Preconditioned Interative Methods for Large Linear Systems

大型稀疏线性代数方程组的数值求解是科学计算中非常重要的一部分，此类线性系统起源于偏微分方程的离散化以及许多其他方面，如非线性方程组和优化问题的数值求解.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。 本项目将解决与使用Krylov空间方法解决线性系统相关的一些理论和实际问题。在这个领域中，一个大的剩余开放问题是发展一个明确的方法来选择非对称问题。非对称Krylov空间方法的分析将是本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是很重要的，这一主题也将得到讨论。 最后，从实际的角度来看，它是最经常的预条件，有最大的影响迭代方法的性能。 本研究所提出的想法将应用于中子输运和地下水流模拟等问题，以检验其实用性。 在所有的工作中，都将是一种奉献。

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.