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)