Ensemble Kalman inversion: mean-field limit and convergence analysis

Abstract

Ensemble Kalman inversion (EKI) has been a very popular algorithm used in Bayesian inverse problems (Iglesias et al. in Inverse Probl 29: 045001, 2013). It samples particles from a prior distribution and introduces a motion to move the particles around in pseudo-time. As the pseudo-time goes to infinity, the method finds the minimizer of the objective function, and when the pseudo-time stops at 1, the ensemble distribution of the particles resembles, in some sense, the posterior distribution in the linear setting. The ideas trace back further to ensemble Kalman filter and the associated analysis (Evensen in J Geophys Res: Oceans 99: 10143–10162, 1994; Reich in BIT Numer Math 51: 235–249, 2011), but to today, when viewed as a sampling method, why EKI works, and in what sense with what rate the method converges is still largely unknown. In this paper, we analyze the continuous version of EKI, a coupled SDE system, and prove the mean-field limit of this SDE system. In particular, we will show that 1. as the number of particles goes to infinity, the empirical measure of particles following SDE converges to the solution to a Fokker–Planck equation in Wasserstein 2-distance with an optimal rate, for both linear and weakly nonlinear case; 2. the solution to the Fokker–Planck equation reconstructs the target distribution in finite time in the linear case, as suggested in Iglesias et al. (Inverse Probl 29: 045001, 2013).