Iterative Methods for Large Scale Nonlinear and Linear Systems

小步行者9727128 线性和非线性方程的大规模系统几乎 普遍存在于物理现象的计算建模中， 偏微分方程，积分方程， Krylov子空间方法构成了一大类迭代线性化方法， 已广泛用于这些大规模线性问题的代数方法 系统.这些方法已适用于大规模非线性系统 通过Newton-Krylov实现，其中它们用于解决 描述牛顿步的线性系统。这项研究是一个 继续进行一项旨在发展更强大和 有效的牛顿-克雷洛夫方法的非线性问题，伴随着， 更有效的Krylov子空间方法。当前 研究主要集中在Newton-Krylov方法和相关方法 解决大规模非线性问题。具体目标是发展 更强大的全球化，以实施和测试更多 有效的Krylov求解器，以改善预处理策略和选项， 探索自动微分等新技术。一个主要 扩展以前的计划目标是制定有效的方法 对于大规模的连续问题。该方法基于某些 Krylov子空间方法和Newton-Krylov方法的改进方法 到允许使用现有预处理器的延续设置 以及其他以前开发的问题技术。 这项工作的一个特别目的是开发改进的方法， 复杂物理现象的高性能计算建模 电脑特定的目标应用包括全物理建模 反应流（燃烧和其他化学反应流）和相关的 计算流体力学中的问题。这些应用的成功 将在工业设计和运行中具有巨大的潜在效益 设备，如用于热和发电和化学蒸汽的熔炉 用于半导体制造的沉积反应器。这项工作将建立 与桑迪亚研究人员先前建立的合作活动 劳伦斯利弗莫尔国家实验室（Lawrence Livermore National Laboratory） 犹他州大学高性能计算中心， 正在开发的方法已经成功地应用于许多 涉及大规模传热传质的实际流动问题 并行计算机未来可能的合作是与研究人员在 犹他州大学对意外火灾和爆炸进行建模， 劳伦斯利弗莫尔国家实验室的污染物传输模型 含水层，并应用于环境管理和生物补救。

9727128 Homer Walker Large-scale systems of linear and nonlinear equations occur almost ubiquitously in the computational modeling of physical phenomena through the discretization of partial differential equations, integral equations, etc. Krylov subspace methods make up a large class of iterative linear algebra methods that have been widely used for these large-scale linear systems. These methods have been adapted to large-scale nonlinear systems through Newton-Krylov implementations, in which they are used to solve the linear systems that characterize Newton steps. This research is a continuation of an ongoing program aimed at developing more robust and efficient Newton-Krylov methods for nonlinear problems and, concomitantly, more effective Krylov subspace methods for linear problems. The current research focusses primarily on Newton-Krylov methods and related methods for large-scale nonlinear problems. Specific objectives are to develop more robust globalizations, to implement and test additional and more effective Krylov solvers, to improve preconditioning strategies and options, and to explore new techniques such as automatic differentiation. A major extension of the previous program goals is to develop effective methods for large-scale continuation problems. The approach is based on certain ways of adapting Krylov subspace methods and Newton-Krylov methodology to the continuation setting that allow the use of existing preconditioners and other previously developed problem technology. A particular aim of this work is to develop improved methods for large-scale computational modeling of complex physical phenomena on high performance computers. Particular target applications include full-physics modeling of reactive flows (combustion and other chemically reacting flows) and related problems in computational fluid dynamics. Success in these applications will have great potential benefits in the design and operation of industrial facilit ies such as furnaces for heat and power generation and chemical vapor deposition reactors for semiconductor manufacturing. This work will build on previously established collaborative activities with researchers at Sandia National Laboratories, Lawrence Livermore National Laboratory, and the University of Utah Center for High Performance Computing, in which the methods under development have been successfully applied to a number of realistic flow problems involving heat and mass transport on massively parallel computers. Likely future collaborations are with researchers at the University of Utah on modeling accidental fires and explosions and at Lawrence Livermore National Laboratory on modeling contaminant transport in aquifers, with applications to environmental management and bioremediation.