Any student attending either DUNY or a NJ or NY community college is eligible for this opportunity. Students from community colleges in upstate NY, Long Island, and even South Jersey participated in our program in 2024 and lived on our beautiful campus at Dominican University in Rockland County, NY. 

I am hiring 4 students for Summer 2025 (May 20 to June 27) who will receive $3500 and free on-campus housing (if desired) to participate in a 6-week computational modeling research internship. My previous student interns continued on to engineering, computer science, and applied math programs at schools such as Stony Brook, NJIT, and Claremont McKenna. 

There are 3 other programs in biochemistry, biodiversity, and chemistry that are likewise recruiting students. See the application page and program flyer for more details!

In the computational modeling program, interns will learn how to apply cutting-edge research methods to build computational models that can be applied to neuroscience, biology, finance, or any other field they might be interested in. More details about my research program can be found on my website. Having some knowledge of physics and calculus is recommended, but nothing is necessarily required. The most important factor is interest in doing scientific research!

Only US citizens/residents (i.e. no international students) can apply. Please email jovan.zigic@duny.edu if you have any questions. We look forward to your application!

1. Internship Description:

I am looking to host a cohort of 20 students at Dominican University during the next academic year that will be part of the DUNY Data Mine virtual internship program. This internship is available to any student at DUNY (regardless of citizenship, major, year of study, undergraduate/graduate, etc.). There is no requirement for data science or computing experience, only a desire and willingness to learn. The following two courses cover (in general) what kind of methods you would learn and apply during this internship:

1. https://www.freecodecamp.org/learn/data-analysis-with-python/

2. https://www.youtube.com/watch?v=hDKCxebp88A

In particular, you may want to consider your knowledge of the following concepts:

• to have some basic ideas about linear regression and tree methods. 

• to be familiar with two Python packages: pandas and scikit-learn. 

This is an incredible opportunity for anyone looking to pursue a STEM career or any field that involves working with data. This is directly beneficial to students pursuing degrees such as Finance, IT, Math, Biology, Health Sciences, Psychology, etc. Students that participate will also be able to receive DUNY internship credits. 

The DUNY Data Mine is a 9-month virtual internship (during the Fall 2024 and Spring 2025 semesters) in collaboration with Purdue University that requires you to attend a 60-min lecture with a corporate mentor and 90-min lab session with your project team every week. In addition to these meetings, you are required to complete weekly data science projects that culminate in a presentation of your lab group's research project to the corporate mentors. This 3-credit internship is funded by a government grant that allows any undergraduate US citizens to receive a $500 monthly stipend (total of $4500 over 9 months) for their participation.

Since the internship is virtual and the meetings are only 2-3 hours a week during school semesters, it is designed so that you can maintain your full-time student status throughout the length of the internship.

See this recent article published by DUNY for more information: https://www.duny.edu/ten-students-chosen-to-participate-in-national-data-mine-network/

Currently, in 2023-24, we have 15 students (majoring in either Business, IT, Biology, Mathematics, Health Sciences, or Nursing) participating in Data Mine projects for the following corporate partners (with a link to the 2022-23 corporate projects):

• Raytheon Technologies (https://datamine.purdue.edu/corporate/raytheon/)

• BASF (https://datamine.purdue.edu/corporate/basf/)

• Bayer (https://datamine.purdue.edu/corporate/bayer/)

• FSSA (https://datamine.purdue.edu/corporate/fssa/)

• Renzoe Box (https://datamine.purdue.edu/corporate/renzoebox/)

• Stratolaunch (New partner in 2023-24: https://www.stratolaunch.com/)

2. Internship Application:

If you are interested, please fill out this application form:

https://forms.office.com/r/deqYVxuPtf

The deadline to apply is March 1, 2024. I will contact you by the first week of March informing you whether or not you have been selected to participate. Please reach out to me if you have any questions about the program, and reach out to the Career Development Center for information on DUNY 3-credit internships. We look forward to reviewing your application!

This is an overview of state-of-the-art momentum-based gradient descent methods on Riemannian manifolds applied to infinite-dimensional dynamical systems. The fundamental question of interest is to find an momentum-based alternative to the Riemannian conjugate gradient (RCG) approach for solving a PDE-constrained optimization problem. It is assumed that the PDE has a smooth solution for smooth initial data. However, the nonlinear optimization problem being considered is nonconvex and thus the initial data can only guarantee a local optimal solution. Riemannian Stochastic Weighted (RSW) gradient methods are defined in this report, and numerical studies suggest that an RSW variant is a suitable alternative to the RCG method.

• github.com/zigicjovan/1DB_MaxEnstrophy contains code to reproduce the results described in this report.

[Link to paper]

The Burgers equation is a fluid dynamics model that is used towards studying the Navier-Stokes system. One important quantity characterizing solutions to fluid mechanics models is the initial enstrophy imposed on the system, whose boundedness over a finite-time interval guarantees a smooth solution and is known to adhere to a certain power law in scale. Through use of numerical optimization and data-driven heuristics, this report demonstrates the convergence of maximizers for the problem of finite-time maximum enstrophy growth when solving the Burgers equation.

• github.com/zigicjovan/1DB_MaxEnstrophy contains code to reproduce the results described in this report.

[Link to paper] [Link to slides]

Class Project for Math 745 (Topics in Numerical Analysis) at McMaster

This report provides an investigation into solving the Kuramoto-Sivashinsky equation in two spatial

dimensions (2DKS) using a pseudo-spectral method on various rectangular periodic domains. The

Kuramoto-Sivashinsky equation is a fluid dynamics model that exhibits dynamical features that are

highly dependent on the length of the periodic domain. The goals of this report are to describe the mathematical problem being studied; explain the details of the chosen numerical method; inspect solutions and dynamical features for varying grid sizes, step sizes, and domains; and summarize the findings.

• github.com/zigicjovan/2DKS_Solver contains code to reproduce the results described in this report.

[Link to paper] [Link to slides]

Master's Thesis, Department of Mathematics at Virginia Tech (Supervisor: Jeff Borggaard)

The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field.

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

This research is partially supported by the National Science Foundation under contract DMS-1819110.

• github.com/zigicjovan/SplitDMD_1DKS contains code to reproduce the results of the Split DMD study.

• github.com/zigicjovan/OptimizedDMD_1DBurgers contains code to reproduce the results of the Optimized DMD study.

[Link to thesis] [Link to paper] [Link to slides]

Class Project for Math 5414 (Model Reduction) at Virginia Tech

This is a summarized report on model order reduction by the balanced truncation method for control of a nonlinear dynamical system. The purpose of this text is to determine the practicality of using balanced truncation for reduced-order modeling of problems in fluid flow control. The main outcome of this text is an understanding of the steps required for balanced truncation of nonlinear PDE problems.

As a summarized report, this text assumes knowledge of basic concepts in model order reduction, systems and control theory, computational modeling and fluid dynamics required to discuss the implementation of the balanced truncation method for a discretized PDE problem.

[Link to paper]

Class Project for Math 5414 (Optimization) at Virginia Tech

An implementation of the Variable Projection method for optimizing the Dynamic Mode Decomposition (DMD) reduced model of a 1-D Burgers' equation. This project shows that the so-called "Optimized DMD" method, which makes use of a Levenberg-Marquardt algorithm to solve the nonlinear least squares problem within the DMD, provides a more accurate reconstruction over the POD (Proper Orthogonal Decomposition) method for this problem. 

[Link to paper] [Link to slides]

Class Project for Math 5524 (Matrix Theory) at Virginia Tech

A systematic construction and analysis of the algebra in defining the "Quadratic-Quadratic Regulator" (QQR) problem. The focus was to understand the role of Kronecker products for solving the QQR problem, whose usefulness and practicality are mainly for computational purposes. This was done through a thorough definition of basic Kronecker product properties, and a simplified explanation of the operator and function essential to solving the problem. This paper provides a useful clarification of some matrix theory properties and their importance to applications involving numerical linear algebra.

[Link to paper]

Research Internship at NISMAT in Manhattan, New York

PURPOSE: Machine learning-based methods, which include Artificial Neural Networks [ANN], have been used successfully in many different classification problems. If these methods can successfully classify those vulnerable to musculoskeletal problems such as LBP, then these may have utility in screening and management of such conditions and aid in identifying what assessment methods provide the most useful information for practitioners. We examined whether ANN techniques could correctly classify whether subjects experienced low back pain [LBP] in a convenience sample of dancers.

RESULTS: Based on the SPM1D analysis, only approximately 10% of data were used for training the ANN. For example, for the walking trials, lower lumbar and lower thoracic axial rotations and upper thoracic coronal plane rotations were used. The ANN classifier was able to correctly identify incidence of LBP with approximately 65% accuracy.

CONCLUSIONS: Based on our small sample, ANN techniques show promise for identifying subjects with LBP based on their movement patterns. A larger training set of data is needed for better results. Future work should optimize feature selection by focusing on areas of difference between data rather than by selecting fixed features [e.g., max value] and examine the effect of different ANN architectures.

My responsibilities:

- Designed and implemented efficient algorithms in MATLAB and Python for analysis and development of Biomechanics research involving 3D motion analysis and electromyography (EMG) data

- Developed robust programs using raw data to conduct statistical analysis/testing, artificial neural network training, and data conversion

- Increased the precision and accuracy of trained neural networks used for independent data testing

The success of this research included submission of work to American College of Sports Medicine's Annual Meeting in May 2020.

[Slides: "An Analysis of Spine Biomechanics in Dancers: 1-D Statistical Parametric Mapping and Pattern Recognition Methods"]

[Journal link: Medicine & Science in Sports & Exercise: July 2020 - Volume 52 - Issue 7S - p 194]