Fast inversion of L-block banded matrices and their inverses

Block banded matrices generalize banded matrices. In this paper, we exploit properties of full matrices whose inverses are L-block banded to derive fast inverses for such matrices, and inverses for matrices that are themselves block banded. We apply these results to design fast implementations for the Kalman-Bucy filter in applications arising often in the physical sciences where the underlying models derive from discretizations of partial differential equations and the observations are sparse.