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A note on augmented unprojected Krylov subspace methods

Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [Soodhalter et al, GAMM-Mitt. 2020]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the R$^3$GMRES method proposed in [Dong et al. ETNA 2014] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [Saad. SIMAX 1997].