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Dettaglio Insegnamento
| Academic Year |
2021/2022 |
| Name |
Dott. - MI (1385) Modelli e Metodi Matematici per l'Ingegneria / Mathematical Models and Methods in Engineering |
| Programme Year |
1 |
| ID Code |
057408 |
| Course Title |
WASSERSTEIN METRICS: FROM DIFFUSION EQUATIONS TO STATISTICAL ISSUES |
| Course Type |
MONODISCIPLINARE |
| Credits (CFU / ECTS) |
5.0 |
| Course Description |
The Kantorovich-Wasserstein metric has been successfully used in several branches of mathe-
matics, including probability, analysis and statistics. Moreover, it has recently found applications
also in computer science, machine learning and image processing.
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. The main topics will include a short
introduction to probability metrics and to the theory of the Kantorovich-Wasserstein metrics, con-
nections with the theory of linear (and possibly nonlinear) PDEs of diusion type seen as metric
gradient
ows, applications to quantitative central limit problems, computational issues and, time
permitting, also to statistics (Wasserstein means and clustering).
program:
An introduction to probability metrics: semi-distances, simple and compound metrics, ideal metrics. [16]
Kantorovich-Wasserstein functional: duality and basic properties. Connections with the Transportation Problem. [2, 20, 21]
Weak convergence of measures and the topology induced by the Kantorovich-Wassersteinmetrics. The Wasserstein distance on the real
line. [2, 16, 20, 21]
Gradient flows generated by lower-semicontinuous convex functionals: from the classical theory in Rn to the EVI (Evolution Variational Inequality) formulation in a metric setting.[17, 18].
An overview of the differentiable structure of the Wasserstein space of probability measures in Rn. [2, 4]
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. [2, 3, 4,7, 9, 13, 14].
Rate of convergence in central limit problems. [8, 15]
Wasserstein Barycenters and some statistical applications. [15, 19]
Computational issues in discrete setting: regularized transport and connection with flow on graphs. [1, 5, 6, 10, 11, 18, 19] |
| Scientific-Disciplinary Sector (SSD)
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SSD Code
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SSD Description
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CFU
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MAT/06
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PROBABILITY AND STATISTICS
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2.5
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Alphabetical group
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Professor
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Course details
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From (included)
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To (excluded)
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A
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ZZZZ
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Bassetti Federico, Muratori Matteo
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