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Scheda Riassuntiva
Anno Accademico 2021/2022
Tipo incarico Dottorato
Insegnamento 057408 - WASSERSTEIN METRICS: FROM DIFFUSION EQUATIONS TO STATISTICAL ISSUES
Docente Bassetti Federico
Cfu 5.00 Tipo insegnamento Monodisciplinare

Corso di Dottorato Da (compreso) A (escluso) Insegnamento
MI (1385) - MODELLI E METODI MATEMATICI PER L'INGEGNERIA / MATHEMATICAL MODELS AND METHODS IN ENGINEERINGAZZZZ057408 - WASSERSTEIN METRICS: FROM DIFFUSION EQUATIONS TO STATISTICAL ISSUES

Programma dettagliato e risultati di apprendimento attesi

Goals

The goal of this course is to provide an overview of the basic theory  of the Kantorovich-Wasserstein metric and its wide context of application.

Topics

The main topics will include: a short introduction to probability metrics and to the theory of the Kantorovich-Wasserstein metrics, connections with  the theory of linear (and possibly nonlinear) PDEs of diffusion type seen as metric gradient flows, applications to  quantitative central limit problems, computational issues and, time permitting, also to statistics (Wasserstein means and clustering)

• An introduction to probability metrics: semi-distances, simple and compound metrics, ideal metrics.

• Kantorovich-Wasserstein functional: duality and basic properties. Connections with the Transportation Problem.

• Weak convergence of measures and the topology induced by the Kantorovich-Wasserstein metrics. The Wasserstein distance on the real line. 

• Gradient flows generated by lower-semicontinuous convex functionals: from the classical theory in R n to the EVI (Evolution Variational Inequality) formulation in a metric setting.

• An overview of the differentiable structure of the Wasserstein space of probability measures in R n.

• The heat equation in the Euclidean space as a gradient flow of the entropy functional in the Wasserstein space, and related stability properties. If time allows, possible extensions to more general (nonlinear) diffusion equations and to non-Euclidean frameworks.

• Rate of convergence in central limit problems. 

• Wasserstein Barycenters and some statistical applications.


Note Sulla Modalità di valutazione

The student can give a seminar either on a specific topic addressed in the course or on a research paper suggested by the lecturers.


Intervallo di svolgimento dell'attività didattica
Data inizio
Data termine

Calendario testuale dell'attività didattica

The detailed timetable will be discussed in a preliminary meeting with the students, to be held in December. Tentatively, we plan to give 2 (2/3 hours) lectures per week, for about 5/6 weeks.


Bibliografia
Risorsa bibliografica facoltativaL. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Editore: Birkhäuser Verlag, Basel
Risorsa bibliografica facoltativaM. Cuturi, G. Peyré, Computational Optimal Transport
Risorsa bibliografica facoltativaC. Villani, Topics in Optimal Transportation, Editore: AMS (American Mathematical Society), 2003.

Software utilizzato
Nessun software richiesto

Mix Forme Didattiche
Tipo Forma Didattica Ore didattiche
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esercitazione
0.0
laboratorio informatico
0.0
laboratorio sperimentale
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progetto
0.0
laboratorio di progetto
0.0

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese

Note Docente
schedaincarico v. 1.7.2 / 1.7.2
Area Servizi ICT
04/10/2022