Abstract

This paper compares the advantages, limitations, and computational considerations of using Finite-Time Lyapunov Exponents (FTLEs) and Lagrangian Descriptors (LDs) as tools for identifying barriers and mechanisms of fluid transport in two-dimensional time-periodic ows. These barriers and mechanisms of transport are often referred to as “Lagrangian Coherent Structures," though this term often changes meaning depending on the author or context. This paper will specifically focus on using FTLEs and LDs to identify stable and unstable manifolds of hyperbolic stagnation points, and the Kolmogorov-Arnold-Moser (KAM) tori associated with elliptic stagnation points. The background and theory behind both methods and their associated phase space structures will be presented, and then examples of FTLEs and LDs will be shown based on a simple, periodic, time-dependent double-gyre toy model with varying parameters.

DOI: 10.1137/20S1367040

Authors

Vishnu Nittoor (The International School Bangalore)

Project Advisors

Dr. Tim Chartier (Davidson)

Abstract

This paper investigates the effect of increasing competitive balance on the reliability of tournament rankings. Reliability of rankings, a previously qualitative property, is quantified in this paper by the closeness between ground truth rankings and the rankings of teams at the end of a tournament. Three metrics are used to measure this closeness: Spearman’s rank correlation coefficient, Kendall’s tau, and a relatively unused algorithm in the field of ranking: Levenshtein distance. Three tournament structures are simulated: round-robin, random pairings, and the Swiss system. The tournaments are simulated across multiple trials and over a varying number of games. It is found that the rate of growth of reliability of a tournament structure falls as the number of games increases. It is also found that there is a positive relationship between competitive imbalance and reliability. The marginal benefit of increasing competitive imbalance falls as it is increased. Unexpectedly, in comparison to random pairings and Swiss pairings, the round-robin tournament structure is seen to achieve the highest reliability score across all metrics and number of games played. The difference in reliability between the tournament structures increases as competitive imbalance is increased. The further work suggested includes investigation of tournament outcome uncertainty in conjunction with reliability and competitive balance, a closer study into Levenshtein distance as a useful algorithm to quantify closeness between two rankings, and an inquiry into the specific factors that bottleneck reliability while the number of games played in a tournament increases.

DOI: 10.1137/20S1357974

Authors

Riley Fisher (Duke University), Mckenzie Garcia (Duke University), Nagaprasad Rudrapatna (Duke University)

Project Advisors

Dr. Saulo Orizaga (New Mexico Tech)

Abstract

This paper considers schemes based on the convexity splitting technique – linear extrapolation, second order backward difference formulas, and implicit-explicit Runge-Kutta methods – for solving the Cahn-Hilliard equation, a model for phase separation, with periodic boundary conditions. The Cahn-Hilliard equation, which is derived from a gradient flow of an energy functional, is a higher-order, parabolic nonlinear partial differential equation. As such, each of the proposed solvers preserve the energy decreasing property of this equation. The solution methods, which were implemented by incorporating fast Fourier transformations, were compared by means of accuracy and computational runtime. Furthermore, the convexity splitting parameter, a, was varied in order to observe its effects on the accuracy and stability of numerical solvers. The best method for modest accuracy was determined to be the first-order linear extrapolation method. The preferred high-order solver was an implicit-explicit Runge-Kutta scheme, which had a relatively low computational cost. Simulations confirmed that this method preserves stability at larger timesteps whilemaintaining a high degree of accuracy at both large and small timesteps.

DOI: 10.1137/19S1300571

Authors

Justin Bennett (Arizona State University), Karissa Gund (Arizona State University), Jingteng iLu (Arizona State University), Xixu Hu (University of Science and Technology of China), Anya Porter (Harvey Mud College)

Project Advisors

Yang Kuang (Arizona State University)

Abstract

Over time, tumor treatment resistance inadvertently develops when androgen de-privation therapy (ADT) is applied to metastasized prostate cancer (PCa). To combat tumor resistance, while reducing the harsh side effects of hormone therapy, the clinician may opt to cyclically alternates the patient’s treatment on and off. This method, known as intermittent ADT, is an alternative to continuous ADT that improves the patient’s quality of life while testosterone levels recover between cycles. In this paper, we explore the response of intermittent ADT to metastasized prostate cancer by employing a previously clinical data validated mathematical model to new clinical data from patients undergoing Abiraterone therapy. This cell quota model, a system of ordinary differential equations constructed using Droop’s nutrient limiting theory, assumes the tumor comprises of castration-sensitive (CS) and castration-resistant (CR) cancer sub-populations. The two sub-populations rely on varying levels of intracellular androgen for growth, death and transformation. Due to the complexity of the model, we carry out sensitivity analyses to study the effect of certain parameters on their outputs, and to increase the identifiability of each patient’s unique parameter set. The model’s forecasting results show consistent accuracy for patients with sufficient data, which means the model could give useful information in practice, especially to decide whether an additional round of treatment would be effective.

DOI: 10.1137/20S1363728

Authors

Elif Sensoy (Wilbur Wright College)

Project Advisors

Dr. Hellen Colman (Wilbur Wright College)

Abstract

We exhibit an algorithm with continuous instructions for two robots moving without collisions on a track shaped as a wedge of three circles. We show that the topological complexity of the configuration space associated with this problem is 3. The topological complexity is a homotopy invariant that can be thought of as the minimum number of continuous instructions required to describe the movement of the robots between any initial configuration to any final one without collisions. The algorithm presented is optimal in the sense that it requires exactly 3 continuous instructions.

DOI: 10.1137/19S1266885

Authors

Justin Park (MIT)

Project Advisors

Dr. Kasso Okoudjou (University of Maryland, College Park)

Abstract

We expand on a prior result about the cardinalities of harmonic tight frames generated from the discrete Fourier transform. Harmonic finite unit-norm tight frames (FUNTFs) constructed from the first two rows of the M ×M discrete Fourier transform have previously been described and characterized as prime or divisible, where M ≥ 2 is an integer. We generalize the result to any choice of two rows b and c for which c−b has up to two distinct prime factors. These new results allow for much more flexibility in constructing harmonic FUNTFs from M-th roots of unity.

DOI: 10.1137/20S1353344

Authors

John Nguyen (University of Minnesota, Twin Cities) , Aileen Zebrowski (University of Minnesota, Twin Cities)

Project Advisors

Dr. Kaitlin Hill (Wake Forest University)

Abstract

Permafrost is a thick layer of soil that is frozen throughout the year and covers significant portions of the northern hemisphere. Currently, there is a large amount of carbon trapped in the permafrost, and as permafrost melts, a significant portion of this carbon will be released into the atmosphere as either carbon dioxide or methane. We use empirical data to estimate that, on average, permafrost currently extends from the arctic to latitude 61_N. We propose an adaption to the Budyko energy balance model to study the impacts of receding permafrost. We track the steady-state latitude of both the permafrost line and the snow line as greenhouse gas emissions, and consequently, global mean temperature increases. Using the change in permafrost surface area, we are able to quantify the total carbon feedback of melting permafrost. Focusing our analysis on scenarios described in recent IPCC reports and the Paris Climate Agreement, we use change in the permafrost line latitude to estimate the amount of carbon dioxide released by the melted permafrost. Similarly, we use the snow line to calculate the minimum average global temperature that would cause the ice caps to completely melt. We find that our adaption of the Budyko model produces estimates of carbon dioxide emissions within the range of projections of models with higher complexity.

DOI: 10.1137/20S1360578

Authors

Foyez Alauddin (Trinity School, NYC)

Project Advisors

Gleb Pogudin (Polytechnique)

Abstract

Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. It has been recently used as a preprocessing step for new model order reduction methods, so it is important to keep the number of new variables small. Several algorithms have been designed to search for a quadratization with the new variables being monomials in the original variables. To understand the limitations and potential ways of improving such algorithms, we study the following question: can quadratizations with not necessarily new monomial variables produce a model of substantially smaller dimension than quadratization with only new monomial variables? To do this, we restrict our attention to scalar polynomial ODEs. Our first result is that a scalar polynomial ODE x_ = p(x) = anxn + an1xn1 + : : : + a0 with n > 5 and an 6= 0 can be quadratized using exactly one new variable if and only if p(x an1 n_an ) = anxn + ax2 + bx for some a; b 2 C. In fact, the new variable can be taken as z := (x +an1 n_an)n1. Our second result is that two new non-monomial variables are enough to quadratize all degree 6 scalar polynomial ODEs. Based on these results, we observe that a quadratization with not necessarily new monomial variables can be much smaller than a monomial quadratization even for scalar ODEs. The main results of the paper have been discovered using computational methods of applied nonlinear algebra (Gröbner bases), and we describe these computations.