Kaczmarz-type inner-iteration preconditioned flexible GMRES methods for consistent linear systems

Abstract

We propose using greedy and randomized Kaczmarz inner-iterations as preconditioners for the right preconditioned flexible GMRES method to solve consistent linear systems, with a parameter tuning strategy for adjusting the number of inner iterations and the relaxation parameter. We also present theoretical justifications of the right-preconditioned flexible GMRES for solving consistent linear systems. Numerical experiments on overdetermined and underdetermined linear systems show that the proposed method is superior to the GMRES method preconditioned by NE-SOR inner iterations in terms of total CPU time.

Convergence of Adaptive, Randomized, Iterative Linear Solvers

“…Iterative linear solvers are often preferred for solving large-scale linear systems, as they can take advantage of problem structure such as sparsity or bandedness, require inexpensive floating point operations, and can be readily paired with preconditioning techniques [19, see preface]. While such iterative linear solvers as Conjugate Gradients (CG) and the Generalized Minimal Residual method (GM-RES) are still dominant solvers in practice, randomized row-action [8,1,14,23] and column-action iterative solvers [10,25] have been growing in interest for several reasons: they (usually) require very few floating point operations per iteration [5,3]; they have low-memory footprints [9]; they can readily be composed with randomization techniques to quickly produce approximate solutions [23,10,24,6,11,2,7,17]; they can be used for solving systems constructed in a streaming fashion (e.g., [15]), which supports emerging computing paradigms (e.g., [13]); and, just like the more popular iterative Krylov solvers, they can be parallelized, preconditioned or combined with other linear solvers [20,16,4,18];…”