Well posedness and convergence analysis of the ensemble Kalman inversion

Abstract

The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing in various areas of application. We present a complete analysis of the ensemble Kalman inversion with perturbed observations for a fixed ensemble size when applied to linear inverse problems. The well-posedness and convergence results are based on the continuous time scaling limits of the method. The resulting coupled system of stochastic differential equations allows to derive estimates on the long-time behaviour and provides insights into the convergence properties of the ensemble Kalman inversion. We view the method as a derivative free optimization method for the least-squares misfit functional, which opens up the perspective to use the method in various areas of applications such as imaging, groundwater flow problems, biological problems as well as in the context of the training of neural networks.AMS classification scheme numbers: 65N21, 62F15, 65N75, 65C30, 90C56 for Hilbert spaces (H 1 , ·, · H 1 ), (H 2 , ·, · H 2 ) and z 1 ∈ H 1 , z 2 ∈ H 2 . The empirical means are given bythe minimum of n and the first exit time of e s at radius n. Then, for all n ∈ N, from (14) (after rebasing the integration interval from [0, t] to [s, s + t]) we obtainAs τ n → ∞, applying Fatou's lemma on the left hand side and applying the monotone convergence theorem on the right hand side givesProof. By Lemma Appendix A.4 we can directly take expectations in (14) to obtains 2 ds.Note that by dropping the non-negative mixed terms j = k and by using Jensen's and Young's inequalityProof. The idea of this proof is based on Theorem 4.6.2 in [33]. We define the stochastic Lyapunov functionThe generator applied to V fulfillsis monotonically decreasing.Proof. The assertions follow by arguments similar to the proof of Proposition 4.11.Thus, for all t, s ≥ 0, it follows similarly to the proof of Lemma 4.1 that,...,J converges to zero almost surely as t → ∞. Proof. We define the Lyapunov function V (r, t) = t β 1 J J j=1 |r (j) | 2 and obtain LV (r, t) ≤ βt β−1 J J j=1 |r (j) | 2 − t β 1 J J j=1 r (j) , C(r) + 1 t α + R B r (j) .Thus, LV (r, t) ≤ 1 J J j=1 |r (j) | 2 β − λ min t t α + R t β−1 .

Continuous limits for constrained ensemble Kalman filter

“…which is the preconditioned gradient flow equation for the least square functional Φ, studied e.g. in [5,30,31]. We expect that (18) still allows each ensemble member to satisfy the linear constraint if the initial condition is feasible.…”

Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators

“…As a consequence of EnKF employing a Gaussian-like update of its particles, the mean-field EnKF will in many cases not be equal to the Bayes filter [35,11]. But, to the best of our knowledge, there does not exist any thorough scientific comparison of the mean-field EnKF and the Bayes Filter, and Figure 1 shows that for the nonlinear dynamics Ψ defined by the SDE du = −(u + π cos(πu/5)/5)dt + σdW (5) and (1), the dissipative/contractive properties of the associated Fokker-Planck equation can produce prediction densities for the respective filtering methods that are indistinguishable to the naked eye. Figure 1.…”

“…Due to this discrepancy between EnKF and the Bayes filter even in the largeensemble limit, due to the large uncertainty in the estimation of the model error and due to the constraints imposed by challening high-dimensional problems and limited computational budgets, a considerable number of works have, instead of studying the large-ensemble limit, focused on the large-time and/or continuoustime limit of the fixed-ensemble-size EnKF cf. [30,46,42,43,5,9,33]. However, a recurring problem for fixed-ensemble-size EnKF is to determine how large the ensemble ought to be to equilibrate the model error with the other error contributions (statistical error and bias).…”

Application of ensemble transform data assimilation methods for parameter estimation in nonlinear problems

“…EKI is an adaptive SMC method that approximates the first two statistical moments of a posterior distribution. For a linear forward model, EKI is optimal in a sense it minimizes the error in the mean (Blömker et al, 2019). For a nonlinear forward model, EKI still provides a good estimation of posterior (e.g., Iglesias et al, 2018).…”