Regularised optimal transport for solving gradient flows

This project focuses on investigating the role of optimal transport as a numerical tool for solving certain (nonlinear) PDEs that can formalised as gradient flows wrt the Wasserstein distance. Numerical schemes of that kind are based on the discretisation of metric-gradient flows by the method of minimizing movements and the JKO scheme [1,2]. They are interesting because they are able to preserve the intrinsic structure of such PDEs such as mass and positivity conservation of the solution. Examples of such PDEs are the heat equation, the porous medium equation, the Fokker-Planck equation and the Keller-Segel model.  We will focus on two numerical approaches in this study, proposed in [3] and [4,5].

[1] Ambrosio, L., Gigli, N., & Savaré, G. (2008). Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media.
[2] Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The variational formulation of the Fokker–Planck equation. SIAM journal on mathematical analysis, 29(1), 1-17.
[3] Burger, M., Franek, M., & Schönlieb, C. B. (2012). Regularized regression and density estimation based on optimal transport. Applied Mathematics Research eXpress, 2012(2), 209-253.
[4] Cuturi, M. (2013). Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems (pp. 2292-2300).
[5] Peyré, G. (2015). Entropic approximation of Wasserstein gradient flows. SIAM Journal on Imaging Sciences, 8(4), 2323-2351.

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