Preconditioners for saddle point linear systems

鞍点线性系统在许多应用领域具有重要意义，近十年来对求解方法的研究激增。约束优化等领域的问题往往需要求解这类系统。相关应用的例子有流体动力学、曲面拟合和图像重建，还有更多。鞍点系统的有效数值解是本方案的核心。我建议开发和研究基于我和我的合作者最近衍生的新技术的方法，并将我的工作扩展到新的方向。本提案中描述的其他研究项目包括开发新的广义Krylov解，研究条件数最小化和特征值敏感性问题，以及马尔可夫链的数值解，重点是谷歌的PageRank算法。

Saddle point linear systems are of great importance in many areas of applications, and research on solution methods has surged in the last decade. Problems in constrained optimization and other areas often require the solution of such systems. Examples of relevant applications are fluid dynamics, surface fitting and image reconstruction, and there are many more.The efficient numerical solution of saddle point systems stands at the centre of this proposal. I propose to develop and study methods based on recent novel techniques that I have derived with my collaborators, and extend my work into new directions.Other research projects that are described in this proposal are the development of new generalized Krylov solvers, the study of the problem of condition number minimization and eigenvalue sensitivity, and the numerical solution of Markov chains, with focus on Google's PageRank algorithm.