NII Technical Report (NII-2007-009E)：GMRES Methods for Least Squares Problems

The standard iterative method for solving large sparse least squares problems minx ∈ R^n || b - A x ||_2, A ∈ Rm 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 ∈ Rn x m and applying the Generalized Minimal Residual (GMRES) method to minz ∈ R^m || b - A B z ||_2 or minx ∈ 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.