One of the limitations of recycled GCRO methods is the large amount of computation required to orthogonalize the basis vectors of the newly generated Krylov subspace for the approximate solution when combined with those of the recycle subspace. Recent advancements in low synchronization Gram-Schmidt and generalized minimal residual algorithms, Swirydowicz et al.~\cite{2020-swirydowicz-nlawa}, Carson et al. \cite{Carson2022}, and Lund \cite{Lund2022}, can be incorporated, thereby mitigating the loss of orthogonality of the basis vectors. An augmented Arnoldi formulation of recycling leads to a matrix decomposition and the associated algorithm can also be viewed as a {\it block} Krylov method. Generalizations of both classical and modified block Gram-Schmidt algorithms have been proposed, Carson et al.~\cite{Carson2022}. Here, an inverse compact $WY$ modified Gram-Schmidt algorithm is applied for the inter-block orthogonalization scheme with a block lower triangular correction matrix $T_k$ at iteration $k$. When combined with a weighted (oblique inner product) projection step, the inverse compact $WY$ scheme leads to significant (over 10$\times$ in certain cases) reductions in the number of solver iterations per linear system. The weight is also interpreted in terms of the angle between restart residuals in LGMRES, as defined by Baker et al.\cite{Baker2005}. In many cases, the recycle subspace eigen-spectrum can substitute for a preconditioner.

翻译：回收型GCRO方法的局限性之一在于，为近似解新生成的Krylov子空间的基向量与回收子空间的基向量进行正交化时，需要大量计算。近年来，低同步Gram-Schmidt算法和广义最小残差算法的进展（Swirydowicz等人~\cite{2020-swirydowicz-nlawa}，Carson等人\cite{Carson2022}，以及Lund\cite{Lund2022}）可被引入，从而减轻基向量正交性的损失。回收过程的增强型Arnoldi公式化导出了一种矩阵分解，相关联的算法也可视为一种{\it 块}Krylov方法。已有研究提出了经典和改进块Gram-Schmidt算法的推广形式，Carson等人~\cite{Carson2022}。本文在迭代步$k$，采用逆紧凑$WY$改进Gram-Schmidt算法进行块间正交化处理，并配合一块下三角校正矩阵$T_k$。当与加权（斜内积）投影步骤结合时，逆紧凑$WY$方案能显著减少每个线性系统求解的迭代次数（某些情况下减少超过10倍）。该权重还可根据Baker等人\cite{Baker2005}定义的LGMRES中重启动残差间的夹角进行解释。在许多情况下，回收子空间的特征谱可替代预处理器的功能。