Nested Krylov methods and preserving the orthogonality

Eric De Sturler

Delft University of Technology, Delft, The Netherlands

Diederik R. Fokkema

University of Utrecht, Utrecht, The Netherlands


Recently the GMRESR inner-outer iteration scheme for the solution of linear systems of equations has been proposed by Van der Vorst and Vuik. Similar methods have been proposed by Axelsson and Vassilevski [AxVas91] and Saad (FGMRES) [Saad93]. The outer iteration is GCR, which minimizes the residual over a given set of direction vectors. The inner iteration is GMRES, which at each step computes a new direction vector by approximately solving the residual equation. However, the optimality of the approximation over the space of outer search directions is ignored in the inner GMRES iteration. This leads to suboptimal corrections to the solution in the outer iteration, as components of the outer iteration directions may reenter in the inner iteration process. Therefore we propose to preserve the orthogonality relations of GCR in the inner GMRES iteration. This gives optimal corrections, however, it involves working with a singular, non-symmetric operator. We will discuss some important properties and we will show by experiments, that in terms of matrix vector products this modification (almost) always leads to better convergence. However, because we do more orthogonalizations, it does not always give an improved performance in CPU-time. Furthermore, we will discuss efficient implementations as well as the truncation possibilities of the outer GCR process. The experimental results indicate, that for such methods it is advantageous to preserve the orthogonality in the inner iteration. Of course we can also use other iteration schemes than GMRES as the inner method. Especially methods with short recurrences like BICGSTAB seem of interest.


[AxVas91] O. Axelsson and P. S. Vassilevski, A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 625-644.

[Saad93] Y. Saad, A flexible inner-outer preconditioned {GMRES} algorithm, SIAM J. Sci. Statist. Comput., 14 (1993), pp. 461-469.