Course Variational Modelling: Gradient flows and optimal transport

Course content: This course has two objectives. The first is to develop the methodology of 'variational modelling' of energy-driven systems. This methodology applies to systems whose evolution in time is driven by the decrease of an energy, in a friction-dominated or strongly damped way. Recent developments have shown that a surprisingly large class of evolutionary systems is of this form, even though the energy and the friction mechanism may not be obvious. Examples of these include linear and nonlinear diffusion equations, nonlocal diffusion equations, higher-order parabolic equations, moving-boundary problems, and many others. In the course we will discuss how to use the mathematical structure as a modelling tool: each choice of an energy and a friction mechanism provide an evolutionary system. The two choices characterize in a remarkably clear way the modelling choices that underlie the resulting differential equations.

The second objective arises from the fact that in the modelling of diffusive systems - PDEs like the diffusion equation - the natural friction mechanism is given by the Wasserstein metric. We will introduce this metric, discuss its origins and its properties, and develop various useful ways of applying it in the variational-modelling methodology. An important element of this part of the course will be the theory of optimal transport, which deals with finding the optimal way of transporting a set of objects from a given starting position to a given end position. This problem lies at the basis of the Wasserstein metric, and we will show how it also is the basic ingredient in the modelling of diffusion processes.

Lecture notes: This is the first time that I teach this course, and I will write the lecture notes as the course progresses. The current version can always be found in the Notes directory. Please give me all your feedback on these notes! I will award a special prize to whoever finds the most mistakes ...

Times: Thursdays 11-13 UK time. In Eindhoven, Michiel Renger will organize the possibility of following this course by internet.

Place: 3W 4.13 (the TCC room)

In Eindhoven, Michiel Renger will organize the possibility of following this course by internet. Current planning is

• 20/01 - AUD08
• 27/01 - AUD08
• 03/02 - AUD12
• 10/02 - AUD12
• 17/02 - AUD09
• 24/02 - HG 8.61
• 03/03 - HG 8.61
• 10/03 - AUD12
• 17/03 - HG 8.61
• 24/03 - HG 8.61

Course assessment: The course will be assessed by viva.

Notes can be found in the Notes directory.

References:

• F. Otto. The geometry of dissipative evolution equations: The porous medium equation. Communications in Partial Differential Equations, 26:101-174, 2001.
• S. Adams, N. Dirr, M. A. Peletier, and J. Zimmer. From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Arxiv preprint arxiv:1004.4076.