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

Figure 1.  Illustration, based on the nonlinear dynamics (5), of the contracting property which can produce almost identical prediction densities (middle panels) for the Bayes filter and MFEnKF even when the preceding updated densities differ notably

Figure 2.  One prediction-update iteration of the MLEnKF estimator described in Section 2.4.1. Green and pink ovals represent fine- and coarse-level prediction-state particles, respectively, sharing the same initial condition and driving noise $ \omega^{\ell} $ and the respective squares represent fine- and coarse-level updated-state particles sharing the perturbed observartions. The MLEnKF estimator is obtained by iid copies of pairwise-coupled samples, cf (9)

Figure 3.  Top row: comparison of the runtime versus RMSE for the QoIs mean (left) and variance (right) over $ \mathcal{N} = 10 $ observation times for the problem in Section 3.1. The solid-crossed line represents MLEnKF, the solid-asterisk line represents the original MLEnKF and the bottom reference triangle with the slope $ \frac{1}{2} $, the solid-bulleted line represents EnKF and the upper reference triangle with the slope $ \frac{1}{3} $. Bottom row: similar plots over $ \mathcal{N} = 20 $ observation times

Figure 4.  Realization of the double-well SDE from Section 3.2 over time $ \mathcal{N} = 20 $ observation times (solid line) and observations (dots)

Figure 5.  Left column: Well transition of the EnKF ensemble when the measurements are located in the opposite well. Right column: Animation of the particle paths of the corresponding EnKF ensemble during the well-transition, and the resulting kernel density estimations of the EnKF prediction and update densities (Section 3.2). For practical purposes, EnKF with only 7 particles is not very robust. We use so few particles in this computation for the sole purpose of obtaining a visually clear illustration of the ensemble transition

Figure 6.  Top row: comparison of the runtime versus RMSE for the QoIs mean (left) and variance (right) over $ \mathcal{N} = 10 $ observation times for the problem in Section 3.2. The solid-crossed line represents MLEnKF, the solid-asterisk line represents the original MLEnKF and the bottom reference triangle with the slope $ \frac{1}{2} $, the solid-bulleted line represents EnKF and the upper reference triangle with the slope $ \frac{1}{3} $. Bottom row: similar plots over $ \mathcal{N} = 20 $ observation times

Figure 7.  (a) The inequality $ \min(\beta s,1)<s $ (green line). (b) The equality $ \min(\beta s,1) = s $ (blue line). (c) The inequality $ \min(\beta s,1)>s $ (red line). The dash lines correspond to the function $ y(s) = \min(\beta s, 1) $ and the dotted lines refer to the function $ y(s) = \beta s $ varying by different cases of $ \beta $ value