Deflated and Augmented Krylov Subspace Techniques

We present a general framework for a number of techniques based on projection methods on 'augmented Krylov subspaces'. These methods include the deflated GMRES algorithm, an inner-outer FGMRES iteration algorithm, and the class of block Krylov methods. Augmented Krylov subspace methods often show a significant improvement in convergence rate when compared with their standard counterparts using the subspaces of the same dimension. The methods can all be implemented with a variant of the FGMRES algorithm.