  Ensemble Kalman Inversion (EKI) has been a very popular algorithm used in
Bayesian inverse problems. 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, 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.
