Self-Organized Network Flows
A model for traffic flow in street networks or material flows in supply networks is presented, that takes into account the conservation of cars or materials and other significant features of traffic flows such as jam formation, spillovers, and load-depend
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Abstract This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.
1 Introduction The opportunity to write down these notes on Optimal Transport has been the CIME course in Cetraro given by the first author in 2009. Later on the second author joined to the project, and the initial set of notes has been enriched and made more detailed, in particular in connection with the differentiable structure of the Wasserstein space, the synthetic curvature bounds and their analytic implications. Some of the results presented here have not yet appeared in a book form, with the exception of [44]. It is clear that this subject is expanding so quickly that it is impossible to give an account of all developments of the theory in a few hours, or a few pages. A more modest approach is to give a quick mention of the many aspects of the theory,
L. Ambrosio () Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy e-mail: [email protected] N. Gigli Universit´e de Nice, Math´ematiques, Parc Valrose, 06108 Nice, France e-mail: [email protected] L. Ambrosio et al., Modelling and Optimisation of Flows on Networks, Lecture Notes in Mathematics 2062, DOI 10.1007/978-3-642-32160-3 1, © Springer-Verlag Berlin Heidelberg 2013
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stimulating the reader’s curiosity and leaving to more detailed treatises as [7] (mostly focused on the theory of gradient flows) and the monumental book [80] (for a—much—broader overview on optimal transport). In chapter “A User’s Guide to Optimal Transport” we introduce the optimal transport problem and its formulations in terms of transport maps and transport plans. Then we introduce basic tools of the theory, namely the duality formula, the c-monotonicity and discuss the problem of existence of optimal maps in the model case costDdistance2 . In chapter “Hyperbolic Conservation Laws: An Illustrated Tutorial” we introduce the Wasserstein distance W2 on the set P2 .X / of probability measures with finite quadratic moments and X is a generic Polish space. This distance naturally arises when considering the optimal transport problem with quadratic cost. The connections between geodesics in P2 .X / and geodesics in X and between the time evolution of Kantorovich potentials and the Hopf–Lax semigroup are discussed in detail. Also, when looking at geodesics in this space, and in particular when the underlying metric space X is a Riemannian manifold M , one is naturally lead to the so-called time-dependent optimal transport problem, where geodesics are singled out by an action minimization principle. This
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