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).

集成卡尔曼反演（Esemble Kalman inversion，EKI）是贝叶斯反问题中非常流行的算法（Iglesias等人，Inverse Probl 29：045001，2013）。它从先验分布中采样粒子，并引入运动以在伪时间内四处移动粒子。随着伪时间趋于无穷大，该方法找到了目标函数的极小值，当伪时间在1处停止时，粒子的总体分布在某种意义上类似于线性设置中的后验分布。这些想法可以追溯到集合卡尔曼滤波器和相关分析（Evensen in J Geophys Res：Oceans 99：10143-10162，1994; Reich in BIT Numer Math 51：235–249，2011），但是直到今天，采样方法，EKI为何起作用以及该方法在什么意义上以何种速率收敛仍然是未知之数。在本文中，我们分析了耦合SDE系统EKI的连续版本，并证明了该SDE系统的平均场限制。特别地，我们将证明1.随着粒子数量达到无穷大，SDE之后的粒子的经验测度以线性和弱的最优速率收敛到Wasserstein 2距离的Fokker-Planck方程的解，具有最优速率。非线性情况 2. Fokker-Planck方程的解在线性情况下在有限时间内重建了目标分布，如Iglesias等人所述。（Inverse Probl 29：045001，2013）。对于线性和弱非线性情况，SDE之后的颗粒的经验测度以最优速率收敛到Wasserstein 2距离Fokker-Planck方程的解；2. Fokker-Planck方程的解在线性情况下在有限时间内重建了目标分布，如Iglesias等人所述。（Inverse Probl 29：045001，2013）。对于线性和弱非线性情况，SDE之后的颗粒的经验测度以最优速率收敛到Wasserstein 2距离Fokker-Planck方程的解；2. Fokker-Planck方程的解在线性情况下在有限时间内重建了目标分布，如Iglesias等人所述。（Inverse Probl 29：045001，2013）。