The purpose of this manuscript is to review the state-of-the-art in convex regularization methods applied to PDE problems and to propose concrete research questions in this direction. The review is accomplished by an introduction to PDE-constrained inverse problems; relevant convex analysis theory; and convex regularization methods. A direction of further research related to convex regularization methods applied to PDE problems is proposed; specifically, this section provides motivation for pursuing these problems and a solution approach that may be useful.

[Available upon request]

Class Project for Numerical Algebraic Geometry (Fall 2024) course at the Fields Institute

Motivated by PDE-constrained optimization problems, we propose a general algorithm that locally solves a polynomial-constrained optimization problem using homotopy continuation. The algorithm is presented in the 2D real space for a linear objective constrained to a circle centered at the origin. In particular, we use homotopy continuation to correct an unconstrained iterative update to one which belongs to the constraint manifold. We provide several numerical studies that display convergence of this algorithm under various conditions.

[Available upon request]

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.

[Available upon request]

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.

[Available upon request]

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.

[Available upon request]

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.

[Available upon request]

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. 

[Available upon request]

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.

[Available upon request]

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]