Filippo Santambrogio
Prof. Dr. Filippo Santambrogio

Technische Universität München
Zentrum Mathematik, M8
Boltzmannstraße 3
85748 Garching bei München

Virtuelle Vorlesung zum Wintersemester 2021/22

Homepage: Filippo Santambrogio

Optimal Transport, Numerics and Sampling

  1. Monge and Kantorovich problems:
    Introduction to OT; the Monge problem and the Monge-Ampère equations; the Kantorovich formulation with proof of existence. The assignment problem as a discrete case of OT (permutations and bistochastic matrices).
  2. Duality and the Brenier theorem
    Formal duality, c-concave functions, existence in the problem. Existence and uniqueness of the Monge solution for twist costs, Brenier's theorem.
  3. Proof of the duality result
    Elements from convex analysis, proof of the strong duality.
  4. Monotonicity and cyclical monotonicity
    c-cyclical monotonicity. The 1D case. The Knothe map.
  5. About the L^1 case
    The L^1 case of the distance cost. Minimal flow formulation. Transport density.
  6. Wasserstein distances
    The Wasserstein distances W_p and the topology they induce. Elements from the analysis of curves in metric spaces.
  7. Curves in the Wasserstein space
    Absolutely continuous curves in W_p and the continuity equation. Geodesics in W_p.
  8. Around geodesics in W_p
    Geodesically convex functionals. Mc Cann's condition. The Benamou-Brenier formulation of OT and its numerical application.
  9. Semidiscrete OT
    Voronoi and Laguerre cells. Newton's algorithms for semidiscrete OT. Uniform and non-uniform optimal quantization, Lloyd algorithms, critical points.
  10. Linear programming and the Sinkhorn algorithm
    LP formulation of OT in the discrete case. Network formulation and network simplex. Improvement for separable costs on grids. Entropic regularization: primal and dual Sinkhorn algorithms and their convergence.
  11. Around Wasserstein distances
    Comparison between Wasserstein and negative Sobolev distances. Sharp bounds for integral sampling. The sliced Wasserstein distance.
  12. Optimal non-uniform quantization
    The optimal location of n points to approach a continuous distribution: asymptotics for large n.

ECTS Credits:
MA1001 Analysis 1, MA1002 Analysis 2, MA2003 Measure and Integration, MA3001 Functional Analysis. Suggested optional: MA2504 Convex Optimization, MA3005 Partial Differential Equations, MA3504 Convex Analysis

Recommended literature:
F. Santambrogio: Optimal Transport for Applied Mathematicians, Birkhauser (2015)
C. Villani: Topics in Optimal Transportation, American Mathematical Society (2003)
M. Cuturi, G. Peyré: Computational Optimal Transport: With Applications to Data Science, Foundations and Trends in Machine Learning, Now Publishers (2018)
G. Bouchitté, C. Jimenez, R. Mahadevan: Asymptotic of an optimal location problem, Comptes Rendus Mathematique 335(10):853-858 (2002)

Lecture Time: Wednesday 16:00 - 17:30 Uhr

First class: 20th October 2021
The course is given online via Zoom. If you are interested, please attend the first lecture on October 20. Day and time for further lectures may be adjusted to the needs of the majority of the participants. Recordings will be made available to those who are unable to attend.

-- MiriamManlik - 11 Jun 2021
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