Iterative Methods and Matrix Analysis (Computer Science)

本研究旨在分析和发展求解非对称线性系统的迭代方法，这些方法出现在科学计算的许多领域。目前解决这类问题的迭代方法有时是有效的，但缺乏坚实的理论基础，可能不够健壮。Krylov空间方法，如GMRES，将进行分析，并考虑从不同空间选择近似值的想法。双共轭梯度和QMR算法将被联系起来，并试图解释它们的成功。我们将研究各种算法中并行的潜力，并为特定的机器架构开发并行实现。计划与康奈尔大学计算机科学系的Nick Trefethen和其他人合作。与学生互动的两项主要活动是：教授关于迭代方法和矩阵分析的研究生研讨会（“CS 722”），使学生和其他感兴趣的科学家熟悉该领域的最新技术；并引入二年级水平的实习课程，结合数值分析课程CS 222，学生将在康奈尔理论中心提供的机器上实现并行数值算法。与此活动相关的各种本科生研讨会将被组织起来。

This research is aimed at analyzing and developing iterative methods for solving the nonsymmetric linear systems that arise in many areas of scientific computing. Current iterative methods for solving such problems ar sometimes effective but lack a firm theoretical foundation and may not be robust. Krylov space methods, such as GMRES, will be analyzed and the idea of choosing approximations from different spaces will also be considered. The biconjugate gradient and QMR algorithms will be related and an attempt made to explain their success. Potential for parallelism in various algorithms will be studied and parallel implementations will be developed for specific machine architectures. Collaborations are planned with Nick Trefethen and others in the Computer Science Department at Cornell. Two main activities for interacting with students are: teaching a graduate seminar ("CS 722") on iterative methods and matrix analysis, to familiarize students and other interested scientists and the state of the art in this field; and introducing a sophomore level practicum where, in conjunction with the numerical analysis course CS 222, students will implement parallel numerical algorithms on the machines available at the Cornell Theory Center. Various undergraduate seminars will be organized related to this activity.