Covariance-modulated optimal transport and gradient flows

Understand

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the 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: one for the evolution of mean and covariance of the interpolating curve and one for its shape.
• The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation.