{Euclidean, metric, and Wasserstein} gradient flows: an overview

Bulletin of Mathematical Sciences, Mar 2017

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan–Kinderlehrer–Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

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{Euclidean, metric, and Wasserstein} gradient flows: an overview

{Euclidean, metric, and Wasserstein} gradient flows: an overview Filippo Santambrogio 0 0 Laboratoire de Mathématiques D'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay , 91405 Orsay Cedex , France This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan-Kinderlehrer-Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces. Cauchy problem; Subdifferential; Analysis in metric spaces; Optimal transport; Wasserstein distances; Heat flow; Fokker-Planck equation; Numerical methods; Contractivity; Metric measure spaces 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 From Euclidean to metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The general theory in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Existence of a gradient flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Uniqueness and contractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Gradient flows in the Wasserstein space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries on optimal transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Wasserstein distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Minimizing movement schemes in the Wasserstein space and evolution PDEs . . . . . . . . . . 4.4 How to prove convergence of the JKO scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Geodesic convexity in W2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Other gradient-flow PDEs and variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The heat flow in metric measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Gradient flows comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Gradient flows, or steepest descent curves, are a very classical topic in evolution equations: take a functional F defined on a vector space X, and, instead of looking at points x minizing F (which is related to the static equation ∇ F(x) = 0), we look, given an initial point x0, for a curve starting at x0 and trying to minimize F as fast as possible (in this case, we will solve equations of the form x (t) = −∇ F(x(t))). As we speak of gradients (which are element of X, and not of X as the differential of F should be), it is natural to impose that X is an Hilbert space (so as to identify it with its dual and produce a gradient vector). In the finite-dimensional case, the above equation is very easy to deal with, but also the infinite-dimensional case is not so exotic. Indeed, just think at the evolution equation ∂t u = u, which is the evolution variant of the static Laplace equation − u = 0. In this way, the heat equation is the gradient flow, in the L2 Hilbert space, of the Dirichlet energy F(u) = 21 ´ |∇u|2, of which − u is the gradient in the appropriate sense (more generally, one could consider equations of the form ∂t u = δF/δu, where this notation stands for the first variation of F). But this is somehow classical… The renovated interest for the notion of gradient flow arrived between the end of the twentieth century and the beginning of the twentyfirst century, with the work of Jordan et al. [54] and then of Otto [78], who saw a gradient flow structure in some equations of the form ∂t − ∇ · ( v) = 0, where the vector field v is given by v = ∇[δF/δ ]. This requires to use the space of probabilities on a given domain, and to endow it with a non-linear metric structure, derived from the theory of optimal transport. This theory, initiated by Monge [77] in the eighteenth century, then developed by Kantorovich in the ’40s [55], is now well-established (many text (...truncated)


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Filippo Santambrogio. {Euclidean, metric, and Wasserstein} gradient flows: an overview, Bulletin of Mathematical Sciences, 2017, pp. 87-154, Volume 7, Issue 1, DOI: 10.1007/s13373-017-0101-1