Relaxation strategies for nested Krylov methods

There are classes of linear problems for which the matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. In recent years different authors have investigated the use of, what is called, relaxation strategies for various Krylov subspace methods. These relaxation strategies aim to minimize the amount of work that is spent in the computation of the matrix-vector product without compromising the accuracy of the method or the convergence speed too much. In order to achieve this goal, the accuracy of the matrix-vector product is decreased when the iterative process comes closer to the solution. In this paper we show that a further significant reduction in computing time can be obtained by combining a relaxation strategy with the nesting of inexact Krylov methods. Flexible Krylov subspace methods allow variable preconditioning and therefore can be used in the outer most loop of our overall method. We analyze for several flexible Krylov methods strategies for controlling the accuracy of both the inexact matrix-vector products and of the inner iterations. The results of our analysis will be illustrated with an example that models global ocean circulation.