Math 217, Spring 2022
Topics in Applied Mathematics: Optimal Transport
Lectures: 1:00 pm  1:50, pm MWF, AP&M 5402
Instructor: Bo Li (Office: AP&M 5723. Office phone: (858) 5346932.
Email: bli@math.ucsd.edu)
Office hours: 2:00  2:45 Wednesdays and Fridays, or by appointment.
Course Description
This course focuses on applied and computational aspects of optimal transport (OT).
It will cover:

Monge and Kantorovich formulations of the OT problem. Duality. Existence results.
Wasserstein metric.

Regularization of the discrete OT problem. Numerical methods, stability and convergence.

Gradient flow with the Wasserstein metric. FokkerPlanck equation and other evolutionary equations.

Some applications in machine learning and molecular dynamics.
While not required, some background of optimization and measure theory will
be helpful. No knowledge in the related application areas is assumed.
Lecture Notes

Lecture 1: Course description. Monge's and Kontorovich's formulations
of the discrete OT problem.

Lecture 2: Kantorovich OT as linear programming. Monge vs. Kantorovich.

Lecture 3: Wasserstein metric (Wmetric) for discrete OT.

Lecture 4: Wmetric with the
discrete metric matrix. Discrete OT: Wmetric is equivalent to Euclid matric.

Lecture 5: Duality  weak and strong. The complementarity condition.

Lecture 6: Measuretheoretical descriptions of the OT problem
in Monge's and Katorovich's forms.

Lecture 7: Entropy regularization. KullbackLeibler divergence. General regularized OT.

Lecture 8: Sinkhorn's Theorem on matrix rescaling. Uniqueness.

Lecture 9: Sinkhorn's algorithm: Convergence.

Lecture 10: Sinkhorn's algorithm: convergence rate via Hilbert's metric.
FranklinLorenz theorem.

Lecture 11: Sinkhorn's algorithm: Overflow instability. Logdomain algorithm.

Lecture 12: Continuous OT: Monge's and Kantorovich's formulations. Exampls. M. vs. K.

Lecture 13: Properties of transport maps and plans. Existence Theorem. Direct methods.

Lecture 14: Narrow topology for probability measures. Weak lower semicontinuity of
Kcost functional.

Lecture 15: Ulam's Lemma. Prokhorov's Theorem. Narrow compactness of the set of transport plans.

Lecture 16: Wasserstein metric. Some examples.

Lecture 17: An elementary proof of the triangle inequality.
The betametric.

Lecture 18:
Completeness and compactness of the Wmetric. An example.

Lecture 19:
Separability of the Wmetric.

Lecture 20:
Characterization of the convergence w.r.t. the Wmetric.

Lecture 21:
The JKO scheme for heat equation and FokkerPlanck equaiton: steepest descent in Wmetric.

Lecture 22:
Proof of the convergence of the JKO scheme.

Lecture 23: Dual problem. Weak duality. Kantorovich (strong) duality.

Lecture 24: Proof of the Kantorovich duality for compact Polish spaces.

Lecture 25: The KantorovichRubinstein Theorem. cconcave functions.
 Lecture 26: Existence and characterization of optimal transport maps:
BrenierKnottSmith theorem.
 Lecture 27: Support of optimal transport plan is ccyclically monotone.

Lecture 28: csubdifferentials. Generalized Rockafellar Theorem.
 Lecture: Dynamic formulation of optimal transport.
References
 L. Ambrosio, E. Brue, and D. Semola, Lectures on Optimal Transport, Springer, 2021.
 L. Ambrosio, N. Gigli, and G. Savare,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Birkhauser, 2008.
(UCSD eversion available.)
 L. Ambrosio and N. Gigli, A User's Guide to Optimal Transport, in
Modelling and Optimisation of Flows on Networks, Editors: B. Piccoli and M. Rascle,
Lecture Notes in Mathematics 2062, Springer, 2010.
(UCSD eversion available.)
 A. Figalli and F. Glaudo, An Invitation to Optimal Transport, Wasserstein Distances,
and Gradient Flows, EMS Press, 2021.
 G. Peyre and M. Cuturi, Computational Optimal Transport,
Foundations and Trends in Machine Learning, Vol. 11, No. 56, pp. 355607, 2019.
(Eversion available for UCSD.)
 S. T. Rachev and L. Ruschendorf, Mass Transportation Problems:
Volume I: Theory, Springer, 1998.
(Eversion available for UCSD.)
 S. T. Rachev and L. Ruschendorf, Mass Transportation Problems:
Volume II: Applications, Springer, 1998.
(Eversion available for UCSD.)
 F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhauser, 2015.
(UCSD eversion available.)
 C. Villani, Topics in Optimal Transportation, Amer. Math Soc., 2003.
 C. Villani, Optimal Transport, Springer, 2009.
(UCSD eversion available.)
Some related research papers will be discussed.
Expected Work for Students
Students are expected to participate in class discussions and possibly read some papers with
brief oral report in class.
Last updated by Bo Li on May 3, 2022.
