Complete Deterministic Dynamics and Spectral Decomposition of the Linear Ensemble Kalman Inversion

The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems of type $y = A u +\varepsilon$, with $u$ being an unknown parameter, $y$ a given datum, and $\varepsilon$ measurement noise, 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, with $t=1$ emulating the posterior, and $t\to\infty$ corresponding to the under-regularized minimum-norm solution of the inverse problem. Using spectral techniques, we provide a complete description of the deterministic dynamics of EKI and its asymptotic behavior in parameter space. In particular, we analyze the dynamics of naive EKI and mean-field EKI with a special focus on their time asymptotic behavior. 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. While the analysis is aimed at the EKI, we believe that it can be applied to understand more general particle-based dynamical systems.