Sparsity-Based Kalman Filters for Data Assimilation

TL;DR

This paper introduces sparsity-based variants of the UKF and EKF designed for high-dimensional data assimilation, utilizing sparse matrix approximations to improve computational efficiency and error covariance estimation.

Contribution

The paper presents novel sparsity-based Kalman filter algorithms that outperform traditional ensemble methods in high-dimensional settings by leveraging sparse covariance matrices.

Findings

Reduced memory requirements due to sparsity

Full-rank error covariance without ensemble limitations

Enhanced parallelization capabilities

Abstract

Several variations of the Kalman filter algorithm, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are widely used in science and engineering applications. In this paper, we introduce two algorithms of sparsity-based Kalman filters, namely the sparse UKF and the progressive EKF. The filters are designed specifically for problems with very high dimensions. Different from various types of ensemble Kalman filters (EnKFs) in which the error covariance is approximated using a set of dense ensemble vectors, the algorithms developed in this paper are based on sparse matrix approximations of error covariance. The new algorithms enjoy several advantages. The error covariance has full rank without being limited by a set of ensembles. In addition to the estimated states, the algorithms provide updated error covariance for the next assimilation cycle. The sparsity of error covariance significantly reduces the required memory size for the numerical computation. In addition, the granularity of the sparse error covariance can be adjusted to optimize the parallelization of the algorithms.