Complete Deterministic Dynamics and Spectral Decomposition of the Ensemble Kalman Inversion

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 \new{deterministic} 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.