Filtering with Randomised Observations: Sequential Learning of Relevant Subspace Properties and Accuracy Analysis
This work addresses state estimation challenges in applications like data assimilation, offering an adaptive method to reduce computational costs while maintaining accuracy, though it is incremental as it builds on existing ensemble Kalman filter techniques.
The paper tackles the problem of state estimation with partial observations by analyzing the signal-tracking performance of continuous ensemble Kalman filters under fixed, randomized, and adaptive observations, establishing rigorous error bounds. It proposes a sequential learning scheme that adaptively determines the dimension of a relevant state subspace to balance observation complexity with estimation accuracy, ensuring bounded filtering error.
State estimation that combines observational data with mathematical models is central to many applications and is commonly addressed through filtering methods, such as ensemble Kalman filters. In this article, we examine the signal-tracking performance of a continuous ensemble Kalman filtering under fixed, randomised, and adaptively varying partial observations. Rigorous bounds are established for the expected signal-tracking error relative to the randomness of the observation operator. In addition, we propose a sequential learning scheme that adaptively determines the dimension of a state subspace sufficient to ensure bounded filtering error, by balancing observation complexity with estimation accuracy. Beyond error control, the adaptive scheme provides a systematic approach to identifying the appropriate size of the filter-relevant subspace of the underlying dynamics.