Complete Deterministic Dynamics and Spectral Decomposition of the Ensemble Kalman Inversion
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The Ensemble Kalman inversion (EKI), proposed by Iglesias et al. for the solution of Bayesian inverse problems of type y=A u^† +ε, 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 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.
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