CAREER: Preconditioning Large, Sparse Linear Systems: Theory and Practice

预条件包含一系列的技术，用于将线性方程组Ax=B转换为某种程度上“更好”的相关系统。 例如，求解器可能会花费更少的时间来计算新系统的解，它可能会计算出更精确的解，或者它可以同时执行这两项操作。 尽管一个好的预条件子对于有效地计算精确解是至关重要的，但是从现有的许多预条件子中选择一个好的预条件子可能是困难的。这个项目的目标是开发工具，帮助用户从大量的预条件子中选择一个可能非常适合他们特定应用的预条件子;最初的重点将放在预条件，这是基于重新排序的行和/或列的矩阵，和那些计算一个不完整的因式分解矩阵。 在开发这些工具的过程中，PI，本科生研究人员和其他合作者将致力于开发更好的算法，提高有关特定预处理器的理论知识，创建一个可扩展的框架来评估不同的预处理器，并使广泛的实验结果以用户友好的形式提供。 所有的工作都将有助于从具体的应用中更好地理解特定预处理器在稀疏系统上的性能。作为一名文科大学的教授，PI对更广泛影响的主要计划包括：监督本科生和学士后的研究，为学生寻找机会在当地社区之外展示他们的工作，更普遍地为他们的研究生工作做准备;通过指导和建立支持网络，向传统上在计算机科学领域代表性不足的群体进行宣传;以及强调数学在计算机科学中的作用的课程的发展、完善和传播，以及那些让非计算机专业的学生对计算机有基本了解的课程，但这些知识仍然会提高他们的教育水平。

Preconditioners comprise a range of techniques for transforming a system oflinear equations Ax=b into a related system that is somehow ``better''. Forexample, a solver may take less time to compute the solution to the new system,it may compute a more accurate solution, or it may do both. Although a goodpreconditioner can be critical to computing an accurate solution efficiently,choosing a good preconditioner from the many that are available can bedifficult.The goal of this project is the development of tools to help users choose fromlarge families of preconditioners one that is likely to be well-suited to theirparticular application; the initial focus will be on preconditioners that arebased on reordering the rows and/or columns of a matrix, and those that computean incomplete factorization of the matrix. In the process of developing thesetools, the PI, undergraduate researchers, and other collaborators will work ondeveloping better heuristics, improving theoretical knowledge about specificpreconditioners, creating an extensible framework for evaluating differentpreconditioners, and making the results of extensive experimentation availablein a user-friendly form. All work will help lead to a better understanding ofthe performance of specific preconditioners on sparse systems from specificapplications.As a professor at a liberal arts college, the PI's primary plans for broaderimpact include: supervising undergraduate and postbaccalaureate research,seeking opportunities for students to present their work outside of the localcommunity and more generally preparing them for graduate work; outreach togroups traditionally underrepresented in computer science through mentoring andthe formation of support networks; and the development, refinement, anddissemination of courses that emphasize the role of mathematics in computerscience, as well as of courses that give a basic understanding of computing tostudents who are not computer science majors but whose education willnevertheless be enhanced by such knowledge.