SIAM Undergraduate Research Online

Volume 14

SIAM Undergraduate Research Online Volume 12

Adapting the Budyko Model to Analyze Permafrost Recession and Potential for Carbon Feedback

Published electronically January 12, 2021
DOI: 10.1137/20S1353344

Authors: John Nguyen and Aileen Zebrowski (University of Minnesota, Twin Cities) 
Sponsor: 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.

Characterization of the Divisibility of DFT HTFs for C2

Published electronically January 19, 2021
DOI: 10.1137/19S1266885

Authors: Justin Park (MIT) 
Sponsor: 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.

Collision Free Motion Planning on a Wedge of Circles

Published electronically January 27, 2021
DOI: 10.1137/20S1363728

Authors: Elif Sensoy (Wilbur Wright College) 
Sponsor: 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.

Quadratization of ODEs: Monomial vs. Non-Monomial

Published electronically January 12, 2021
DOI: 10.1137/20S1360578

Authors: Foyez Alauddin (Trinity School, NYC)
Sponsor: 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.

Clinical Data Validated Mathematical Model for Intermittent Abiraterone Response in Castration-Resistant Prostate Cancer Patients

Published electronically February 22, 2021
DOI: 10.1137/19S1300571

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

Comparing Numerical Solution Methods for the Cahn-Hilliard Equation

Published electronically February 25, 2021


Authors: Riley Fisher, Mckenzie Garcia, Nagaprasad Rudrapatna (Duke University)
Sponsor: 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.

A Monte-Carlo analysis of competitive balance and reliability across tournament structures

Published electronically March 1, 2021
DOI: 10.1137/20S1367040

Authors: Vishnu Nittoor (The International School Bangalore)
Sponsor: 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.

The convexity of optimal transport-based waveform inversion for certain structured velocity models

Published electronically March 18, 2021
DOI: 10.1137/20S1361870

Authors: Srinath Mahankali (Stuyvesant High School)
Sponsor: 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.

Comparing Finite-Time Lyapunov Exponents and Lagrangian Descriptors for identifying phase space structures in a simple two-dimensional, time-periodic double-gyre model

Published electronically March 18, 2021
DOI: 10.1137/20S137208110.1137/20S1372081

Author: Timothy Getscher (Woods Hole Oceanographic Institution)
Project Advisor: 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.

Numerical Analysis of the Parabolic 1-D Optimal Transport Problem

Published electronically March 23, 2021
DOI: 10.1137/20S1367155

Authors: Manuel Santana (Utah State University), Abby Brauer (Lewis and Clark University), and Megan Krawick (Youngstown State University)
Project Advisor: 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.

Predicting the Spread of COVID-19 in Ireland Using an Age-Cohort SEIRD Model

Published electronically April 16, 2021
DOI: 10.1137/20S1362930

Author: Shane Doyle (National University of Ireland, Galway)
Project Advisor: 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.

Analytical Solution to the Large Deformation of Cantilever Beams with Two Angled Point Force

Published electronically May 11. 2021
DOI: 10.1137/20S1357342

Author: Jennifer Lew (Palos Verdes Peninsula High School)
Project Advisor: 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.

SIR Model of Time Dependent Drug and Vaccine Distribution on COVID-19

Published electronically May 11, 2021
DOI: 10.1137/20S1369841

Authors: Henry Stewart, Megan Johnston, Jesse Sun, and David Zhang (Emory University)
Project Advisor: 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

Analyzing Epidemic Thresholds on Dynamic Network Structures

Published electronically July 12, 2021
DOI: 10.1137/20S1368227

Authors: Keegan Kresge (Rochester Institute of Technology) and Natalie Petruzelli (St. John Fisher College)
Project Advisor: 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.

Defeating the Digital Divide

Published electronically July 28, 2021
DOI: 10.1137/21S1417922 
M3 Introduction

Authors: Edward Wang, Charles Yu, Aditya Desai, Sidhant Srivastava, and Leo Stepanewk (Livingston High School, Livingston, NJ) 
Sponsor: 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.

First Occurrence and Frequency of Invisible Lattice Point Patterns

Published electronically July 28, 2021
DOI: 10.1137/20S1364047

Authors: Lauren Schmiedeler, Ellen Stonner, and Nolan Murphy (St Louis University)
Project Advisor: 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.