An invitation to optimal transport, Wasserstein distances, and gradient flows
type: Item
instance of: scholarly article
title: An invitation to optimal transport, Wasserstein distances, and gradient flows
zbMATH Open document ID: 1472.49001
DOI: 10.4171/ETB/22
author: Alessio Figalli
author: Federico Glaudo
published in: EMS Textbooks in Mathematics
publication date: 2021-07-27T00:00:00Z
review text: This graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices. Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry. Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs. Chapter 3 utilizes the \([1,\infty)\ni p\)-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations. Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures. Chapter 5 suggests several applied topics of the optimal transport theory. Appendix A includes a set of some interesting exercises and their solutions. Appendix B outlines a proof of the disintegration theorem.
reviewed by: Jie Xiao
Mathematics Subject Classification ID: 49-01
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 49Q22
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 28A50
zbMATH DE Number: 7375481
zbMATH Keywords: optimal transport
zbMATH Keywords: Wasserstein distance
zbMATH Keywords: duality
zbMATH Keywords: gradient flows
zbMATH Keywords: measure theory
zbMATH Keywords: displacement convexity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/etb/22
OpenAlex ID: W3175831284
description: scientific article; zbMATH DE number 7375481