Inner-iteration preconditioning with a symmetric splitting matrix for rank-deficient least squares problems
This work provides theoretical justification for using inner-iteration preconditioned Krylov methods on rank-deficient least squares problems, which is an incremental extension of existing convergence theories.
The authors provide conditions for inner-iteration preconditioning with a symmetric splitting matrix to be definite, and prove that CG and MINRES preconditioned by inner iterations can solve singular symmetric linear systems. They apply these results to CGLS, LSMR, and CGNE methods for rank-deficient least squares problems, complementing previous convergence theories.
Stationary iterative methods with a symmetric splitting matrix are performed as inner-iteration preconditioning for Krylov subspace methods. We give conditions such that the inner-iteration preconditioning matrix is definite, and show that conjugate gradient (CG) method preconditioned by the inner iterations determines a solution of symmetric and positive semidefinite linear systems, and the minimal residual (MINRES) method preconditioned by the inner iterations determines a solution of symmetric linear systems including the singular case. These results are applied to the CG and MINRES-type methods such as the CGLS, LSMR, and CGNE methods preconditioned by inner iterations, and thus justify using these methods for solving least squares and minimum-norm solution problems whose coefficient matrices are not necessarily of full rank. Thus, we complement the convergence theories of these methods presented in [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), pp.1-22], [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 36 (2015), pp. 225-250], and give bounds for these methods.