The Ensemble Kalman inversion ( EnKI ), proposed by Iglesias et al. for the solution of Bayesian inverse problems of type y= Au +\varepsilon, with u  being an unknown parameter and y a given datum, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles \{u^j(t)\}_{j=1}^J, whose initial empirical measure is sampled from the prior, evolving over an artificial time t towards an approximate solution of the inverse problem. Using spectral techniques, we provide a complete description of the dynamics of EnKI and their asymptotic behavior in parameter space. In particular, we debunk a common working assumption in the field, saying that in the linear Gaussian regime, the empirical measure at time t=1 is a surrogate for the posterior , i.e., the correct Bayesian answer to the inverse problem . Furthermore, we show that  residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix.
The Ensemble Kalman inversion ( EKI ), proposed by Iglesias et al. for the solution of Bayesian inverse problems of type y= A u^\dagger +\varepsilon, with u ^\dagger being an unknown parameter and y a given datum, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles \{u^j(t)\}_{j=1}^J, whose initial empirical measure is sampled from the prior, evolving over an artificial time t towards an approximate solution of the inverse problem. Using spectral techniques, we provide a complete description of the dynamics of EKI and their asymptotic behavior in parameter space. In particular, we analyze dynamics of deterministic EKI, averaged quantities of stochastic EKI, and mean-field EKI. We show that in the linear Gaussian regime, the Bayesian posterior can only be recovered with the mean-field limit and not with finite sample sizes or deterministic EKI . Furthermore, we show that -- even in the deterministic case -- residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix.
R
EnKI
EKI
R
Au
A u^\dagger
A
^\dagger
R
dynamics of EnKI
dynamics of EKI
R
debunk a common working assumption in the field, saying
analyze dynamics of deterministic EKI, averaged quantities of stochastic EKI, and mean-field EKI. We show
R
empirical measure at time t=1 is a surrogate for the posterior , i.e., the correct Bayesian answer to the inverse problem
Bayesian posterior can only be recovered with the mean-field limit and not with finite sample sizes or deterministic EKI
A
-- even in the deterministic case --