Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and ModelingBirkhäuser, 17 oct. 2015 - 353 pages This monograph presents a rigorous mathematical introduction to optimal transport as a variational problem, its use in modeling various phenomena, and its connections with partial differential equations. Its main goal is to provide the reader with the techniques necessary to understand the current research in optimal transport and the tools which are most useful for its applications. Full proofs are used to illustrate mathematical concepts and each chapter includes a section that discusses applications of optimal transport to various areas, such as economics, finance, potential games, image processing and fluid dynamics. Several topics are covered that have never been previously in books on this subject, such as the Knothe transport, the properties of functionals on measures, the Dacorogna-Moser flow, the formulation through minimal flows with prescribed divergence formulation, the case of the supremal cost, and the most classical numerical methods. Graduate students and researchers in both pure and applied mathematics interested in the problems and applications of optimal transport will find this to be an invaluable resource. |
À l'intérieur du livre
Résultats 1-5 sur 74
Page xi
... inequalities) and outside mathematics (histogram equalization in image processing). Chapter 3 deals with some limit cases, not covered in Chapter 1. Indeed, from the results of the first chapter, we know how to handle transport costs of ...
... inequalities) and outside mathematics (histogram equalization in image processing). Chapter 3 deals with some limit cases, not covered in Chapter 1. Indeed, from the results of the first chapter, we know how to handle transport costs of ...
Page xii
... inequalities but also equilibria in urban regions. Chapter 8 gives an original presentation of one of the most striking applications of optimal transport: gradient flows in Wasserstein spaces, which allow us to deal with many evolution ...
... inequalities but also equilibria in urban regions. Chapter 8 gives an original presentation of one of the most striking applications of optimal transport: gradient flows in Wasserstein spaces, which allow us to deal with many evolution ...
Page xvii
... inequalities that can be proven (or reproven) in this way. In this book, we only present the isoperimetric inequality (Section 2.5.3) and the Brunn-Minkowski inequality (Section 7.4.2). We stress that one of the most refined advances in ...
... inequalities that can be proven (or reproven) in this way. In this book, we only present the isoperimetric inequality (Section 2.5.3) and the Brunn-Minkowski inequality (Section 7.4.2). We stress that one of the most refined advances in ...
Page xxii
... . . . . . . . . . . . . . . . 81 2.5.3 Isoperimetric inequality via Knothe or Brenier maps......... 83 3 L1 and L1 theory ............................................................ 87 3.1 The Monge case, with cost jx yj.........
... . . . . . . . . . . . . . . . 81 2.5.3 Isoperimetric inequality via Knothe or Brenier maps......... 83 3 L1 and L1 theory ............................................................ 87 3.1 The Monge case, with cost jx yj.........
Page 9
... inequalities. Hence, an important tool will be duality theory, which is typically used for convex problems. We will find a dual problem (DP) for (KP) and exploit the relations between dual and primal. The first thing we will do is ...
... inequalities. Hence, an important tool will be duality theory, which is typically used for convex problems. We will find a dual problem (DP) for (KP) and exploit the relations between dual and primal. The first thing we will do is ...
Table des matières
1 | |
2 Onedimensional issues | 58 |
3 L1 and L theory | 87 |
4 Minimal flows | 120 |
5 Wasserstein spaces | 177 |
6 Numerical methods | 219 |
7 Functionals over probabilities | 249 |
8 Gradient flows | 285 |
Exercises | 324 |
References | 339 |
Index | 351 |
Autres éditions - Tout afficher
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs ... Filippo Santambrogio Aucun aperçu disponible - 2016 |
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs ... Filippo Santambrogio Aucun aperçu disponible - 2015 |
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs ... Filippo Santambrogio Aucun aperçu disponible - 2015 |
Expressions et termes fréquents
absolutely continuous algorithm apply approximation assumption atomic barycenter boundary bounded Brenier Chapter compact compute concave condition consider constraint continuity equation continuous function convex function countable curves defined definition denote density differentiable displacement convexity domain dual duality equality equilibrium existence fact finite function f geodesic given gives gradient flow hence Hint to Ex implies inequality infimum jx yj Kantorovich potential Knothe Lebesgue Lebesgue measure Lemma linear Lipschitz continuous lower semi-continuous Lp norms mass Math metric space Monge Monge-Ampère equation monotone Moreover norm Note obtained optimal map optimal transport map optimal transport plan particles particular probability measures Proof properties Proposition prove quadratic cost regularity result Santambrogio satisfies Section semi-continuity sequence ſº solution solve strictly convex Suppose Theorem triangle inequality unique variational vector field Wasserstein distances weak convergence