nach oben

2009 | Buch

Kapitel lesen Erstes Kapitel lesen

Optimal Transport

Old and New

verfasst von: Cédric Villani

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results.

PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject.

Inhaltsverzeichnis

Frontmatter

Introduction

Frontmatter

1. Couplings and changes of variables

Abstract

Couplings are very well-known in all branches of probability theory, but since they will occur again and again in this course, it might be a good idea to start with some basic reminders and a few more technical issues.

Cédric Villani

2. Three examples of coupling techniques

Abstract

In this chapter I shall present three applications of coupling methods. The first one is classical and quite simple, the other two are more original but well-representative of the topics that will be considered later in these notes. The proofs are extremely variable in difficulty and will only be sketched here; see the references in the bibliographical notes for details.

Cédric Villani

3. The founding fathers of optimal transport

Abstract

Like many other research subjects in mathematics, the field of optimal transport was born several times. The first of these births occurred at the end of the eighteenth century, by way of the French geometer Gaspard Monge.

Cédric Villani

Qualitative description of optimal transport

Frontmatter

4. Basic properties

Abstract

The proof relies on basic variational arguments involving the topology of weak convergence (i.e. imposed by bounded continuous test functions).

Cédric Villani

5. Cyclical monotonicity and Kantorovich duality

Abstract

To go on, we should become acquainted with two basic concepts in the theory of optimal transport. The first one is a geometric property called cyclical monotonicity; the second one is the Kantorovich dual problem, which is another face of the original Monge—Kantorovich problem. The main result in this chapter is Theorem 5.10.

Cédric Villani

6. The Wasserstein distances

Abstract

Assume, as before, that you are in charge of the transport of goods between producers and consumers, whose respective spatial distributions are modeled by probability measures.

Cédric Villani

7. Displacement interpolation

Abstract

As in the previous chapter I shall assume that the initial and final probability measures are defined on the same Polish space (X, d).

Cédric Villani

8. The Monge—Mather shortening principle

Abstract

Monge himself made the following important observation. Consider the transport cost c(x, y) = |x - y| in the Euclidean plane, and two pairs (x1, y1), (x2, y2), such that an optimal transport maps x1 to y1 and x2 to y2. (In our language, (x1, y1) and (x2, y2) belong to the support of an optimal coupling p.)

Cédric Villani

9. Solution of the Monge problem I: global approach

Abstract

In the present chapter and the next one I shall investigate the solvability of the Monge problem for a Lagrangian cost function. Recall from Theorem 5.30 that it is sufficient to identify conditions under which the initial measure µ does not see the set of points where the c-subdifferential of a c-convex function ψ is multivalued.

Cédric Villani

10. Solution of the Monge problem II: Local approach

Abstract

In the previous chapter, we tried to establish the almost sure singlevaluedness of the c-subdifferential by an argument involving “global” topological properties, such as connectedness. Since this strategy worked out only in certain particular cases, we shall now explore a different method, based on local properties of c-convex functions. The idea is that the global question “Is the c-subdifferential of ψ at x single-valued or not?” might be much more subtle to attack than the local question “Is the function ψ differentiable at x or not?” For a large class of cost functions, these questions are in fact equivalent; but these different formulations suggest different strategies. So in this chapter, the emphasis will be on tangent vectors and gradients, rather than points in the c-subdifferential.

Cédric Villani

11. The Jacobian equation

Abstract

There are two important things that one should check before writing the Jacobian equation: First, T should be injective on its domain of definition; secondly, it should possess some minimal regularity.

Cédric Villani

12. Smoothness

Abstract

The smoothness of the optimal transport map may give information about its qualitative behavior, as well as simplify computations. So it is natural to investigate the regularity of this map.

Cédric Villani

13. Qualitative picture

Abstract

This chapter is devoted to a recap of the whole picture of optimal transport on a smooth Riemannian manifold M. For simplicity I shall not try to impose the most general assumptions. A good understanding of this chapter is sufficient to attack Part II of this course.

Cédric Villani

Optimal transport and Riemannian geometry

Frontmatter

14. Ricci curvature

Abstract

Curvature is a generic name to designate a local invariant of a metric space that quantifies the deviation of this space from being Euclidean. (Here “local invariant” means a quantity which is invariant under local isometries.) It is standard to define and study curvature mainly on Riemannian manifolds, for in that setting definitions are rather simple, and the Riemannian structure allows for “explicit” computations. Throughout this chapter, M will stand for a complete connected Riemannian manifold, equipped with its metric g.

Cédric Villani

15. Otto calculus

Abstract

One of the reasons for the popularity of Riemannian geometry (as opposed to the study of more general metric structures) is that it allows for rather explicit computations.

Cédric Villani

16. Displacement convexity I

Abstract

It is a natural problem to identify functionals that are convex on the Wasserstein space. In his 1994 PhD thesis, McCann established and used the convexity of certain functionals on P2(Rn) to prove the uniqueness of their minimizers.

Cédric Villani

17. Displacement convexity II

Abstract

In Chapter 16, a conjecture was formulated about the links between displacement convexity and curvature-dimension bounds; its plausibility was justified by some formal computations based on Otto's calculus. In the present chapter I shall provide a rigorous justification of this conjecture. For this I shall use a Lagrangian point of view, in contrast with the Eulerian approach used in the previous chapter. Not only is the Lagrangian formalism easier to justify, but it will also lead to new curvature-dimension criteria based on so-called “distorted displacement convexity”.

The main results in this chapter are Theorems 17.15 and 17.37.

Cédric Villani

18. Volume control

Abstract

Controlling the volume of balls is a universal problem in geometry. This means of course controlling the volume from above when the radius increases to infinity; but also controlling the volume from below when the radius decreases to 0. The doubling property is useful in both situations.

Cédric Villani

19. Density control and local regularity

Abstract

The following situation occurs in many problems of local regularity: Knowing a certain estimate on a certain ball Br(x0), deduce a better estimate on a smaller ball, say B _r _/2(x0). In the fifties, this point of view was put to a high degree of sophistication by De Giorgi in his famous proof of Hölder estimates for elliptic second-order partial differential equations in divergence form; and it also plays a role in the alternative solutions found at the same time by Nash, and later by Moser. When fine analysis on metric spaces started to develop, it became an important issue to understand what were the key ingredients lying at the core of the methods of De Giorgi, Nash and Moser. It is now accepted by many that the two key inequalities are:

a doubling inequality for the reference volume measure;
a local Poincaré inequality, controlling the deviation of a function on a smaller ball by the integral of its gradient on a larger ball.

Cédric Villani

20. Infinitesimal displacement convexity

Abstract

The goal of the present chapter is to translate displacement convexity inequalities of the form “the graph of a convex function lies below the chord” into inequalities of the form “the graph of a convex function lies above the tangent” — just as in statements (ii) and (iii) of Proposition 16.2. This corresponds to the limit t → 0 in the convexity inequality.

Cédric Villani

21. Isoperimetric-type inequalities

Abstract

It is a fact of experience that several inequalities with isoperimetric content can be retrieved by considering the above-tangent formulation of displacement convexity. Here is a possible heuristic explanation for this phenomenon. Assume, for the sake of the discussion, that the initial measure is the normalized indicator function of some set A. Think of the functional U_v as the internal energy of some fluid that is initially confined in A. In a displacement interpolation, some of the mass of the fluid will have to flow out of A, leading to a variation of the energy (typically, more space available means less density and less energy). The decrease of energy at initial time is related to the amount of mass that is able to flow out of A at initial time, and that in turn is related to the surface of A (a small surface leads to a small variation, because not much of the fluid can escape). So by controlling the decrease of energy, one should eventually gain control of the surface of A.

Cédric Villani

22. Concentration inequalities

Abstract

The theory of concentration of measure is a collection of results, tools and recipes built on the idea that if a set A is given in a metric probability space (X, d, P ), then the enlargement Ar := {x; d(x,A) = r} might acquire a very high probability as r increases.

Cédric Villani

23. Gradient flows I

Abstract

Take a Riemannian manifold M and a function F : M ? R, which for the sake of this exposition will be assumed to be continuously differentiable.

Cédric Villani

24. Gradient flows II: Qualitative properties

Abstract

Theorem 23.19 provides an interpretation of (24.1) as a gradient flow in the Wasserstein space P2(M). What do we gain from that information? A first possible answer is a new physical insight.

Cédric Villani

25. Gradient flows III: Functional inequalities

Abstract

In the preceding chapter certain functional inequalities were used to provide quantitative information about the behavior of solutions to certain partial differential equations. In the present chapter, conversely, the behavior of solutions to certain partial differential equations will help establish certain functional inequalities.

Cédric Villani

Synthetic treatment of Ricci curvature

Frontmatter

26. Analytic and synthetic points of view

Abstract

The present chapter is devoted to a simple pedagogical illustration of the opposition between the “analytic” and “synthetic” points of view.

Cédric Villani

27. Convergence of metric-measure spaces

Abstract

The central question in this chapter is the following:What does it mean to say that a metric-measure space (X, d _x, _vx) is “close” to another metric-measure space (Y, d _y, _vy)? We would like to have an answer that is as “intrinsic” as possible, in the sense that it should depend only on the metric-measure properties of X and Y.

So as not to inflate this chapter too much, I shall omit many proofs when they can be found in accessible references, and prefer to insist on the main stream of ideas.

Cédric Villani

28. Stability of optimal transport

Abstract

This chapter is devoted to the following theme: Consider a family of geodesic spaces Xkich converges to some geodesic space X; does this imply that certain basic objects in the theory of optimal transport on Xk “pass to the limit”? In this chapter I shall show that the answer is affirmative: One of the main results is that the Wasserstein space P2(Xk) converges, in (local) Gromov—Hausdorff sense, to the Wasserstein space P2(X). Then I shall consider the stability of dynamical optimal transference plans, and related objects (displacement interpolation, kinetic energy, etc.). Compact spaces will be considered first, and will be the basis for the subsequent treatment of noncompact spaces.

Cédric Villani

29. Weak Ricci curvature bounds I: Definition and Stability

Abstract

How to generalize this definition in such a way that it would make sense in a possibly nonsmooth metric-measure space?

Cédric Villani

30. Weak Ricci curvature bounds II: Geometric and analytic properties

Abstract

In the previous chapter I introduced the concept of weak curvaturedimension bound, which extends the classical notion of curvaturedimension bound from the world of smooth Riemannian manifolds to the world of metric-measure geodesic spaces; then I proved that such bounds are stable under measured Gromov—Hausdorff convergence.

Cédric Villani

Backmatter

Titel: Optimal Transport
verfasst von: Cédric Villani
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-71050-9
Print ISBN: 978-3-540-71049-3
DOI: https://doi.org/10.1007/978-3-540-71050-9