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

Abstract

We introduce a new multilevel ensemble Kalman filter method (MLEnKF) which consists of a hierarchy of independent samples of ensemble Kalman filters (EnKF). This new MLEnKF method is fundamentally different from the preexisting method introduced by Hoel, Law and Tempone in 2016, and it is suitable for extensions towards multi-index Monte Carlo based filtering methods. Robust theoretical analysis and supporting numerical examples show that under appropriate regularity assumptions, the MLEnKF method has better complexity than plain vanilla EnKF in the large-ensemble and fineresolution limits, for weak approximations of quantities of interest. The method is developed for discrete-time filtering problems with finite-dimensional state space and linear observations polluted by additive Gaussian noise.

Improved Efficiency of Multilevel Monte Carlo for Stochastic PDE through Strong Pairwise Coupling

“…Pathwise correct coupling is on the other hand not necessary to obtain consistency, and there are many examples of performant MLMC methods that only are weakly correct, cf. [17,25]. Pathwise correct coupling is however often an easy way to ensure the needed weakly correct coupling.…”

A Multi‐Model Ensemble Kalman Filter for Data Assimilation and Forecasting

“…The low-resolution state forecast was not used. Hoel et al (2016Hoel et al ( , 2020) combined forecasts at different resolutions in an EnKF, but they did not weight them differently. Popov et al (2021) combined models of different fidelities in an EnKF with a control variate approach.…”

Continuous time limit of the stochastic ensemble Kalman inversion: Strong convergence analysis

“…In [47] the authors study the mean field limit of the closely related ensemble square root filter (ESRF). The EnKF has been formulated in various multilevel formulations [31,14,32,8]. A long time and ergodicity analysis are presented in [40,62,41], including uniform bounds in time and the incorporation of covariance inflation.…”