Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting

TL;DR

This paper analyzes the accuracy of the ensemble Kalman filter in near-linear settings, providing error bounds for finite particle systems with unbounded dynamics, extending previous theoretical results.

Contribution

It establishes error bounds for the finite ensemble Kalman filter with unbounded vector fields, broadening the theoretical understanding beyond linear Gaussian assumptions.

Findings

Error bounds are derived for the ensemble Kalman filter with unbounded dynamics.

The analysis applies to finite particle systems, not just mean field limits.

The results extend the validity of the filter in more realistic, complex models.

Abstract

The filtering distribution captures the statistics of the state of a dynamical system from partial and noisy observations. Classical particle filters provably approximate this distribution in quite general settings; however they behave poorly for high dimensional problems, suffering weight collapse. This issue is circumvented by the ensemble Kalman filter which is an equal-weight interacting particle system. However, this finite particle system is only proven to approximate the true filter in the linear Gaussian case. In practice, however, it is applied in much broader settings; as a result, establishing its approximation properties more generally is important. There has been recent progress in the theoretical analysis of the algorithm, establishing stability and error estimates in non-Gaussian settings, but the assumptions on the dynamics and observation models rule out the unbounded vector fields that arise in practice and the analysis applies only to the mean field limit of the ensemble Kalman filter. The present work establishes error bounds between the filtering distribution and the finite particle ensemble Kalman filter when the dynamics and observation vector fields may be unbounded, allowing linear growth.