Research
Program
Catherine A. Roberts
My training is as an applied mathematician. Below you can read descriptions of my research. As you will see, some of these projects provide opportunities for students to participate in research. If you are interested in any of these projects, please email me.
You can also read about several projects that were conducted by students prior to my arrival at Holy Cross in Fall 2001. Brief descriptions are included here at the bottom of this page.
Explosion, Quenching and Absorption
Some reprints are available on my vita.
The objective of these studies is to investigate the phenomena of explosion, quenching and absorption within a diffusive medium in the context of nonlinear Volterra integral equations. These problems are characterized by spatially localized nonlinearities, which distinguishes them from previous work. Such nonlinearities are motivated, for example, by applications where the event occurs within a very confined area (e.g. from an internal electrode energy stimulation within a flame or on a heated surface).
When explosion, quenching or absorption occurs within a diffusive medium, the system undergoes dramatic change. In explosion, the solution becomes unbounded in finite time. In quenching, the solution remains bounded, whereas its derivative in time blows-up at some finite moment. In absorption, the solution vanishes in finite time in a region called a dead-core. These phenomena are of wide importance in applications.
Studies of explosion, and to a lesser extent quenching and absorption, have been conducted on theoretical models that are originally presented as nonlinear partial differential equations. Unlike previous studies, which have required a smoothness property for spatially dependent nonlinearities, future research replaces such requirements by strongly localized behavior. In this case, conversion of the theoretical models from partial differential equations to integral equations represents a very effective format for the analysis. This reformulation permits a more direct and efficient inquiry into the challenging scenario of spatially localized nonlinear behavior.
Prior work (see vita) on the phenomena of explosion solutions for Volterra equations inform the development of viable techniques to address the newer circumstances of quenching and absorption. We have treated systems of Volterra equations, as well as the influence of non-local nonlinearities on explosion. In addition, we have examined several classes of quenching problems using similar analytic techniques. To date, for certain classes of nonlinear Volterra equations, we have demonstrated the existence of a quenching solution, established lower and upper estimates for the critical time to quenching, and characterized the solution growth rates for various nonlinearities. The resulting integral equations have corresponded to one-dimensional models with a spatially localized nonlinearity as either the source or boundary condition.
Future research will derive the appropriate Volterra equation formulation for a variety of new problems. For example, we will consider quenching problems posed in more general domains and in two or three spatial dimensions. A wider class of nonlinearities, such as the inclusion of a logarithmic component and the incorporation of spatial derivative dependence, will be examined. The influence of several source terms will also be explored. We will characterize the classes of nonlinearities leading to various quenching or absorption solutions. Such models are physically relevant yet have not been explored mathematically. Moreover, the analytic techniques under development for explosion problems are modifiable for use in these new problems. Future research will characterize sufficient conditions to guarantee quenching and estimate absorption rates for a variety of nonlinearities. The results will characterize the spatial results for each of the physical applications because it is useful to determine conditions for when quenching or absorption occurs at a single point, a finite region or everywhere in the domain. Hyperbolic partial differential equations with a similar strongly spatially localized nonlinearity have not been considered in the explosion literature.
The methods used above are currently being modified to address problems of this sort. Interestingly, analysis leads to a nonlinear Volterra equation that can also be represented in terms of a delay differential equation. This class of delay differential equations has not been previously examined for unbounded solutions. It is anticipated that contributions along these lines will naturally follow from examination of the hyperbolic problem. The primary tools for this research come from functional, numerical and asymptotic analyses. A new direction of interest for us is to examine these equations from a numerical point of view. Contraction mappings are used to establish a lower limit on existence of a continuous bounded solution. Techniques of comparison and contradiction are applied to prove nonexistence of a continuous bounded solution beyond a certain point in time. Asymptotic expansions of integrals are used to determine the growth rate behavior of the solutions in the key limit. New tools will be developed in this research to strengthen the bounds for the critical time and to examine new questions regarding the solutions of these integral equations.
This work is conducted collaboratively. Current collaborators include W. Okrasinski (Univ. of Zielona Gora, Poland), W. E. Olmstead (Northwestern University), Colleen Kirk (Montclair State University), and Kelly Fuller (Nazarene College).
STUDENT PARTICIPATION: One doctoral student, Kelly Fuller, received her Ph.D. from me in 1996. She started working on her project when I was a faculty member at the University of Rhode Island. She is now a professor at Nazareth College in New York.
Mathematical Modeling: Human-Environment Interactions
Some reprints are available on my vita.
This research seeks to characterize complex human-environment interactions on a river. Using mathematics and statistics, the intent of this research is to develop a novel computer program that simulates the responses of humans to an ever-changing natural environment.
Already, we have developed the Grand Canyon River Trip Simulator. The program uses artificial intelligence and statistical techniques to capture the unique nature of this interaction. The computer simulation program is being used by managers at the Grand Canyon National Park as part of their Colorado River Management Plan. This project is the result of a cooperative effort between researchers at Northern Arizona University and the University of Arizona.
The next step in this research is to re-engineer our program for other environmental settings, sharpen the artificial intelligence algorithms and examine more carefully the robustness of these mathematical approaches to modeling the real-world. The Grand Canyon River Trip Simulator Project (GCRTSim) involves a team at Northern Arizona University working in cooperation with a team from the University of Arizona to assist the Grand Canyon National Park in the management of rafting traffic on the Colorado River. Taking over three years to develop and test, our software was recently turned over to the park (version 1.0 in October 1999 and version 2.0 in August 2000). This computer-implemented model estimates the movement of and interactions among raft-trip parties on the Colorado River within the Grand Canyon National Park. This modeling system employs complementary statistical analyses and mathematical models based upon exiting raft trip itinerary data, as well as data collected during the 1998 rafting season. GCRTSim is an interactive computer simulation system providing dynamic visual displays of individual raft-trip progress as well as interactions among the multiple trips on the river. The goal of this model system is to provide Park managers with an effective decision support tool for representing and evaluating alternative trip/launch scheduling scenarios.
Current research efforts include expanding the database by incorporating additional river trip diaries collected in 1999 and 2000. In addition, the computer program will be generalized to enable simulations to be run under two different Glen Canyon dam flow regimes -- the "typical" flow rate represented by the 1998 and 1999 seasons, and the recent "low" flow rate represented by the 2000 season. Current and future work consists of a re-direction of the Grand Canyon River Trip Simulation Project to new settings. I intend to investigate the plethora of mathematical questions that have arisen out of this earlier work. This research project will move away from the specific setting of the Grand Canyon and, instead focus on the development of new integrated mathematical and statistical techniques for modeling complex human-environment interaction using artificial intelligence and other methods.
Mathematical modeling of human-environment interactions is quite new -- from our perspective of simulation modeling, we are confident that this work will represent a significant new contribution to science. Expanding this project to encompass the broader question of exploring the interaction between humans and the environment is, clearly, a new direction for my research. I plan to capitalize on what I learned with the Colorado River project, but I also recognize that the demands of this project require that I gain expertise in new methodologies and new mathematical modeling techniques. My aim is to accelerate my learning in this new area of research. This summer I spoke at an international conference addressing new progress in Complex Modeling Systems. I have also joined the Resource Modeling Association to help expand my network. Social scientists and ecosystem management science has addressed some of the issues related to this project, but the mathematical and statistical artificial-intelligence based computer programming perspective is entirely new.
One aim of this research is to continue to involve undergraduates in
mathematical research. This project has already given six undergraduates (as
well as two graduate students) the valuable experience of working on a real-world
project. One student was so inspired by his work on this project that he is
now pursuing his doctoral degree in the applied math program at the University
of Colorado. Another student just had an article published by the Association
for Women in Science in a special issue devoted to young scientists. Techniques
used in developing this model include intelligent agent design, artificial
intelligence, fuzzy logic and statistical optimization algorithms. The simulation
engine uses object oriented programming (VisualBasic). The objective is to
develop a robust computational tool that not only simulates the current human-environment
scenarios but that can also be modified to consider the implications of various
policy changes in managing the resource. While we develop a simulation engine
along with an adaptive learning and statistical analysis module, we are developing
the artificial intelligence algorithms required for such complex, multi-variable
scenarios.
STUDENT PARTICIPATION: The computer simulation has been designed and coded by undergraduates at Northern Arizona University including Doug Stallman (Computer Science & Engineering, May 2000), Rian Bogle (Mathematics, May 2001, and a master's in applied mathematics from University of Colorado, Boulder in May 2003), Rob Allred (Computer Science & Engineering, May 2001) and Joanna Bieri (Mathematics and Physics & Astronomy, May 2002 and a graduate student in applied mathematics at Northwestern University). A graduate student in Geology, Gary O'Brien, conducted a study of beach site capacity in 2001. Funding is from the Grand Canyon National Park, the NAU/NASA Space Grant and the NAU Organized Research and Hooper Awards programs.
Previous and Completed Research Projects
Northern Arizona University's Space Allocation Project
As part of the University effort to inventory its space and to understand how the classrooms are being utilized, the Modeling and Simulation Lab is assisting with data analysis. In addition, we are helping with the implementation of a suite of software packages called Resource25•Schedule25•Model25 that will eventually comprise the University's centralized scheduling and inventory.
STUDENT PARTICIPATION: Undergraduate students Ruben Kovalcik (NAU, Environmental Engineering, May 2001) and Brody Guion (NAU, Business, May 2001) worked on this project. Funding was from the Office of the Vice President for Business Affairs.
Course Rotation Project
The Department of History at Northern Arizona University faces the common challenge of devising an optimal course rotation. Not only does the Department want all of its courses arranged so that the students can graduate in the appropriate number of years, but it also wants to utilize the talents of its faculty in the best way.
View Final Report
STUDENT PARTICIPATION: This project was conducted in Spring 2000 at NAU by Jamie Lindeman and Stephanie Hammond with funding from the NAU Hooper Award Program and the Department of History.