Block Krylov subspace methods (KSMs) comprise building blocks in many state‐of‐the‐art solvers for large‐scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well‐explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs.

中文翻译：

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Sylvester矩阵方程的压缩并重新启动块Krylov子空间方法

块Krylov子空间方法（KSM）包含许多先进求解器中的大型矩阵方程的构造块，这些方程例如是由偏微分方程的离散化引起的。尽管扩展和有理块Krylov子空间方法比多项式块KSM减少了迭代次数，但它们也需要可靠的系数矩阵求解器，而这些求解器本身通常也是迭代方法。不难设计一种方案，在这种方案中，可用内存以及Krylov子空间的大小受到限制。在线性系统和特征值问题的这种情况下，重新启动是缓解内存限制的一种经过充分研究的技术。在这项工作中 此类重启技术应用于具有压缩步骤的矩阵方程式的多项式KSM，以控制残差的增长秩。还执行错误分析，从而在每个重新启动周期中动态调整基准大小的启发式方法。一组数值实验证明了新方法相对于扩展块KSM的有效性。