Augmented unprojected Krylov subspace methods

TL;DR

This paper introduces an unprojected formulation of augmented Krylov subspace methods, including a new unprojected augmented FOM, enhancing recycling techniques and broadening practical implementation options.

Contribution

It derives the first unprojected augmented FOM and shows how existing algorithms like R^3GMRES fit within the new framework, expanding the theoretical understanding of augmented Krylov methods.

Findings

Unprojected augmented FOM effectively recycles subspace information.

Unprojected methods can be implemented under specific conditions on the augmentation space.

Demonstrated the practical effectiveness of the unprojected augmented FOM.

Abstract

Augmented Krylov subspace methods aid in accelerating the convergence of a standard Krylov subspace method by including additional vectors in the search space. A residual projection framework based on residual (Petrov-) Galerkin constraints was presented in [Gaul et al. SIAM J. Matrix Anal. Appl 2013], and later generalised in a recent survey on subspace recycling iterative methods [Soodhalter et al. GAMM-Mitt. 2020]. The framework describes augmented Krylov subspace methods in terms of applying a standard Krylov subspace method to an appropriately projected problem. In this work we show that the projected problem has an equivalent unprojected formulation, and that viewing the framework in this way provides a similar description for the class of unprojected augmented Krylov subspace methods. We derive the first unprojected augmented Full Orthogonalization Method (FOM), and demonstrate its effectiveness as a recycling method. We then show how the R$^{3}$GMRES algorithm fits within the framework. We show that unprojected augmented short recurrence methods fit within the framework, but can only be implemented in practice under certain conditions on the augmentation subspace. We demonstrate this using the Augmented Conjugate Gradient (AugCG) algorithm as an example.