Convergence of Inner-Iteration GMRES Methods for Rank-Deficient Least Squares Problems

0438625 - ÚI 2015 RIV US eng J - Journal Article
Morikuni, Keiichi - Hayami, K.
Convergence of Inner-Iteration GMRES Methods for Rank-Deficient Least Squares Problems.
SIAM Journal on Matrix Analysis and Applications. Roč. 36, č. 1 (2015), s. 225-250. ISSN 0895-4798. E-ISSN 1095-7162
Institutional support: RVO:67985807
Keywords : least squares problem * iterative methods * preconditioner * inner-outer iteration * GMRES method * stationary iterative method * rank-deficient problem
Subject RIV: BA - General Mathematics
Impact factor: 1.883, year: 2015

We develop a general convergence theory for the generalized minimal residual method preconditioned by inner iterations for solving least squares problems. The inner iterations are performed by stationary iterative methods. We also present theoretical justifications for using specific inner iterations such as the Jacobi and SOR-type methods. The theory improves previous work [K. Morikuni and K. Hayami, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1--22], particularly in the rank-deficient case. We also characterize the spectrum of the preconditioned coefficient matrix by the spectral radius of the iteration matrix for the inner iterations and give a convergence bound for the proposed methods. Finally, numerical experiments show that the proposed methods are more robust and efficient compared to previous methods for some rank-deficient problems.
Permanent Link: http://hdl.handle.net/11104/0242032