Optimal transportation and Monge-Ampère equations

Lecturer

Associate Professor Jiakun Liu, University of Wollongong

Synopsis

This course offers an introductory exploration of the optimal transport problem, a multifaceted concept integral to various fields including fluid mechanics, partial differential equations (PDE), optimisation, and financial mathematics. Recently, optimal transport theory has been successfully applied in machine learning, particularly in areas related to the Wasserstein geometry and image recognition. As the relevance of this theory continues to expand, it is an opportune time for Australian students to engage in training and familiarise themselves with fundamental concepts in this domain.

The course is structed into three parts. The first part covers the foundational theory of optimal mappings, laying a solid groundwork for understanding the basic principles of optimal transport. The second part advances to more complex topics, focusing on the advanced theory including regularity results and offering an in-depth analysis of Monge-Ampère type PDEs. The final section of the course addresses recent applications in fields such as image recognition and reflector design, among other interdisciplinary area.

This comprehensive approach not only provides students with a robust theoretical foundation but also exposes them to the latest developments and practical applications in the field of optimal transport.

Course Overview

Week 1:

  • Introduction to optimal transportation
  • Kantorovich’s duality and linear optimisation
  • Existence of optimal mappings

Week 2:

  • Geometric characterisation of optimal maps
  • Basics of convex analysis
  • Brenier’s factorisation theorem

Week 3:

  • Introduction to Monge-Ampère equations
  • Pogorelov type estimates
  • Regularity of optimal mappings

Week 4:

  • Application on image recognition
  • Application on reflector design
  • Other interdisciplinary applications

Prerequisites

Multivariable calculus, Linear algebra, PDEs, basic probability and optimisation

The course is designed for students at the Honours/masters level, who has acquired with multivariable calculus, linear algebra, and preferably some basic knowledge in PDEs. After successful completion of this subject, students should be able to understand some key concepts about the optimal transport problems and their applications, and to appreciate some fundamental theorems and proofs in fully nonlinear PDEs.

Assessment

  • 3 Assignments due on Tuesdays of Week 2-4 (20% each)
  • Take home exam (40%)

Attendance requirements

TBA

Resources/pre-reading

  • Villani, C., 2003. Topics in Optimal Transportation. Graduate Studies in Mathematics 58, American Mathematical Society, Providence, Rhode Island.
  • Villani, C., 2009. Optimal Transportation, Old and New. Grundlehren der Math. Wiss. 338. Springer-Verlag, Berlin.
  • Figalli, A., 2017. The Monge-Ampère equation and its applications. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS).
  • Gilbarg, D. and Trudinger, N. S., 1983. Elliptic partial differential equations of second order. Springer-Verlag, Berlin.

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Associate Professor Jiakun Liu, University of Wollongong

Jiakun Liu is an Associate Professor and ARC Future Fellow at the University of Wollongong. His research primarily revolves around nonlinear elliptic and parabolic partial differential equations and their multifaceted applications, such as in optimal transport, geometry, optics, image processing, engineering, linear and nonlinear programming.