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. |
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Page xxv
... finite measures, positive finite measures, and vector measures (valued in R") on X. / (Q): on 2 c R", the space of measures we ./"(2) with V. we .4(2). R", T', S': the d-dimensional Euclidean space, flat torus, and sphere. C(X), Ch(X) ...
... finite measures, positive finite measures, and vector measures (valued in R") on X. / (Q): on 2 c R", the space of measures we ./"(2) with V. we .4(2). R", T', S': the d-dimensional Euclidean space, flat torus, and sphere. C(X), Ch(X) ...
Page 4
... finite signed measure on a metric space X is a map associating to every Borel subset A X a value .A/ 2 R (we will see in Chapter 4 the case of vector measures, where is valued in Rd) such that, for every countable disjoint union A D S i ...
... finite signed measure on a metric space X is a map associating to every Borel subset A X a value .A/ 2 R (we will see in Chapter 4 the case of vector measures, where is valued in Rd) such that, for every countable disjoint union A D S i ...
Page 8
... finite regular measure (i.e., any Borel set A X satisfies .A/ D supf .K/ W K A; K compactg D inff.B/ W B A; B openg). The arrival space Y will be supposed to be second countable (i.e., it admits a countable family .Bi/i of open sets ...
... finite regular measure (i.e., any Borel set A X satisfies .A/ D supf .K/ W K A; K compactg D inff.B/ W B A; B openg). The arrival space Y will be supposed to be second countable (i.e., it admits a countable family .Bi/i of open sets ...
Page 9
... finite-dimensional spaces, which is not the case here. To handle infinite-dimensional situations, one needs to use a more general mini-max theorems2. For now, we prefer not to investigate anymore the question of obtaining the duality ...
... finite-dimensional spaces, which is not the case here. To handle infinite-dimensional situations, one needs to use a more general mini-max theorems2. For now, we prefer not to investigate anymore the question of obtaining the duality ...
Page 11
... (finite, infinite, countable, uncountable, etc.) of functions, all satisfying the same condition lf, (x) —f, (x)| s o(d(x, x')). Consider f defined through f(x) := inf.f., (x). Then f also satisfies the same estimate. This can be easily ...
... (finite, infinite, countable, uncountable, etc.) of functions, all satisfying the same condition lf, (x) —f, (x)| s o(d(x, x')). Consider f defined through f(x) := inf.f., (x). Then f also satisfies the same estimate. This can be easily ...
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