The use of block Krylov subspace methods for computing the solution to a sequence of shifted linear systems using subspace recycling was first proposed in [Soodhalter, SISC 2016], where a recycled shifted block GMRES algorithm (rsbGMRES) was proposed. Such methods use the equivalence of the shifted system to a Sylvester equation and exploit the shift invariance of the block Krylov subspace generated from the Sylvester operator. This avoids the need for initial residuals to span the same subspace and allows for a viable restarted Krylov subspace method with recycling for solving sequences of shifted systems. In this paper we propose to develop these types of methods using unprojected Krylov subspaces. In doing so we show how one can overcome the difficulties associated with developing methods based on projected Krylov subspaces such as rsbGMRES, while also allowing for practical methods to fit within a well known residual projection framework. In addition, unprojected methods are known to be advantageous when the projector is expensive to apply, making them of significant interest for High-Performance Computing applications. We develop an unprojected rsbFOM and unprojected rsbGMRES. We also develop a procedure for extracting shift dependent harmonic Ritz vectors over an augmented block Krylov subspace for shifted systems yielding an approach for selecting a new recycling subspace after each cycle of the algorithm. Numerical experiments demonstrate the effectiveness of our methods.
翻译:在[Soodhalter, SISC 2016] 中,首次提议使用Cret Krylov 子空间来计算转换线性系统序列的解决方案。在[Soodhalter, SISC 2016] 中,首次提议使用Crept Krylov 子空间来计算使用子空间回收利用的转换线性系统序列的解决方案。这些方法使用Sylvester 方程式的等同系统,并利用Sylvester 操作者产生的Krylov 子空间块的变换。这避免了初始残渣跨越同一子空间的需要,并允许采用可行的重新启用Krylov 子空间方法,用于解决转移系统的序列。在本文件中,我们提议利用未预测的Krylov 子空间来开发这些类型的方法。我们如何克服基于预测的Krylov 子空间(如 rsbGMRES) 的开发方法,同时允许实际方法适应众所周知的残余预测框架。此外,在投影器应用费用昂贵时,已知的Krylovallov 子空间再利用这些方法,从而对高精度的高级计算系统产生重大的兴趣分析程序。我们每个变压变压的系统后,我们再变换的系统将一个不动的系统。