Numerical Solution of Large-Scale Sparse Linear Systems Arising from Problems with Constraints

*计算科学和工程领域的最新进展，加上计算能力的提高和新的计算范式的出现，要求科学计算领域的专家开发新的算法和方法来解决比以往任何时候都更大的问题，并且必须利用其特征来生成健壮、高效和可靠的数值解技术。这一建议旨在解决结构线性系统的数值解的这些挑战，这些挑战贯穿于各种有约束的问题的计算过程中。*从有约束的问题中产生的线性系统是高度结构化的，但它们以难以求解而臭名昭著。现代方法基于模型简化技术(即，试图将原始问题转变为保持大部分原始属性的较小问题)。对于由约束问题产生的线性代数问题，保持结构和其他属性是至关重要的，这是一个巨大的挑战。*我的主要目标将是开发快速和健壮的求解器，这些求解器是可扩展的，并且尊重结构。为了实现这一目标，我将致力于从约束优化和偏微分方程数值解领域为各种问题生成一个统一的框架。为了使这项任务成功，应该探索和利用对称性、矩阵秩属性和光谱分布等属性。一个同样重要的目标将是为可伸缩的、灵活的和可移植的求解器开发数值软件，以允许解决大规模问题。*这里有许多相关的应用程序，以及与其他领域的许多联系。首先，任何可以被假定为约束优化问题的问题都需要求解本提案中讨论的形式的线性系统。机器学习、计算机图形学、机器人学、医学成像和许多其他计算领域的问题提供了丰富的来源。此外，偏微分方程组的数值解也提供了大量有约束的问题；计算电磁学或流体动力学只是其中的两个例子。总而言之，需要解决的大量问题和有效处理大规模问题的重要性使这一研究领域变得非常活跃和重要。*鉴于大量应用导致了有约束的数学模型，并且鉴于我们的社会需要处理越来越多的数据，针对本提案中相关问题的快速数值求解器的进步可能会在计算科学和工程中产生重要的积极影响。

***Recent advances in computational science and engineering, combined with the availability of increased computing power and new computational paradigms, require experts in the scientific computing domain to develop new algorithms and methodologies for solving problems that are larger than ever before, and whose features must be exploited in order to be able to generate robust, efficient, and reliable numerical solution techniques. This proposal aims to take on these challenges for the numerical solution of structured linear systems that arise throughout the computation of a large variety of problems with constraints.******Linear systems that arise from problems with constraints are highly structured, yet they are notorious for being difficult to solve. Modern methods are based on model reduction techniques (namely, the attempt to turn the original problem into a smaller problem that maintains most of the original properties). For linear algebra problems arising from problems with constraints, preserving the structure and other attributes is critical and it presents a significant challenge.******My primary goal will be to develop fast and robust solvers, which are scalable and respect the structure. To accomplish this goal, I will aim to generate a unified framework for a variety of problems from the area of constrained optimization and numerical solution of partial differential equations. Attributes such as symmetry, matrix rank property, and spectral distribution, to name just a few, should be explored and exploited for this mission to be successful. An equally important goal will be to develop numerical software for scalable, flexible, and portable solvers that allow for solving large-scale problems.******There are many relevant applications here and many connections to other areas. Primarily, any problem that can be posed as a constrained optimization problem requires solving linear systems of the form discussed in this proposal. Problems in machine learning, computer graphics, robotics, medical imaging, and many other computational areas, provide a rich source. In addition, the numerical solution of partial differential equations provides a large collection of problems with constraints, too; computational electromagnetics or fluid dynamics are just two examples. Altogether, the large collection of problems to be solved and the importance of dealing efficiently with large-scale problems, make this area of research extremely active and important.******Given the large number of applications that lead to mathematical models with constraints, and given the increasing amounts of data that our society needs to deal with, advances in fast numerical solvers for the problems concerned in this proposal may have a significant positive impact in computational science and engineering.