Restarted minimum residual methods are a popular solution approach for solving non-Hermitian linear systems, but one that is fraught with the basic difficulty that convergence of the restarted method may be considerably impeded compared with the full (unrestarted) method. We provide an overview of existing strategies which compensate for this deterioration in convergence due to restarting for the class of minimum residual (MR) Krylov subspace methods. The key theoretical device for comparing different strategies is their abstract formulation as repeated orthogonal projections with respect to general correction spaces. We further evaluate the popular practice of using nearly invariant subspaces to either augment Krylov subspaces or to construct preconditioners which invert on these subspaces. In the case where these spaces are exactly invariant, the augmentation approach is shown to be superior. Moreover, we show how a strategy recently introduced by de Sturler for truncating the approximation space of an MR method can be interpreted as a controlled loosening of the condition for global MR approximation based on the canonical angles between subspaces.
This is joint work with Michael Eiermann and Olaf Schneider (TU Freiberg)