ACM Transactions on Mathematical Software (TOMS)
Journal of Computational Physics
Milestones in the development of iterative solution methods
Journal on Image and Video Processing - Special issue on iterative signal processing in communications
Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
Original article: Macro-elementwise preconditioning methods
Mathematics and Computers in Simulation
Hierarchical Krylov and nested Krylov methods for extreme-scale computing
Parallel Computing
Flexible global generalized Hessenberg methods for linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
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Flexible Krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next. Given a Krylov subspace method, such as CG, GMRES, QMR, etc. for the solution of a linear system Ax=b, instead of having a fixed preconditioner M and the (right) preconditioned equation AM-1 y = b (Mx =y), one may have a different matrix, say Mk, at each step. In this paper, the case where the preconditioner itself is a Krylov subspace method is studied. There are several papers in the literature where such a situation is presented and numerical examples given. A general theory is provided encompassing many of these cases, including truncated methods. The overall space where the solution is approximated is no longer a Krylov subspace but a subspace of a larger Krylov space. We show how this subspace keeps growing as the outer iteration progresses, thus providing a convergence theory for these inner-outer methods. Numerical tests illustrate some important implementation aspects that make the discussed inner-outer methods very appealing in practical circumstances.