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 87
Page 6
... implies in particular ( ~ +co. For every k, let us choose a pointy, such that f(y) + kd(y,x) < f(x) + 1/k. We get d(y,x) = oil-sou s £, thanks to the lower bound on f and to £ 3 co. Hence, we know y → x. Yet, we have f(x) + 1/k > f(y) ...
... implies in particular ( ~ +co. For every k, let us choose a pointy, such that f(y) + kd(y,x) < f(x) + 1/k. We get d(y,x) = oil-sou s £, thanks to the lower bound on f and to £ 3 co. Hence, we know y → x. Yet, we have f(x) + 1/k > f(y) ...
Page 7
... implies lim upscab dyn is | local) dy + 2Mö < scab dy + 2M6 Pl and, since 6 is arbitrary, lim sup, sc(a, b) dy, s sc(a, b) dy. This proves upper semi-continuity of the integral functional when c is bounded by M. An analogous computation ...
... implies lim upscab dyn is | local) dy + 2Mö < scab dy + 2M6 Pl and, since 6 is arbitrary, lim sup, sc(a, b) dy, s sc(a, b) dy. This proves upper semi-continuity of the integral functional when c is bounded by M. An analogous computation ...
Page 11
... implies a bound on the modulus of continuity. Box 1.8. Memo: Continuity of functions defined as an infor Sup Proposition. Let (f, ), be a family (finite, infinite, countable, uncountable, etc.) of functions, all satisfying the same ...
... implies a bound on the modulus of continuity. Box 1.8. Memo: Continuity of functions defined as an infor Sup Proposition. Let (f, ), be a family (finite, infinite, countable, uncountable, etc.) of functions, all satisfying the same ...
Page 13
... implies equality -a.e. These functions being continuous, the equality is satisfied on a closed set, i.e., on the support of the measure (let us recall the definition of support of a measure, not to be confused with sets where it is ...
... implies equality -a.e. These functions being continuous, the equality is satisfied on a closed set, i.e., on the support of the measure (let us recall the definition of support of a measure, not to be confused with sets where it is ...
Page 18
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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