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Discrete-to-Continuum Limits of Transport Problems and Gradient-Flow Evolutions

Lorenzo Portinale
IST Austria
Friday, November 6, 2020 - 16:00
Zoom Meeting
In the seminal work of Jordan, Kinderlehrer and Otto (’98) the authors showed
that the heat flow on Rd can be seen as gradient flow of the relative entropy functional in the space of probability measures with respect to the Wasserstein distance $\mathbb W_2$. The correspondent discrete counterpart was proposed in the work of Maas (2011) and Mielke (2011), where a new notion of discrete dynamical optimal transport has been introduced and a similar result has been obtained.
This talk consists of an introductory overview on this topic. I will first recall the classical notion of 2-Wasserstein distance in euclidean spaces and discuss the correspondent discrete counterpart on probability measures on finite spaces. Secondly, I will introduce and explain the classical gradient flow structure of the Fokker-Planck equation in $\mathbb R^d$ and the connection to the optimal transport problem. Associate to it, one can consider a finite volume discretisation of the problem, to which it is possible to associate a natural graph structure, together with a correspondent discrete transport distance.
Finally, I will present an overview of the state of the art of the problem, discussing various convergence results, including limit behaviors of discrete transport costs and the convergence of discrete gradient-flow structures to the correspondent continuous ones.
This allows in particular for a variational proof of the convergence of finite-volume approximations of Fokker–Planck equations in $\mathbb R^d$.
This talk includes some results obtained in collaboration with Dominik Forkert, Eva Kopfer, Peter Gladbach, and Jan Maas.

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