论文标题
增强未投影的Krylov子空间方法
Augmented unprojected Krylov subspace methods
论文作者
论文摘要
增强Krylov子空间方法有助于加速标准Krylov子空间方法的收敛性，通过在搜索空间中包含其他向量。 [Gaul等人在[Gaul等人中提供了基于残留（PETROV-）Galerkin约束的残留投影框架。暹罗J.矩阵肛门。 Appl 2013]，后来在最近关于子空间回收迭代方法的调查中进行了概括[Soodhalter等。加姆·米特。 2020]。该框架描述了将标准Krylov子空间方法应用于适当投影的问题的增强Krylov子空间方法。 在这项工作中，我们表明预计的问题具有等效的未调整公式，并且以这种方式查看框架为未投影的增强的Krylov子空间方法提供了类似的描述。我们得出了第一个未投影的增强完整正交法（FOM），并证明了其作为回收方法的有效性。然后，我们展示r $^{3} $ gmres算法如何适合框架。我们表明，未调查的增强的简短复发方法适合该框架内，但只能在某些条件下在增强子空间的某些条件下实施。我们以增强的共轭梯度（AUGCG）算法为例，证明了这一点。
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.
