Sparsity-Based Kalman Filters for Data Assimilation

Abstract

Several variations of the Kalman filter, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are widely used in science and engineering applications. However, traditional UKFs or EKFs cannot assimilate big data sets associated with models that have high dimensions, such as those in operational numerical weather prediction. In this chapter, we introduce two sparsity-based Kalman filters, namely the sparse-UKF and the progressive-EKF. The filters are designed specifically for problems with high dimensions. Different from ensemble Kalman filters (EnKFs) in which the error covariance is approximated using a set of dense ensemble vectors, the algorithms developed in this chapter are based on the sparse matrix approximation of error covariance. The new algorithms enjoy several advantages. The error covariance has full rank without being limited within a subspace generated by a set of ensembles. In addition to the estimated states, the algorithms provide updated error covariance in every assimilation cycle. Taking the advantage of sparsity, the required memory size and computational load can be significantly reduced.

This work was supported in part by U.S. Naval Research Laboratory—Monterey, CA.