Complete Dynamics and Spectral Decomposition of the Ensemble Kalman Inversion

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.