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 fine-resolution 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.

40 pages, 7 figures. The computer code of the method introduced in the work is written in the Julia programming language, and it can be downloaded from https://github.com/GaukharSH/mlenkf