Preconditioned Interative Methods for Large Linear Systems

大型稀疏线性代数方程的数值解是科学计算中非常重要的一部分。这类线性系统源于偏微分方程组的离散化，以及在许多其他情况下，如非线性方程的数值解和最优化问题。在过去的二十年里，迭代Krylov子空间方法的发展和分析取得了重大进展。这个项目将解决与使用Krylov空间方法来解线性系统相关的一些理论和实践问题。该领域中仍然存在的一个大问题是发展一种明确的非对称问题的选择方法。非对称Krylov空间方法的分析将是该项目的一个组成部分。了解误差对对称和非对称迭代方法的性能的影响也是很重要的，这一主题也将被讨论。最后，从实际的角度来看，对迭代方法的性能影响最大的往往是前置条件。这项研究中提出的想法将被应用于涉及中子输运和地下水流动模拟的问题，以检验其实用性。并行性将是所有工作中的一个考虑因素。

The numerical solution of large sparse linear algebraic equationsrepresents a very significant portion of scientific computing.Such linear systems arise from the discretization of partialdifferential equations as well as in many other contexts, such asthe numerical solution of nonlinear equations and optimizationproblems. Over the past two decades, significant progress hasbeen made in the development and analysis of iterative Krylovsubspace methods. This project will address a number of theoreticaland practical issues associated with the use of Krylov spacemethods for solving linear systems.One of the large remaining open problems in this field is the development of a clear method of choice for nonsymmetric problems.Analysis of nonsymmetric Krylov space methods will be a componentof this project. It is also important to understand the effect ofrounding errors on the performance of both symmetric and nonsymmetriciterative methods and this topic will also be addressed. Finally,from a practical point of view, it is most often the preconditionerthat has the greatest effect on the performance of an iterative method. Ideas developed in this research will be applied in problems involving neutron transport and ground-water flow simulationto test their practicality. Parallelism will be a considerationin all of the work.