Schneider, Michael K.; Willsky, Alan S. Krylov subspace estimation. (English) Zbl 0980.65045 SIAM J. Sci. Comput. 22, No. 5, 1840-1864 (2001). Authors’ abstract: Computing the linear least-squares estimate of a high-dimensional random quantity 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 difficult because the error variances are the diagonal elements of a matrix expression involving the inverse of a given matrix.This paper presents a method for using 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 estimation-theoretic interpretation of the CG algorithm. This paper discusses this interpretation and convergence issues and presents numerical examples. The examples include a \(10^5\)-dimensional estimation problem from oceanography.Reviewer’s remark: He wonders that the editors admitted this title since it does not give any information about the contents of the paper. Reviewer: Dietrich Braess (Bochum) Cited in 1 Document MSC: 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F10 Iterative numerical methods for linear systems 62F10 Point estimation 62J05 Linear regression; mixed models 65C60 Computational problems in statistics (MSC2010) Keywords:linear least-squares estimation; error variances; conjugate gradient algorithm; high-dimensional random quantity; Krylov subspace method; convergence; numerical examples; oceanography PDFBibTeX XMLCite \textit{M. K. Schneider} and \textit{A. S. Willsky}, SIAM J. Sci. Comput. 22, No. 5, 1840--1864 (2001; Zbl 0980.65045) Full Text: DOI