NC State Research Experience
for Undergraduates in Mathematics:
Modeling and Industrial Applied Mathematics
REU arrive on 5/25 and depart on 7/31 (First day of program 5/26)
REU+ (Under-represented Undergraduate students)
have different application sites.
The Online Application is available online.
Program Director: Loek
The NSF and the NSA provide generous funding and support for this
Participant background, requirements
and selection: Participants are expected to meet the
Where: NC State University, Raleigh, NC .
Stipend and support: $4500 for ten weeks, all
housing provided, as well as a partial meal allowance. Travel
funds up to $300 per participant provided as needed.
Topics for 2015 REU program: Project 1: Sensitivity of Sensitivitie, Project 2: Portfolio Optimization with Mixed-Gaussian Model, Project 3: Accurate Gradient Computation and Applications, Project 4: First-Order Methods for Linear and Conic Optimization, Project 5: Modeling Cardiovascular Dynamics During Blood Withdrawal, Project 6: Data-Based Behavior Modeling, and Project 7: Evaluating the Performance of Image Feature Descriptor Methods.
- must be a citizen or permanent resident of the
United States or its possessions,
- must be a full-time undergraduate mathematics
major as of September 2014,
- must be committed to devote their full time to the
program and not engage in any other course work or employment
during the program,
Participants will be selected on the basis of demonstrated mathematical
creativity, motivation, and good work habits as well as meeting
the above requirements, as determined from the application materials
and recommendation letters.
This program will be similiar to those offered in the past. Past program homepages Summer 2005 , Summer
2006 , Summer 2007, Summer 2008, Summer 2009, Summer 2010, Summer 2011, Summer 2012, Summer 2013, and Summer 2014.
Project 1: Sensitivity of Sensitivities
Faculty advisor: Pierre Gremaud
Graduate mentor: Joey Hart
The ability to identify a few important variables or factors-and discard unimportant ones-is a fundamental need in many areas of scientific computing.
In Mathematics, it is customary to assume a functional dependency "y = f(x) + noise" between predictors, i.e., input variables corresponding the p-vector x and the response variable y. Many methods to gauge the relative importance of the components of x on y (sensitivities) are based on the assumption that f can be evaluated at will or that an accurate surrogate model can be constructed. This is however not always the case as applications are often only described through a finite amount of data with no possibility to "sample for more". In addition, it is not clear that the important variables of an approximate problem are roughly the important variables of the original problem.
The group will explore these issues through analysis and computations. The students will become familiar with techniques and methods from Mathematics and Uncertainty Quantification (approximation theory, Sobol indices, Morris screening), Machine Learning (regression trees and random forests) and Statistics (analysis of variance, multivariate adaptive regression). Most of the computational work will be done in R (the previous knowledge of which is not required).
Project 2: Portfolio Optimization with Mixed-Gaussian Model
Faculty advisor: Tao Pang
Graduate mentor: Cagatay Karan
Portfolio optimization is a very important area of financial mathematics. An investor will choose the optimal asset allocation to maximize the expected return or minimize the associated risk given by variance, value at risk (VaR) or conditional value at risk (CVaR). The classical mean-variance optimization method uses the historical mean returns and co-variance matrix as input, but the historical performance may not reflect the future performance. On the other hand, in Black-Litterman model, the subjective view of the asset returns can be integrated into the optimization process by virtue of the Bayesian estimation. The resulting portfolio usually has a better performance than the solution obtained in the classical mean-variance model. Current models usually assume Gaussian distributions for asset returns, but the true market returns always show heavier tail behavior than a Gaussian distribution. In this project, we will consider mixed Gaussian distribution which can have heavier tail than a standard Gaussian distribution. We will consider the classical mean-variance optimization as well as the Black-Litterman model under the mixed Gaussian assumption. The solution performance will be evaluated by virtue of market data or simulated data.
Qualified students should have strong background in linear algebra, probability and statistics. Background in financial mathematics is a plus but not required.
Project 3: Accurate Gradient Computation and Applications
Faculty advisor: Zhilin Li
Graduate mentor: Zhaohui Wang
Many important applications involve ordinary or partial differential equations (ODE/PDE) of boundary value problems. For example, the Navier-Stokes equations for bubble simulations and the blood flow in human hearts. For many such free boundary and moving interface problems, the free boundary or moving interface depends on the gradient of the solution and the curvature of the boundary. So it is important to obtain not only the accurate solution but also accurate first order and second order derivatives of the solution. There are many efficient numerical methods such as finite difference or finite element methods for numerical ODE/PDEs. Many methods can produce accurate solution but less accurate derivatives.
In this project, we will explore ways to compute the solution and its derivative to an ODE/PDE accurately at the same time, particularly using a finite difference discretization. We will start with two-point boundary value problems in one space dimensions. One idea is to treat the derivative as a separate variable to get an enlarged system of equations. A least squares problem needs to be solved to get both the solution and its derivative. The next step would focus on reducing the computational cost by computing the accurate derivative only near the boundary or interface. We also wish to have the convergence proof of the proposed method.
Once we have some promising results for one dimensional problems, then we would like to generalize the results to 2D problems which can have many variations such as irregular boundary or interface, much large system of equations, more challenging theoretical analysis.
If time allows, we will apply the developed methods for some free boundary and moving interface problems such as Peskin's Immersed Boundary method, mean curvature flow, or two phase flow problems.
Project 4: First-Order Methods for Linear and Conic Optimization
Faculty advisor: David Papp
Graduate mentor: to be determined
Emerging applications of very large scale optimization, such as the analysis of ``big data'', radiotherapy treatment planning utilizing a large number of beam directions, and stochastic optimization with a large number of scenarios modeling various sources of uncertainty all call for novel approaches to linear and nonlinear optimization that can identify crude, roughly optimal solutions quickly, rather than computing optimal solutions with high precision.
The most efficient methods for medium-to-large scale linear and conic programming are interior point methods. These are second-order methods, requiring the computation of a Newton step in every iteration. They are not suitable for very large scale applications, because even the computation of a single Newton step requires too much time and memory.
The aim of this REU project is to identify and compare easily implementable first-order methods for linear optimization, and explore possible generalizations to conic optimization, especially second-order cone programming and semidefinite programming.
Project 5: Modeling Cardiovascular Dynamics During Blood Withdrawal
Advisor: Dr. Andrea Arnold
Faculty co-advisor: Mette Olufsen
Graduate mentor: Michael Frank
The body continuously regulates system properties to maintain blood pressure at homeostasis, which prevents us from fainting or feeling light-headed during everyday activities. This project aims at understanding how system properties are controlled during blood withdrawal. It is of importance to understand not only how the body reacts to injury, but also the effect that giving blood has on the body. One question that could be addressed is how long you should rest before driving home after donating blood.
More specifically, we plan to develop a compartment model predicting cardiovascular dynamics. This model will be validated against experimental pressure and volume data measured in the hearts of rats and pigs. Data will be made available from collaborators at the University of Michigan. During blood withdrawal blood volume is reduced, and to maintain blood pressure, heart rate is increased along with vascular resistance and cardiac contractility (i.e., the heart will pump faster and use more power to do so, causing the muscles to contract more). These physiological properties are represented by model parameters, and the objective is to understand how these parameters change in time to predict experimental observations. We plan to use Bayesian filtering techniques (such as the Kalman filter) and piecewise linear splines to estimate the time-varying dynamics. In addition to determining how parameters change, Bayesian filtering provides a measure of uncertainty in the parameter estimates.
This project requires knowledge of differential equations and some linear algebra techniques, especially matrix and vector computations. Some experience with programming (preferably in MATLAB) is also required.
Project 6: Data-Based Behavior Modeling
Advisor: Dr. James M. Keiser (Laboratory for Analytic Sciences)
Graduate mentor: Glenn Sidle
The goal of this REU is to develop data-based models of the behavior of individual entities and build aggregate (joint) models of teams of such entities. The underlying data comes from game play logs of the online game Defense of the Ancients 2 (DotA2). DotA2 is a two team, five (human) person per team game that is essentially Capture the Flag. In addition to the ten human players, there are other semi- autonomous and autonomous entities in the game. During play game logs are captured that make measurements on the state of the "game world" at the rate of 30 observations per second. These logs can be represented by a spreadsheet consisting of rows of measured events and columns of feature values that have been measured. The entity and team models are built from an analysis of this measured data and the accuracy of the models is based on the ability of the model to predict future behavior in a game of either an entity or the team consisting of multiple entities. This REU seeks to develop means for building the models, scoring the performance of the models and suggesting alternative modeling frameworks: Hidden Markov Models (HMMs), Hierarchical HMMs , Finite State Machines, Bayesian Networks, etc. If there is time, the students may (1) develop a game playing mechanism that runs the models against each other; (2) that learns strategies for game play; (3) apply the modeling formalism developed for DotA2 to other game logs, e.g. NFL Play-by-Play (NFL PbP).
Project 7: Evaluating the Performance of Image Feature Descriptor Methods
Advisor: Joseph Murray
Faculty co-advisor: Hien Tran
Graduate mentor: Phuong Hoang
The field of image analysis has many useful applications in areas like object recognition, image matching and panorama stitching. Techniques like these have been significantly improved in recent years by better feature descriptor algorithms, which describe image regions in clever ways. We will compare the performance of several feature descriptor methods, such as geometric blur and scale-invariant feature transforms, for the task of image categorization. For this project we will be using the popular Caltech 101 data set, which contains over 9,000 images in 101 categories.You should feel comfortable with Linear Algebra and expect to write plenty of MATLAB code.
Calendar and Info for REU participants
Week 1: REU Workshop on modeling.
REU students and faculty.
Week 2: Introduction to projects
and mentors. Project teams are determined and begin to work
Weeks 3 - 9: Work on projects.
Progress reports are due each Friday. Every other week there will be presentations given by each group. In addition there may be seminars on:
- mathematics related to the student projects
- research ethics
- applying to graduate school
- how to give poster presentations
- how to give research talks
Extra-curricular activities may include weekly teas, organized game and movie nights, a trip to see the AAA Durham Bulls play a baseball game, as well as an excursion to the beach. The North Carolina beaches and North Carolina mountains are within 2 to 4 hours drive from Raleigh.
Week 10: Students complete their final reports and do poster presentations of
Places to visit:
Sunday in the Park
All applicants will be notified by email about the
completeness of their application a couple of days after the deadline
date. Unless previously notified, a final notification that the
search is closed will be emailed after all positions have been
filled and confirmed (this could take a month). If you have any questions about the status of your application,
especially if you are trying to make a decision on accepting another
summer position, please email the program
director who will be happy to send you a prompt response.
Comments and suggestions to firstname.lastname@example.org