Krylov subspace estimation

Computing the linear least-squares estimate of a high-dimensional random qu antity given noisy data requires solving a large system of linear equations . In many situations, one can solve this system efficiently using a Krylov subspace method, such as the conjugate gradient ( CG) algorithm. Computing the estimation error variances is a more intricate task. It is di cult beca use the error variances are the diagonal elements of a matrix expression in volving the inverse of a given matrix. This paper presents a method for usi ng the conjugate search directions generated by the CG algorithm to obtain a convergent approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a new estimatio n-theoretic interpretation of the CG algorithm. This paper discusses this i nterpretation and convergence issues and presents numerical examples. The e xamples include a 10(5)-dimensional estimation problem from oceanography.