A Flexible Inner-Outer Preconditioned GMRES Algorithm

A variant of the GMRES algorithm is presented that allows changes in the preconditioning at every step. There are many possible applications of the new algorithm, some of which are briefly discussed. In particular, a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner. For example, the standard GMRES algorithm itself can be used as a preconditioner, as can CGNR (or CGNE), the conjugate gradient method applied to the normal equations. However, the more appealing utilization of the method is in conjunction with relaxation techniques, possibly multilevel techniques. The possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness. A few numerical experiments are reported to illustrate this fact.