Multilevel Estimation of Normalization Constants Using the Ensemble Kalman-Bucy Filter

TL;DR

This paper develops a multilevel Monte Carlo method using the ensemble Kalman-Bucy filter to efficiently estimate normalization constants, with theoretical analysis and applications to atmospheric models.

Contribution

It introduces the multilevel ensemble Kalman-Bucy filter (MLEnKBF) for normalization constant estimation and provides Lq-bounds analysis in the linear setting.

Findings

MLEnKBF achieves favorable error-to-cost rates compared to EnKBF.

Numerical results demonstrate the efficiency of MLEnKBF on linear Gaussian models.

Application to atmospheric models shows the method's practical utility.

Abstract

In this article we consider the application of multilevel Monte Carlo, for the estimation of normalizing constants. In particular we will make use of the filtering algorithm, the ensemble Kalman-Bucy filter (EnKBF), which is an N-particle representation of the Kalma-Bucy filter (KBF). The EnKBF is of interest as it coincides with the optimal filter in the continuous-linear setting, i.e. the KBF. This motivates our particular setup in the linear setting. The resulting methodology we will use is the multilevel ensemble Kalman-Bucy filter (MLEnKBF). We provide an analysis based on deriving Lq-bounds for the normalizing constants using both the single-level, and the multilevel algorithms. Our results will be highlighted through numerical results, where we firstly demonstrate the error-to-cost rates of the MLEnKBF comparing it to the EnKBF on a linear Gaussian model. Our analysis will be specific to one variant of the MLEnKBF, whereas the numerics will be tested on different variants. We also exploit this methodology for parameter estimation, where we test this on the models arising in atmospheric sciences, such as the stochastic Lorenz 63 and 96 model.