Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling
Birkhäuser, 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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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