Gradient flow in the Wasserstein metric

Optimal transport not only provides powerful techniques for comparing probability measures, but also for analyzing their evolution over time. For a range of partial differential equations arising in physics, biology, and engineering, solutions are gradient flows in the Wasserstein metric: each equation has a notion of energy for which solutions dissipate energy as quickly as possible, with respect to the Wasserstein structure. Steady states of the equation correspond to minimizers of the energy, and stability properties of the equation translate into convexity properties of the energy. In this talk, I will compare Wasserstein gradient flow with more classical gradient flows arising in optimization and machine learning. I’ll then introduce a class of particle blob methods for simulating Wasserstein gradient flows numerically.