Balanced data assimilation for highly-oscillatory mechanical systems

TL;DR

This paper addresses the limitations of the ensemble Kalman filter in highly oscillatory Hamiltonian systems by proposing modifications to improve the preservation of balance relations and prevent divergence.

Contribution

It introduces blended time-stepping schemes and ensemble-based penalty methods to enhance the ensemble Kalman filter for oscillatory systems with balance constraints.

Findings

Modified schemes improve filter stability in oscillatory systems

Proposed methods better preserve physical balance relations

Numerical tests demonstrate effectiveness in meteorology and molecular dynamics

Abstract

Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, the ensemble Kalman filter also has limitations due to its inherent Gaussian and linearity assumptions. These limitations can manifest themselves in dynamically inconsistent state estimates. We investigate this issue in this paper for highly oscillatory Hamiltonian systems with a dynamical behavior which satisfies certain balance relations. We first demonstrate that the standard ensemble Kalman filter can lead to estimates which do not satisfy those balance relations, ultimately leading to filter divergence. We also propose two remedies for this phenomenon in terms of blended time-stepping schemes and ensemble-based penalty methods. The effect of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for two model scenarios. First, we consider balanced motion for highly oscillatory Hamiltonian systems and, second, we investigate thermally embedded highly oscillatory Hamiltonian systems. The first scenario is relevant for applications from meteorology while the second scenario is relevant for applications of data assimilation to molecular dynamics.