Research Experience for Undergraduates in
Applied and Computational Mathematics

Projects

Project 1: Diffusion Models – Partial Differential Equations
Faculty Mentor: Prof. Eduardo Teixeira

Several PDE questions related to diffusion models and free boundary problems will be presented. Prof. Eduardo Teixeira will be the leading researcher for this theme. Simplified optimal design problems will be presented and investigated collaboratively. They comprise important and insightful applied problems, with budget constraint. The focus will be on constructive solutions, both from the analytical and computational perspectives.

Some classical free boundary problems, including the obstacle problem and the Bernoulli problem will be presented. Students will have the opportunity to work on specific (original) examples of solutions. For each constructed explicit solution, the group will investigate the regularity of solutions as well as their geometric behavior along their free interfaces. One-dimensional global profiles will also be investigated by the construction of explicit examples.

As for the theory of PDEs, questions regarding regularity of solutions will also be discussed. Teixeira envisions projects designed to introduce the notion of viscosity solutions to non-variational elliptic PDEs. Problems involving more complicated operators, like the p-Laplacian and the infinity-Laplacian will be presented and investigated by constructive examples. One of the goals is to build up richer families of two- dimensional solutions presenting the (conjectured) maximal regularity. Ideas coming from geometric tangential analysis will be explored in low dimensional models.

Project 2: Inverse Scattering Problems – Computational Mathematics
Faculty Mentor: Dr. Carlos Borges

Inverse scattering problems have several applications in the different fields of science and engineering, as some examples, we have medical imaging, remote sensing, ocean acoustics, nondestructive testing, geophysics, radar, and sonar. In those problems, given the scattered data generated when a wave impinges in an unknown domain, on tries to obtain different properties of the domain, such as the density, the sound profile, the shape of the boundary, and many others.
In this project, we will concentrate in the solution of the inverse scattering problem for sound-soft impenetrable obstacles, meaning, obstacles with a
Dirichlet boundary condition. In this problem, we try to recover the shape of the obstacle given measurements of the scattered data. This problem presents a great and simple opportunity to give a general knowledge of how-to solve an inverse problem to undergraduate students, covering both theoretical and computational aspects.
Regarding the theoretical part, the students would learn basic functional analysis, as well as some potential theory and theory of integral equations to solve the forward scattering problem. Moreover, since the inverse scattering problem is ill-posed and nonlinear, the students will learn optimization methods to deal with the non-linearity and regularization methods to treat the ill-posedness of the problem.
Regarding the computational part, the students will be trained in the use of MATLAB, the numerical solution of integral equations, the implementation of numerical optimization methods, the use of fast analysis-based methods such as the Fast Fourier Transform and the Fast Multipole Method, and in numerical linear algebra topics, such as direct solvers and iterative solvers for the solution of systems, randomized algorithms, and fast direct solvers.

Project 3: Financial Mathematics
Faculty Mentor: Dr. Yukun Li

The risk-free market includes bonds, notes, treasury bills, etc., and the risky market includes risky assets and various financial derivatives, i.e., options, forward contract, futures contract, swaps. It involves intricate relations between those concepts, and they play important roles in the financial market. To explore the applications in the financial market, we would like to consider the problems based on two different cases: equation-based models and data-based models.
Part 1. Equation-based models. We start from the discrete cases since their limits approach the continuous equation-based models. The binomial tree model and the trinomial tree model serve as basic discrete models for the pricing of options, and it is good practice to study and apply the no-arbitrage balance in the financial market. For deterministic ordinary differential equations, we study the interest rate model and interest rate sensitivity of bond prices model. For deterministic partial differential equations, we study various Black- Scholes models for vanilla options and exotic options such as binary options, chooser option, barrier options, compound options, look-back options, and Asian options. The learning/research outcomes include calibrating/revising the existing models, mastering the numerical methods to solve differential equations, and enhancing the skills of coding. For stochastic cases, we study continuous stock price behavior model including the geometric Brownian motion; the calibration of simple continuous price models including jump diffusion models and Markov switching models; the stochastic volatility models including Heston model, Hull and White model, Ornstein Uhlenbeck process model, and so on. These models will help to apply the theory to real financial applications, and hence connecting theory to the real market.
Part 2. Data-based models. We consider long-time price change and short-time price change of the financial securities. We focus on sentiment analysis for stocks and financial derivatives. The sentiment analysis is based on the positive or negative statements of the companies. The source of positive or negative sentiment could be driven from social media, company articles, the U.S. SEC (Securities and Exchange Commission) filings, various newswires, and so on. The data could be either structured or unstructured, depending on type of news sources. This project will equip the students with machine learning skills including how to design supervised and unsupervised learning algorithms, how to pre-process and post-process the datasets, how to clean and prepare training sets, how to improve the accuracy of the algorithms, and so on.

Project 4: Mathematical Biology
Faculty Mentor: Dr. Zhisheng Shuai

Infectious diseases have created huge global challenges to human society and the recent COVID-19 pandemic has caused for alarm for an improved prediction and understanding in the transmission and spread of infectious diseases. This alarm is not only for current mathematical modelers and public health workers, but also for a further workforce with strong quantitative and analytical ability.
In this project, we will formulate complex mathematical models to investigate specific epidemiological problems and enhance our understanding in the transmission and spread of infectious diseases such as COVID, HIV, malaria, and cholera. Model formulation and validation will be conducted in consultation with our collaborators from public health and medicine.
Students will learn mathematical approaches and techniques to rigorously analyze the disease models, including the derivation of the basic reproduction number, final size relation or stability analysis of disease- free and endemic equilibria, and assessment of disease control strategies. Disease surveillance data will be collected and employed for the model calibration.