Iterative Solvers for Saddle-Point Systems

这个建议涉及的调查，分析和实施的数值求解方法的一个家庭的线性系统所产生的问题，一般可以构成为最小化问题的约束。这类问题极其重要，而且它们经常出现在范围大得惊人的科学模型中。相关应用的示例包括计算机图形学、数据挖掘、图像处理、医学成像、流体流动、电磁学以及更多实例。 该提案提出了一系列目标，包括为约束优化问题和约束微分方程设计新的求解器，推导将联合收割机各种求解方法结合到一个优化求解器中的新算法，以及调查与解决问题的最佳方式有关的问题，重点在于各种重新表达在多大程度上影响求解过程的鲁棒性和可靠性的问题。长期目标包括追求一个统一的框架，阐明不同的解决方法之间的联系，以及导致所考虑的系统的不同应用程序之间的联系。 本提案中所涉及的问题不仅具有理论意义，而且具有很高的实用价值，包括实施各种解决方案和高素质人员的参与。 这一建议的及时性是由新的学科和问题的不断发展所驱动的，这些学科和问题涉及大规模线性代数系统的解决方案，以及处理速度、并行计算架构和高性能计算已经达到的新前沿，在可以解决的问题的大小方面推动了包络。

This proposal concerns the investigation, analysis and implementation of numerical solution methods for a family of linear systems that arise from problems that can be generally posed as minimization problems with constraints. Such problems are extremely important, and they frequently appear in a surprisingly large scope of scientific models. Examples of relevant applications include computer graphics, data mining, image processing, medical imaging, fluid flow, electromagnetics, and many more instances. The proposal puts forward a list of objectives that include the design of new solvers for constrained optimization problems and constrained differential equations, the derivation of new algorithms that combine various solution methodologies into one optimized solver, and the investigation of questions pertaining to the best way to solve the problem, with focus on the question to what extent various reformulations affect the robustness and reliability of solution procedures. Long-term objectives include the pursuit of a unified framework that sheds light on connections among different solution methodologies and among the different applications that lead to the systems that are considered. The issues addressed in this proposal are not just of theoretical interest but also of a high practical value, including the implementation of various solvers and the involvement of highly qualified personnel. The timeliness of this proposal is driven by the continual development of new disciplines and problems that involve the solution of large-scale linear algebra systems, and the new frontiers that processing speed, parallel computing architectures and high performance computing have reached, pushing the envelope in terms of the size of problems that can be tackled.