NII Technical Report (NII-2007-009E)

Title GMRES Methods for Least Squares Problems
Authors Ken HAYAMI, Jun-Feng YIN, and Tokushi ITO
Abstract The standard iterative method for solving large sparse least squares problems min_{x \in R^n} || b - A x ||_2, A \in R^{m x n} is the CGLS method, or its stabilized version LSQR, which applies the (preconditioned) conjugate gradient method to the normal equation A^T A x = A^T b. In this paper, we will consider alternative methods using a matrix B \in R^{n x m} and applying the Generalized Minimal Residual (GMRES) method to min_{z \in R^m} || b - A B z ||_2 or \min_{x \in R^n} || B b - B A x ||_2. Next, we give a sufficient condition concerning B for the GMRES methods to give a least squares solution without breakdown for arbitrary b, for over-determined, under-determined and possibly rank-deficient problems. We then give a convergence analysis of the GMRES methods as well as the CGLS method. Then, we propose using the robust incomplete factorization (RIF) for B. Finally, we show by numerical experiments on over-determined and under-determined problems that, for ill-conditioned problems, the GMRES methods with RIF give least squares solutions faster than the CGLS and LSQR methods with RIF.
Language English
Published Jul 11, 2007
Pages 28p
PDF File 07-009E.pdf

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