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：消缩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底层块结构，为其开发了一种消缩策略。用于消缩的向量来源于基于广义Golub-Kahan双对角化过程的椭圆奇异值分解。消缩的目标块是非对角块，因为在某些应用中，该块的特征值分布存在特殊问题。以细长通道中的斯托克斯流为例，其非对角块存在若干孤立的小奇异值，且奇异值数量取决于通道长度。当使用CRAIG等求解器时，对鞍点系统的特定部分应用消缩策略具有重要意义——这类求解器针对的是矩阵的独立子块而非整个系统。该理论通过扩展现有的方阵消缩框架推导得出，该框架应用于MINRES等Krylov子空间方法之前。数值实验验证了所提策略的有效性，并为使用近似向量进行消缩提出了有趣的研究问题。