This course will introduce you to the fascinating world of transportation optimization, a field that has important applications in many areas of science and engineering, such as economics, image processing, and machine learning. We will start by exploring the discrete Optimal Transport (OT) problem, which involves finding the most efficient way to transport a set of objects from one location to another. While the discrete OT problem can be formulated as a linear programming problem, finding an optimal solution to this problem can be computationally expensive, especially for large-scale problems. To overcome this computational challenge, a popular approach is to use entropy regularization. We will also investigate the entropy regularized OT problem, which provides us with an approximation of optimal transport, with lower computational complexity and easy implementation. In the second half of the course, we will delve into the continuous case, which allows us to consider transport between infinitely many locations. We will study the famous Monge-Kantorovich problem, which involves finding the optimal transportation plan that minimizes the total cost of moving a given amount of mass from one location to another, subject to various constraints. Throughout the course, we will use a combination of theoretical and practical approaches to understand and apply the concepts we cover. By the end of the course, you will have a strong foundation in OT theory, which will prepare you for further studies in this exciting and rapidly evolving field. Recommended Textbooks / Articles: Topics in Optimal Transportation – Cédric Villani Optimal Transport for Applied Mathematicians – Filippo Santambrogio Computational Optimal Transport – Gabriel Peyré, Marco Cuturi (https://arxiv.org/abs/1803.00567)
The Class: Format: lecture
Requirements/Evaluation: Weekly homework, open book/open notes take-home exams.
Prerequisites: Math 350/351 or permission of instructor
Enrollment Preferences: Mathematics and Statistics majors
Distributions: Division III Quantitative/Formal Reasoning
QFR Notes: This is a senior seminar course in mathematics and will require students to use advanced quantitative and formal reasoning skills.