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

***计算科学和工程的最新进展，加上不断增强的计算能力和新的计算范式的可用性，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大的问题，这些问题的特征必须被利用，以便能够产生健壮、高效和可靠的数值解决技术。本提案旨在为结构化线性系统的数值解提出这些挑战，这些系统在各种约束问题的计算中出现。******由约束问题产生的线性系统是高度结构化的，但它们因难以解决而臭名昭著。现代方法是基于模型简化技术（即，试图将原始问题变成一个保持大多数原始属性的小问题）。对于由约束问题引起的线性代数问题，保持结构和其他属性是至关重要的，这是一个重大的挑战。******我的主要目标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，我将致力于为约束优化和偏微分方程数值解领域的各种问题生成一个统一的框架。对称性、矩阵秩性质和光谱分布等属性，仅举几例，应该被探索和利用，以使这项任务取得成功。一个同样重要的目标是开发可扩展、灵活和便携的求解器的数值软件，以解决大规模问题。******这里有许多相关的应用程序，并且与其他领域有许多联系。首先，任何可以作为约束优化问题的问题都需要解决本提案中讨论的形式的线性系统。机器学习、计算机图形学、机器人技术、医学成像和许多其他计算领域的问题提供了丰富的资源。此外，偏微分方程的数值解也提供了大量有约束的问题；计算电磁学和流体动力学就是两个例子。总之，需要解决的大量问题和有效处理大规模问题的重要性，使这一领域的研究非常活跃和重要。******考虑到大量的应用会导致有约束的数学模型，考虑到我们社会需要处理的数据量的增加，本提案中涉及的问题的快速数值解算器的进步可能会对计算科学和工程产生重大的积极影响。

***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.