Gradient Structure Of The Ensemble Kalman Flow With Noise

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 study a number of variants on Ensemble Kalman Inversion (EKI) algorithms, with goal being the construction of methods which generate approximate samples from the Bayesian posterior distribution that solves the inverse problem. Furthermore we establish a mathematical framework within which to study this question. Our starting point is the continuous time limit of EKI. We introduce a specific form of additive noise to the deterministic flow, leading to a stochastic differential equation (SDE), and consider an associated SDE which approximates the original SDE by replacing function evaluation differences with exact gradients. We demonstrate that the nonlinear Fokker-Planck equation defined by the mean-field limit of the associated 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 SDE in the case of a linear observation map, and convergence to an invariant measure which coincides with the Bayesian posterior distribution for the underlying inverse problem. We numerically study the effect of the noisy perturbation of the original derivative-free EKI algorithm, illustrating that it gives good approximate samples from the posterior distribution.