SIAM Undergraduate Research Online (SIURO)
Volume 6
http://dx.doi.org/10.1137/siuro.2013v6
Click on title to view PDF of paper or right click to download paper.
Undergraduate Student Research
Optimal Control in Discrete Pest Control Models
[PDF, 1.29MB]
Published electronically January 18, 2013.
http://dx.doi.org/10.1137/11S011250
Author: Kathryn Dabbs (University of Tennessee)
Sponsor: Suzanne Lenhart (University of Tennessee)
Abstract: We use discrete time models to represent the dynamics of two interacting populations, a “valuable" population and a “pest" population. We investigate optimal control in the form of decreasing the growth rate of the “pest" population with the goal of maximizing the “valuable" population while minimizing the cost of the control. We compare different types of growth functions for the “valuable" population and their impact on the optimal control.
Assessment of Statistical Methods for Water Quality Monitoring in Maryland's Tidal Waterways
[PDF, 1.84MB]
Published electronically April 17, 2013.
http://dx.doi.org/10.1137/12S012070
Author: Rosemary K. Le (Brown University), Christopher V. Rackauckas (Oberlin College), Anne S. Ross (Colorado State University) and Nehemias Ulloa (California State University, Bakersfield)
Sponsor: Matthias K. Gobbert (University of Maryland, Baltimore County)
Abstract: The Chesapeake Bay and its surrounding tributaries are home to over 3,600 species of plants and animals. In order to assess the health of the region, the Maryland Department of Natural Resources (DNR) monitors various parameters, such as dissolved oxygen, with monitoring stations located throughout the tidal waterways. Utilizing data provided by DNR, we assessed the waterways for areas of water quality concern. We analyzed the percentage of the readings taken for each parameter that failed to meet the threshold values and used the Wilcoxon Signed-Rank Test to determine the statuses of the stations. In order to assess the applicability of the Wilcoxon Test given the positive skew in the data, a simulation was performed. This simulation demonstrated that log-transforming the data prior to performing the Wilcoxon Test was not enough to reduce the Type I Error to reasonable levels. Thus, our team developed a relative ranking using a set of multiple comparison methods: a version of the Tukey Test on variance-transformed proportions, the Bonferroni adjustment method, a Bayesian method, and the Benjamini-Hochberg rejection method. From the ranking results we identified when each ranking technique is most applicable to our data.
Modeling Learning and Cooperation in Iterative Games
[PDF, 2MB]
Published electronically May 8, 2013.
http://dx.doi.org/10.1137/12S011866
Authors: Aleksey Chernobelskiy, Vineet Dixit, Agostino Cala, Siddharth Pandya, and Hector Javier Rosas (University of Arizona)
Sponsor: Scott Hottovy (University of Arizona)
Abstract: In this paper, we describe the general framework of neural networks and how such frameworks can be adapted to model human game play and the learning that takes place during iterative games. We introduce a method of pre-processing game matrices in an effort to produce cooperative strategies in games with non-cooperative dominant strategies, such as the Nash Equilibrium solution to the Prisoner's Dilemma. We find that the introduction of the pre-processed matrix increases the probability that the network plays a cooperative strategy significantly when compared to the network behavior without pre-processing.
Finding a Needle in a Haystack: An Image Processing Approach
[PDF, 2MB]
Published electronically May 28, 2013.
http://dx.doi.org/10.1137/12S0119008
Author: Emily Beylerian (University of California, Los Angeles)
Sponsor: Hayden Schaeffer (University of California, Los Angeles)
Abstract: Image segmentation (also known as object/edge detection) is the process of dividing an image into its constituent parts using information about the boundaries between objects, edges within objects, variations in intensity, et cetera. Often, the human eye can easily recognize salient information from an image; however, background variations in intensity, noise and other degradations, and other highly oscillatory features make the process of image segmentation challenging. This work is unique because we propose using a cartoon-texture-noise separation to remove highly oscillatory features from the image prior to segmentation. The cartoon and texture components can be used to analyze important information from the original image; specifically, by applying a segmentation algorithm on the cartoon component, we can extract objects from the original image. A new numerical implementation is provided for one of the two decompositions used as well as various experimental results. The method is applied to the classic example of finding a needle in a haystack, as well as real images where the texture component and noise causes problems for standard techniques.
Sensitivity to Noise in Particle Filters for 2-D Tracking Algorithms
[PDF, 5MB]
Published electronically May 28, 2013.
http://dx.doi.org/10.1137/12S012136
Authors: Dong-Hyeon Park, Stephanie Porter, and Sarah Warkentin (Harvey Mudd College)
Sponsors: Erin Byrne and Rachel Levy (Harvey Mudd College)
Abstract: A particle filter algorithm was used to simulate a Remotely Operated underwater Vehicle (ROV) tracking a moving target in 2-D space. The simulation modeled the behavior of a Sea Perch ROV modified with mounted cameras to perform blob-tracking on the target. Thirteen different noise levels were sampled for both distance and angle, with 100 trials per noise level. The angular noise demonstrated an exponential effect on the performance of the particle filter algorithm, while distance noise had minimal impact on the accuracy of the tracking.
The Effects of Spatial and Temporal Grids on Simulations of Thin Films with Surfactant
[PDF, 2MB]
Published electronically May 30, 2013.
http://dx.doi.org/10.1137/12S011878
Authors: Greg Kronmiller, Eric Autry, and Celeste Conti (Harvey Mudd College)
Sponsor: Rachel Levy (Harvey Mudd College)
Abstract: In this work, we investigated a numerical solver that combines Alternating Direction Implicit (ADI) methods with CLAWPACK to address mixed-type equations, such as the parabolic-hyperbolic system of PDEs describing surfactant spreading on a thin liquid film. In particular, we probed the effects of the spatial and temporal grid on the results of the simulations. Spatial grid effects were studied by rotating a controlled set of initial conditions relative to the grid, while temporal grid effects were studied by varying the time step and spatial resolution.
Spread of a Rumor
[PDF, 1.5MB]
Published electronically May 30, 2013.
http://dx.doi.org/10.1137/12S011829
Authors: Nickolas Fedewa, Emily Krause, and Alexandra Sisson (Central Michigan University)
Sponsor: James Angelos (Central Michigan University)
Abstract: Modelling the random spread of a rumor has a long history. In this article we consider a random process that is based on sampling without replacement leading to the use of the discrete hypergeometric distribution. First considered is the model with only spreaders and ignorants followed by more general models where there are spreaders, ignorants, and stiers. In this case a multivariate hypergeometric model is applied. It is shown that, as in the traditional case, not all ignorants hear the rumor.
M3 Challenge Introduction
[PDF, 80KB]
Waste Not, Want Not: Putting Recyclables in Their Place
[PDF, 1.5MB]
Published electronically June 5, 2013.
http://dx.doi.org/10.1137/13S012509
Authors: Jenny Lai, Abram Sanderson, Amy Xiong, Lynn Zhang, and Roy Zhao (Wayzata High School, Plymouth, MN)
Sponsor: Thomas Kilkelly (Wayzata High School, Plymouth, MN)
Summary: The increased usage of plastic, paper, and other recyclable materials, due to convenience and efficiency, has not been matched by available recycling methods. These readily disposable goods have replaced reusable products such as glassware, resulting in landfills inundated by wastes—such as plastic and Styrofoam—that are not biodegradable (Rogers). While the immense consumption of plastics is harsh on the environment, these synthetic polymers are too integrated in modern-day society to be suspended or discontinued. How might we reconcile the use of these goods with cost-efficient recycling methods for every state and township in the United States?
Our team has been asked to predict the production rate of plastic waste over time, and to forecast the amount of plastic waste present in landfills in ten years. To begin, we assumed that while an increase in population over the next ten years will increase plastic waste output, and that there is a limit on the total amount of plastic generated that is discarded. Thus our model for production rate of plastic is sigmoidal in nature, with a carrying capacity (maximum amount of plastic discarded) of 30,000 tons/year. By integrating our sigmoid function, we predicted the amount of plastic waste present in landfills in 2023 to be 1,026,000 tons.
We were also asked to design a mathematical model that could determine which recycling method is most appropriate for a city, and apply it to Fargo, ND; Price, UT; and Wichita, KS. Our approach began with the assumptions that geographic location has a negligible impact on recycling rate for each method of recycling; each city will have at least one recycling facility; the use by citizens of drop-off and curbside pickup recycling is mutually exclusive; people will recycle in the correct manner; every household has recyclable wastes; and cities may be modeled as circles. Thus our first model considered the probability that a person would recycle at a drop-off center based on distance to the center. Our second model then determined the costs of collecting and operating curbside pickup, taking into account area, population density, and total household units of each city. Analysis led to the conclusion that Price, UT should employ drop-off recycling only, while Fargo, ND and Wichita, KS should employ curbside pickup as the most cost-efficient methods. On a national scale, we must report to the EPA how our model can lead to a municipal recycling guideline policy to govern all states and townships in the United States in an effort to mitigate the problem of recyclables not being recycled. Our model is best applied to cities and townships, as the factors considered—population, area, and household density—are specified on a city and township level. Furthermore, our model should not be used on a state level as states include cities and townships of varying sizes and development, including rural and urban regions. We conducted a cost-benefit analysis of each recycling method based on city population and area. Based on our analysis, we determined that it is more cost-efficient for cities with relatively small populations to adopt drop-off recycling only, while cities with larger populations to adopt curbside pickup recycling. Therefore we recommend that the EPA allows each municipality to determine their own recycling method based on our mathematical model because the variables involved in costs of recycling are unique to each municipality. However, as a general standard, the EPA should require all cities and townships beginning in 2016 to recycle by the method best for them, in order to put recyclables in their place so that future generations are not left to deal with a world wasted away.
Computing Complex Singularities of Differential Equations with Chebfun
[PDF, 3MB]
Published electronically June 24, 2013.
http://dx.doi.org/10.1137/12S011520
Author: Marcus Webb (University of Cambridge)
Sponsor: Lloyd N. Trefethen (Oxford University)
Abstract: Given a solution to an ordinary differential equation (ODE) on a time interval, the solution for complex-valued time may be of interest, in particular whether the solution is singular at some complex time value. How can the solution be approximated in the complex plane using only the data on the interval? A polynomial approximation of the solution always fails to capture singularities; to extrapolate solutions with singularities, approximation with rational functions is more appropriate. In this paper, a robust form of rational interpolation and least-squares approximation, due to Pachón, Gonnet et al., is discussed and tested. It is found that the method avoids the issue of spurious poles found by many standard rational approximations, but that it is not suitable when a high degree of accuracy is required.
