Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub-Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method like MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.

翻译：消去技术通常用于移动孤立的小特征值簇，以获得更紧凑的分布和更小的条件数。这种变化对Krylov子空间方法的收敛行为产生积极影响，而Krylov子空间方法是求解大型稀疏线性系统最流行的迭代求解器之一。我们利用对称鞍点矩阵的块状结构，为其开发了一种消去策略。用于消去的向量来自基于广义Golub-Kahan双对角化过程的椭圆奇异值分解。消去针对的目标是非对角块，因为在特定应用中，该块呈现有问题的奇异值分布。一个例子是细长通道中的斯托克斯流，其中非对角块具有若干孤立的小奇异值，其数量取决于通道长度。在使用CRAIG等求解器时，对鞍点系统的特定部分应用消去策略尤为重要，因为这类求解器作用于单个块而非整个系统。该理论通过扩展现有框架而建立，该框架在应用MINRES等Krylov子空间方法前对方阵进行消去。数值实验验证了我们策略的优势，并引出了关于使用近似向量进行消去的有趣问题。