Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a maximum of t vectors per iteration based on the domain decomposition of the graph of A. As for the s-step versions, s iterations of the enlarged Conjugate Gradient methods are merged in one iteration. The Enlarged CG methods and their s-step versions converge in less iterations than the classical CG, but at the expense of requiring more memory storage than CG. Thus in this paper we explore different options for reducing the memory requirements of these enlarged CG methods without affecting much their convergence.

翻译：扩展Krylov子空间方法及其s步版本在文献[7]中被提出，旨在减少求解线性方程组Ax = b时的通信开销。这些扩展共轭梯度方法通过基于矩阵A的图域分解，每迭代步最多扩展t个向量来扩充Krylov子空间。对于s步版本，将扩展共轭梯度方法的s次迭代合并为一次迭代。扩展CG方法及其s步版本比经典CG收敛所需迭代次数更少，但代价是需要比CG更多的存储空间。因此，本文探讨了减少这些扩展CG方法内存需求的不同方案，同时尽量不影响其收敛性。