Research

"Research is to see what everybody else has seen, and to think what nobody else has thought."------Albert Szent-Gyorgy, Nobel Prize in Medicine, 1934

My research interests span a broad range of fields in Applied Mathematics: Numerical Analysis, History of Statistics, Data Science. The interdisciplinary nature of my work, allows me to employ a range of mathematical tools for real-world physical modeling which include numerical approximation through finite difference, finite volume and multigrid methods, and machine learning algorithms.

Fractional Diffusion Equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equation. For instance, in contaminant transport in groundwater flow the solutes moving through aquifiers do not generally follow a Fickian, second-order partial differential equation because of large deviations from the stochastic process of Brownian motion. Instead, a governing equation with a fractional-order anomalous diffusion provides a more adequate and accurate description of the movement of the solutes.

Applications of Fractional Diffusion Equations. Fractional diffusion equations are used in modeling turbulent flow, chaotic dynamics of classical conservative systems, groundwater contaminant transport, and applications in biology, physics, chemistry, and even finance. In the past analytic methods have been used to find closed form solution for fractional partial differential equations. However, there are very few cases in which the closed-form analytic solutions are available.

Salient Features of Fractional Diffusion Equations. In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, using finite difference methods, finite element methods and spectral methods. The fractional diffusion equation has the following salient features:

• Fractional differential operators are non-local and so raise subtle stability issues on the corresponding numerical approximations;
• Numerical methods for fractional diffusion equations tend to yield full coefficient matrices, which have computational cost of O(N^3) and storage of O(N^2) with N being the number of unknowns.

This is in contrast to numerical methods for second-order diffusion equations which usually generate banded matrices of O(N) non-zero entries and can be solved by fast solution methods such as multigrid methods, domain decomposition methods, and wavelet methods.

The One and Two Dimensional Space-Fractional Diffusion Equation. The goal of my research in this area has been to develop a fast finite difference method for space-fractional diffusion equations. By reducing the computational cost from O(N^3) to O(N logN) and storage requirement from O(N^2) to O(N) my research has allowed me to successfully obtained numerical solutions that are of the same order accuracy as traditional numerical schemes with drastically reduce run time! For example, for a problem of size N=2^10, our proposed fast method can compute a numerical solution in 10 minutes and 5 seconds versus a traditional Crank-Nicolson scheme with Gaussian elimination that takes 8 hours and 15 minutes. This means that our proposed numerical scheme can compute the numerical solution 50 times faster than traditional numerical methods.

History of Statistics

My interest in the history of Statistics stemmed from my teaching the introductory courses in Statistics and from interaction with my students. My work in this subject has resulted in a joint publication and discusses historical developments of Statistics and its applications as well as modern information about research in analytical and computational aspects of Statistics. Simple concepts such as the difference between the term ``average” and ``mean” are described along with more complex topics such as Bernoulli’s Law of Large Numbers. Topics that a student would typically encounter in an introductory Statistics course such as Regression and Correlation are also described. We outline the contributions in Statistics of Florence Nightingale in the field of public health, for which she was an elected fellow of the Royal Statistical Society in 1860, the first woman to be selected for this prestigious honor.

Data Science

As a discipline, Data Science is a multi-disciplinary field that uses scientific methods, processes, algorithms and systems to extract knowledge and insights from structured and unstructured data (Wikipedia) and lies in the intersection of several major fields of study: Mathematics, Statistics and Computer Science. Below I describe the interesting and practical problem I worked on.

The Oxy Acceptance Problem. My co-authors and I, applied several supervised machine learning techniques to four years of data on 11,001 students, each with 35 associated features, admitted to Occidental College to predict student college commitment decisions. By treating the question of whether a student offered admission will accept it as a binary classification problem, we implemented a number of different classifiers and then evaluated the performance of these algorithms using the metrics of accuracy, F-measure and area under the receiver operator curve. The results from this study indicate that the logistic regression classifier performed best in modeling the student college commitment decision problem, i.e., predicting whether a student will accept an admission offer, with an AUC score of 79.6%. Further, we were able to identify the top five features that have the most predictive power in determining whether an admitted student will accept the admission offer. These features are: GPA, Campus Visit Indicator, High School Class Size, Reader Academic Rating, and Gender. The significance of this research is that it demonstrates that institutions could potentially use machine learning algorithms to improve the accuracy of their estimates of entering class sizes, thus allowing more optimal allocation of resources.

Future Research and Endeavors

Fractional Diffusion Equation and Predicting Market Behavior. Most financial modeling systems rely on the hypothesis known as the Efficient Market Hypothesis (EMH) including the famous Black-Scholes formula for placing an option. The EMH is based on the assumption that economic processes are normally distributed and it has long been known that this is not the case. This assumption leads to a number of shortcomings which includes failure to predict the future volatility of a market share value. I am interested in analyzing financial signals that are based on a non-stationary fractional diffusion equation derived under the assumption that the data are Levy distributed.

Reaction-Diffusion Equation. Developing some solution techniques to reaction-diffusion equation especially those arising from biological or population dynamical systems is another one of my goals. Particularly, I wish to investigate patch models where the population migrates from one patch to another and also adaptive mesh refining.

Predicting Summer Melt Phenomenon Using Hybrid Machine Learning Methods. Summer melt is the phenomenon in which college-intending students who, after already committing to attend a college or university do not actually enroll due to a variety of reasons including financial challenges or personal preferences. In this project, I would like to develop and test a mathematical model that predicts which admitted students will ``melt" away over the summer, i.e., they will not attend college in the fall in spite of committing to attend the college, using several machine learning techniques. Since only 1 in 20 students ``melt", this ``extreme class imbalance problem" needs to be approached using sampling techniques such as over-sampling and under-sampling, combined with ensemble techniques such as bagging and boosting. I also intend to explore hybrid methods, in particular data variation-based ensembles. The goal of this research is to be able to better predict which students we are likely to lose after the student has committed to attending Occidental College.

I maintain an active research profile by attending and presenting at various conferences. I am eager to contribute to interdisciplinary research and perform collaborative research with other faculty. I also look for creative ways to incorporate research in my classroom and students in my research. For further details about my work please do not hesitate to contact me.

Note: My paper (with co-author) On Analytical, Computational and Historical Developments of Statistics and Its Applications appeared in the International Journal of Applied and Computational Mathematics in 2015. Please note the following typographical error in the paper:

• page 14, Equation 38: which should read as: