Gradient flow structures for discrete porous medium equations

By Matthias Erbar, Jan Maas. 2012

85 citations recorded by the source corpus.

We consider discrete porous medium equations of the form \partial_t \rho_t = \Delta \phi(\rho_t), where \Delta is the generator of a reversible continuous time Markov chain on a finite set X, and \phi is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in R^n discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/06a67dd20829b827b7386aa7306956697cc83a00

Citations in the corpus

• On the difference between entropic cost and the optimal transport cost (2019, unknown): “See articles [21, 10, 8, 11, 6, 9].”