Program

The school will mainly consist of

3 courses, of 4 lectures of 1.5 hours each,

starting from Monday afternoon and lasting on Friday morning in a informal and collaborative atmosphere.

  1. Computational Optimal Transport and its Applications
    Gabriel Peyré
    , CNRS and Ecole Normale Supérieure
  2. Economic applications of optimal transport
    Alfred Galichon, New York University, FAS and Courant Institute
  3. Conservation laws with nonlocal flux
    Andrea Tosin, Politecnico di Torino

On Monday morning, the course organizers will also offer a couple of introductory lectures on Optimal Transport, intended for young people not fully acquainted with the topic.

DETAILED PROGRAM

#1: Computational Optimal Transport and its Applications
Gabriel Peyré
, CNRS and Ecole Normale Supérieure

Abstract: In this short course, I will review numerical approaches for the approximate resolution of optimization problems related to optimal transport. I will also give some insight on how to apply these methods to imaging sciences and machine learning problems.

Lecture 1: Algorithmic foundations
– Overview of applications in imaging and learning
– Special cases: 1-D, Gaussians
– Network simplex
– Semi-discrete, auction

Lecture 2: Entropic regularization
– Regularization and approximation
– Sinkhorn’s algorithm
– Hilbert’s metric, Perron-Frobenius
– Extensions: multimarginal, unbalanced

Lecture 3: Variational Wasserstein problems
– Wasserstein barycenters
– Gradient flows
– Gromov-Wasserstein

Lecture 4: Density fitting and generative modeling
– Statistical divergences
– Sample complexity
– Minimum Kantorovich Estimator
– Deep learning and generative models

Reference:
Gabriel Peyré and Marco Cuturi,
Computational Optimal Transport
https://optimaltransport.github.io/

#2: Economic applications of optimal transport
Alfred Galichon, New York University, FAS and Courant Institute

Abstract: These lectures will cover applications of optimal transport to economics. The first lecture will cover discrete choice models, from the classical theory to more recent advances. The classical Generalized Extreme Value (GEV) specification will be recalled, as well as Maximum Likelihood estimation in the parametric case. Comparative statics results will be derived using tools from convex analysis, and nonparametric identification will be worked out using optimal transport theory.
The second lecture will be devoted to matching models with unobservable heterogeneity. Equilibrium computation and identification will be worked out using techniques from general equilibrium. The more specific, but empirically relevant logit case, will be efficiently addressed using the Iterative Fitting Proportional Procedure.
The third lecture will cover the statistical estimation of these models, in link with generalized linear models and pseudo-Poisson maximum likelihood estimation. An application to the econometrics of the marriage market will be provided.
The fourth lecture will introduce the “equilibrium transport problem”, which is a generalization of the Monge-Kantorovich problem in which one relaxes the very strong assumption that the utilities should be quasi linear in payments, that is, everybody has a valuation expressed in the same monetary unit, which can be transferred without losses. That assumption is, of course, very strong as various nonlinearities may arise in practice; these might be induced by taxes, by regulations such as price caps, by risk aversion, or by other various inefficiencies. Removing this strong assumption requires moving beyond optimal transport theory, and moving into “Equilibrium transport theory”. We shall give existence and uniqueness results in the regularized case, as well as computational methods.

Lecture 1. Multinomial choice models and their inversion
Based on:
– Chiong, Galichon, Shum (2016). Duality in dynamic discrete choice models. Quantitative Economics.
– Bonnet, Galichon, Shum (2017). Yogurts choose consumers? Identification of Random Utility Models via Two-Sided Matching. Preprint.

Lecture 2. Separable matching models with heterogeneity 
Based on:
– Galichon, Salanié (2010) Matching with Trade-offs: Revealed Preferences over Competing Characteristics. Technical report.
– Galichon, Salanié (2017) Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models. Preprint.

Lecture 3. Affinity estimation: a framework for statistical inference in matching models
Based on:
– Dupuy, Galichon (2014) Personality traits and the marriage market. Journal of Political Economy.
– Dupuy, Galichon, Shum (2017) Estimating matching affinity matrix under low-rank constraints. Preprint.

Lecture 4. Equilibrium transport: incorporating taxes in matching models
Based on:
– Galichon, Kominers, Weber (2017) Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility. Preprint.
– Dupuy, Galichon, Jaffe, Kominers (2017) Taxation in matching markets. Preprint.

#3: Conservation laws with nonlocal flux
Andrea Tosin, Politecnico di Torino

Abstract: Conservation laws with nonlocal flux arise naturally in the mathematical modelling of multi-agent systems to represent interactive dynamics at the macroscopic scale. These models describe the reorganisation of the agent distribution as a conservative transport of mass via a velocity field determined by the interactions among the agents in suitable neighbourhoods or in the whole space.
In this course we will first derive such models starting from a microscopic description of the stochastic particle system via methods typical of the collisional kinetic theory. Then we will abstract them for agent distributions represented by generic measures, of which we will discuss a multiscale conservative transport possibly also on networks. Reference applications will be models of crowd dynamics and vehicular traffic.

Lecture 1: Boltzmann-type kinetic equations

Lecture 2: Vlasov-Fokker-Planck equations

Lecture 3: Multiscale conservation laws

Lecture 4: Transport of measures on networks.

References:

F. Camilli, R. De Maio, A. Tosin.
Transport of measures on networks, Netw. Heterog. Media, 12(2):191-215, 2017

E. Cristiani, B. Piccoli, A. Tosin.
Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, 2014

L. Pareschi, G. Toscani.
Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2013