Mathematics Colloquium

This talk gives an introduction to the theory of optimal transport and gradient flows in probability space. We present a variant of the dynamical optimal transport problem in which the energy to be minimized is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems up to degrees of freedom given by rotations: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. On the level of the gradient flow, a similar splitting into the evolution of moments and shapes of the distribution can be observed. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of a Gaussian target.