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 42
Page xv
... jx yj and more general measures and spaces. The main idea by Kantorovich is that of looking at Monge's problem as connected to linear programming. Kantorovich indeed decided to change the point of view, and to describe the movement of ...
... jx yj and more general measures and spaces. The main idea by Kantorovich is that of looking at Monge's problem as connected to linear programming. Kantorovich indeed decided to change the point of view, and to describe the movement of ...
Page xvi
... jx yj has the following property: the space Rd can be decomposed in an essentially disjoint union of segments that are preserved by (i.e., is concentrated on pairs .x;y/ belonging to the same segment). These segments are built from a ...
... jx yj has the following property: the space Rd can be decomposed in an essentially disjoint union of segments that are preserved by (i.e., is concentrated on pairs .x;y/ belonging to the same segment). These segments are built from a ...
Page xviii
... jx yj) is also equivalent to the problem of minimizing the L1 norm of a vector field w under the condition r w D . This very last problem is presented in Chapter 4, and the traffic congestion and branched transport models are presented ...
... jx yj) is also equivalent to the problem of minimizing the L1 norm of a vector field w under the condition r w D . This very last problem is presented in Chapter 4, and the traffic congestion and branched transport models are presented ...
Page xxii
... jx yj..................................... 87 3.1.1 Dualityfordistancecosts....................................... 88 3.1.2 Secondaryvariationalproblem............
... jx yj..................................... 87 3.1.1 Dualityfordistancecosts....................................... 88 3.1.2 Secondaryvariationalproblem............
Page xxvii
... jxyj or the convex potential T D ru in the jx yj2 case. • Velocity fields are typically denoted by v, momentum fields by w (when they are not time dependent) or E (when they could be time dependent). Chapter 1 Primal and dual problems ...
... jxyj or the convex potential T D ru in the jx yj2 case. • Velocity fields are typically denoted by v, momentum fields by w (when they are not time dependent) or E (when they could be time dependent). Chapter 1 Primal and dual problems ...
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