  Solving inverse problems without the use of derivatives or adjoints of the
forward model is highly desirable in many applications arising in science and
engineering. In this paper, we propose a new version of such a methodology, a
framework for its analysis, and numerical evidence of the practicality of the
method proposed. Our starting point is an ensemble of over-damped Langevin
diffusions which interact through a single preconditioner computed as the
empirical ensemble covariance. We demonstrate that the nonlinear Fokker-Planck
equation arising from the mean-field limit of the associated stochastic
differential equation (SDE) has a novel gradient flow structure, built on the
Wasserstein metric and the covariance matrix of the noisy flow. Using this
structure, we investigate large time properties of the Fokker-Planck equation,
showing that its invariant measure coincides with that of a single Langevin
diffusion, and demonstrating exponential convergence to the invariant measure
in a number of settings. We introduce a new noisy variant on ensemble Kalman
inversion (EKI) algorithms found from the original SDE by replacing exact
gradients with ensemble differences; this defines the ensemble Kalman sampler
(EKS). Numerical results are presented which demonstrate its efficacy as a
derivative-free approximate sampler for the Bayesian posterior arising from
inverse problems.
