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MA957 - Topics in Optimal Transport

Lecturer: Marie-Therese Wolfram


Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Delivery: There will be 2 live lectures (Tuesdays and Fridays at 10:00) plus some pre-recorded lectures til (at least) the end of January.

Prerequisites: Measure theory and functional analysis as well as some background on elliptic and parabolic PDEs.

Content : The aim of this module is to give an an overview on the theory of optimal transport (OT), its connection to PDEs and how it OT techniques be used in applications such as medical imaging, economics and data science.

  • Formulation of Optimal Transport: from Monge to Kantorovich: existence and characterisation of transportation maps
  • Kantorovich Duality
  • Wasserstein Spaces and Geodesics: Wasserstein distance and the topology of Wasserstein spaces
  • Gradient Flows and Nonlinear Partial Differential Equations
  • Computational Optimal Transport: entropic regularisation and Sinkhorn's algorithm, variational Wasserstein problems, auction algorithm

References: I will follow the books of Santambrogio and Villani (2003) and use Peyre and Cuturi to discuss computational aspects.

  • L. Ambogio, N. Gigli and G. Savare, Gradient flows in metric spaces and the space of probability measures, Lectures in Mathematics, ETH Zuerich, Birkhaeuser, Basel, 2008 Encore
  • G. Peyre and M. Cuturi, Computational optimal transport: with applications to data science, Foundations and Trends in Machine Learning, 2019 Arxiv 
  • F. Santambrogio, Optimal Transport for applied mathematicians, Birkhaeuser Springer, Basel, 2015 Encore
  • C. Villani, Topics in optimal transportation, American Mathematical Society, 2003 Encore
  • C. Villani, Optimal transport: old and new, Springer Berlin, 2009 Encore

The lecture notes will be supplemented with Python and Jupyter scripts, which use the Python OT libary and Scikit