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一个非常重要的部分。这种线性系统产生于偏微分方程的离散化以及许多其他情况下，例如非线性方程的数值解和优化问题。在过去的二十年中，迭代krylov子空间方法的开发和分析取得了重大进展。该项目将解决与使用克雷洛夫空间方法求解线性系统相关的一些理论和实践问题。这一领域的一个悬而未决的大问题是发展一种明确的非对称问题的选择方法。非对称克雷洛夫空间方法的分析将是这个项目的一个组成部分。理解舍入误差对对称和非对称迭代方法性能的影响也很重要，这个主题也将得到解决。最后，从实际的角度来看，对迭代方法的性能影响最大的通常是前置条件。本研究中发展的思想将应用于中子输运和地下水流动模拟等问题，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.