Ensemble Kalman inversion: mean-field limit and convergence analysis

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