Doctoral Candidate & Teaching Assistant
Curriculum Vitae (Last Updated: November 2023)
Email: domarov3@gatech.edu
Office: 151 Skiles Building,
School of Mathematics,
Georgia Institute of Technology,
Atlanta, Georgia, USA
Research
My primary research interest revolves around the numerical analysis of Optimal Transport (OT) problems. In general, the OT problem consists in seeking an optimal transport plan, that is the joint probability density function 𝝅(x,y) , having the source density 𝝁(x) and the target density 𝝂(y) as marginals, which realizes the minimum of the functional J(𝝅) below:
J(𝝅) = ∫c(x,y)d𝝅
where the cost function c(x,y) represents the cost of moving one unit of mass from x to y.
There are three main types of OT problem, which are based on the type of marginal densities: discrete, semi-discrete, and continuous. See Figure 1 below.
Figure 1. Three types of OT problems
Continuous OT Problems
For the first project [2], we have analyzed existing OT algorithms in the continuous setting with squared Euclidean cost function. In this case, it is well known that the solution to the OT problem exists and is unique.
The main goal of the project was to compare existing OT algorithms for continuous problems and provided thorough numerical implementation details.
In addition, we have discovered the special class of continuous OT problems that we called separable. The continuous OT problem in 2-d is called separable if source and target densities can be written as a product of one-variable dependent functions. Then it is possible to reformulate the 2-d problem into two 1-d problems, which can be solved much faster and more accurately compared to the 2-d problems.
Semi-Discrete OT Problems
For the next project [3] we have focused on semi-discrete OT problems. In this situation, the target density is discrete and it is given at specific target points and the problem is to find a partition, called Laguerre tessellation, of a domain with respect to these target points.
The problem has been well-studied in the literature when the cost function is the squared 2-norm. In this case, each Laguerre tessellation forms a polygonal region. However, computational techniques do not appear to be very well developed for other cases because of the lack of a simple geometrical shape of Laguerre tessellations. As a result, our goal has been to devise an accurate and fast algorithm for these cases.
Firstly, we proved that all Laguerre cells are star-shaped with respect to the corresponding target point when the cost function is given by a vector norm, c(x,y) = ||x - y||. This result does not hold for 2-norm squared case since this cost function does not satisfy all norm properties. See Figures 2 and 3 below.
Figure 2. Problem with 2-norm cost
Figure 3. Problem with 2-norm squared cost
Above result implies that the boundary of any Laguerre cell in 2-d can be parameterized by an angular coordinate. Then we were able to develop and provide thorough implementation details of an efficient algorithm in 2-d, which computes the gradient and the Hessian of the minimization function up to a desired accuracy by exploiting star-shapedness, smoothness, and structure of the Hessian matrix.
In addition, we programmed and tested ours method to solve several semi-discrete OT problems: different initial guesses, different source densities, different target densities, different p-norm cost functions, and different domains. For example, see Figures 4-7 below.
Figure 4. Problem with
32-norm cost function
Figure 5. Problem with non-uniform source density
Figure 6. Problem with circular domain
Figure 7. Problem with triangular domain
Future Research
OT Problem with Obstacle. In applications, regions of interest can contain natural obstacles, such as rivers, mountains, and lakes, as well as human-made obstacles like cities, dams, and roads. The question then arises: how can this information be incorporated into the abstract definition of domain and included in solving the OT problem? What will be the specific characteristics and challenges of this type of problems?
Applications of OT to Machine Learning. Having built a strong background in both numerical analysis and OT theory, I aim to explore the potential of using OT to improve the performance of machine learning models with emphasis on numerical analysis. This includes to develop new algorithms that incorporate OT principles to better handle complex data distributions and improve the accuracy and robustness of machine learning models.
Optimal Placing Problem. Specifically, in the semi-discrete OT problem, the location of target points is typically predetermined. However, in practice, there may be flexibility in placing the target points. For example, when building a new electricity grid for a city, one may need to determine where to place electricity substations with specific capacities to ensure an efficient electricity grid for transferring power to buildings. The question then becomes how to accurately and efficiently select the locations of these target points.
Publications
Daniyar Omarov et al. “On the application of Sturm’s theorem to analysis of dynamic pull-in for a graphene-based MEMS model”. In: Applied and Computational Mechanics 12.1 (2018).
Luca Dieci and Daniyar Omarov. “Techniques for continuous optimal transport problem”. In: Computers & Mathematics with Applications 146 (2023), pp. 176–191.
Luca Dieci and Daniyar Omarov. “Solving semi-discrete optimal transport problems: star-shapedeness and Newton’s method”. Manuscript submitted for publication. arXiv: 2310.07489.
Luca Dieci and Daniyar Omarov. “A mathematical electoral map of Georgia: is there a reason for gerrymandering?” Manuscript in preparation.
Teaching
Instructor: taught lectures and managed teaching assistants.
Head Teaching Assistant: taught recitations and managed teaching assistants.
Lecture Assistant: assisted the course instructor with lectures and held weekly office hours.
Spring 2021 - Lecture Assistant - A Second Course on Linear Algebra - Math 3406
Teaching Assistant: taught recitations and held weekly office hours.
Fall 2018 - Teaching Assistant - Differential Equations - Math 2552
Spring 2019 - Teaching Assistant - Finite Mathematics - Math 1711
Summer 2019 - Teaching Assistant - Computations in Dynamics- Math 8873
Fall 2019 - Teaching Assistant - Differential Equations - Math 2552
Spring 2020 - Teaching Assistant - Linear Algebra - Math 1554
Summer 2020 - Teaching Assistant - Linear Algebra - Math 1554
Spring 2021 - Teaching Assistant - Differential Equations - Math 2552
Fall 2021 - Teaching Assistant - Precalculus - Math 1113
Spring 2022 - Teaching Assistant - Linear Algebra - Math 1554
Summer 2022 - Teaching Assistant - Differential Equations - Math 2552