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.

翻译：增广Krylov子空间方法通过在搜索空间中引入额外向量，有助于加速标准Krylov子空间方法的收敛。基于残差（Petrov-）Galerkin约束的残差投影框架由Gaul等人提出（《SIAM矩阵分析与应用杂志》，2013年），后经Soodhalter等人在关于子空间循环迭代方法的近期综述（《GAMM Mitteilungen》，2020年）中推广。该框架通过将标准Krylov子空间方法应用于适当投影后的问题来描述增广Krylov子空间方法。本文证明该投影问题存在等价的非投影形式，且这种视角为非投影增广Krylov子空间方法类提供了类似描述。我们推导了首个非投影增广完全正交化方法（FOM），并展示了其作为循环方法的有效性。随后阐明R$^{3}$GMRES算法在该框架中的定位，并证明非投影增广短递推方法虽可纳入该框架，但实际实现必须满足增广子空间的特定条件，并以增广共轭梯度（AugCG）算法为例进行验证。