Preconditioning Techniques for Linear Systems of Equations

NSF提案编号：0431068标题：线性方程组的预处理技术PI： Vivek Sarin，部门摘要：线性方程组的求解是科学和工程各个领域必须解决的基本问题.迭代法用于求解偏微分方程的大型稀疏方程组和积分方程的大型稠密方程组，为了加快收敛速度，必须采用预处理技术。一个理想的预条件子应该是鲁棒的、有效的、可并行化的，并且计算和应用于线性系统是廉价的。在求解稠密系统时，系数矩阵的不可用性往往限制了预条件子的选择。该项目的目标是开发新的预处理技术的稀疏和密集的线性系统。这些技术将使用一系列的线性变换，以获得一个预处理系统从原来的一个。技术将被开发来分析预处理系统，并计划将开发，以提高有效性的预处理。由此产生的预处理器将是强大的，有效的，和廉价的。这些预处理技术将被实施，分析和测试各种问题。这些努力将与跨学科的合作研究活动相结合，以确保对应用领域产生更大的影响。

NSF Proposal Number: 0431068Title: Preconditioning techniques for linear systems of equationsPI: Vivek Sarin, Dept. of Computer Science, Texas A&M UniversityAbstract:The solution of linear systems of equations is a fundamental problem thatmust be tackled in various areas of science and engineering. Iterativemethods are used to solve large sparse systems from partial differentialequations as well as large dense systems from integral equations.Preconditioning techniques are necessary to accelerate the rate ofconvergence of these solvers. An ideal preconditioner should be robust,effective, parallelizable, and inexpensive to compute and apply to thelinear system. When solving dense systems, the unavailability of thecoefficient matrix often limits the choice of preconditioners. The goal ofthe project is to develop novel preconditioning techniques for sparse anddense linear systems. These techniques will use a sequence of lineartransformations to obtain a preconditioned system from the original one.Techniques will be developed to analyze the preconditioned system, andschemes will be developed to improve the effectiveness of thepreconditioner. The resulting preconditioners will be robust, effective,and inexpensive. These preconditioning techniques will be implemented,analyzed, and tested on a variety of problems. These efforts will beintegrated with interdisciplinary collaborative research activities toensure a greater impact on application areas.