Krylov space approximate Kalman filtering

The Kalman filter is a technique for estimating a time-varying state given a dynamical model for and indirect measurements of the state. It is used, for example, on the control problems associated with a variety of navigation systems. Even in the case of nonlinear state and/or measurement models, standard implementations require only linear algebra. However, for sufficiently large-scale problems, such as arise in weather forecasting and oceanography, the matrix inversion and storage requirements of the Kalman filter are prohibitive, and hence, approximations must be made. In this paper, we describe how the conjugate gradient iteration can be used within the Kalman filter for quadratic minimization, as well as for obtaining low-rank approximations of the covariance and inverse-covariance matrices required for its implementation. The approach requires that we exploit the connection between the conjugate gradient and Lanczos iterations.