Abstract Recycling of Krylov subspaces is often used to obtain an augmentation subspace in the context of iterative algorithms for the solution of sequences of linear systems. However, it still remains difficult to quantify the effect of subspaces recycling and thus to determine the dimension of the subspaces to be recycled targeting a specific accuracy. In that context, this work proposes the Automatic Krylov subspaces Recycling algorithm (AKR) that automates the selection of Krylov subspaces to be recycled and generates a basis that can provide sufficiently accurate approximations of the solution for a parametric system on a predefined interval Ψ . The constructed basis is employed as a Galerkin projection basis for a model order reduction (MOR) scheme in the context of non-affine parametric systems. In the offline phase of the MOR scheme, AKR constructs a projection subspace W by sampling Krylov subspaces at an iteratively built set of parameter values Ω . Keeping a balance between the solution accuracy and the memory required, the algorithm, apart from guaranteeing a predefined residual level r tol , also permits the predetermination of a threshold regarding the maximum memory employed. Nevertheless, following the unpreconditioned Krylov methods effectiveness criteria, the proposed technique proves to be efficient for systems with relatively clustered eigenvalues such as the ones encountered in the conventional Boundary Element Method. The performance of the proposed AKR algorithm is assessed in comparison with an alternative version of the reduced basis method, which is based on the same assumptions as the AKR and is specifically designed to provide a good benchmark. These techniques are deployed for a randomly generated complex system and an acoustic BEM system. The advantage of employing AKR is demonstrated as fewer system assemblies are required for the construction of the projection basis.

中文翻译：

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一种用于构建非仿射参数线性系统全局解基的自动 Krylov 子空间回收技术

摘要 在求解线性系统序列的迭代算法的上下文中，Krylov 子空间的循环经常用于获得增广子空间。然而，仍然很难量化子空间回收的影响，从而确定要回收的子空间的维度，以达到特定的精度。在这种情况下，这项工作提出了自动 Krylov 子空间回收算法 (AKR)，该算法可以自动选择要回收的 Krylov 子空间并生成一个基础，该基础可以为预定义间隔 Ψ 上的参数系统提供足够准确的解近似值。在非仿射参数系统的上下文中，构造的基被用作模型降阶 (MOR) 方案的 Galerkin 投影基。在MOR方案的离线阶段，AKR 通过在一组迭代构建的参数值 Ω 处对 Krylov 子空间进行采样来构建投影子空间 W。在求解精度和所需内存之间保持平衡，该算法除了保证预定义的残差水平 r tol 之外，还允许预先确定关于所使用的最大内存的阈值。然而，遵循未经预处理的 Krylov 方法有效性标准，所提出的技术证明对于具有相对聚类特征值的系统（例如在传统边界元方法中遇到的特征值）是有效的。所提出的 AKR 算法的性能是通过与缩减基方法的替代版本进行比较来评估的，该版本基于与 AKR 相同的假设，并且专门设计用于提供良好的基准。这些技术用于随机生成的复杂系统和声学 BEM 系统。使用 AKR 的优势体现为构建投影基础所需的系统组件更少。