Iterative Methods for Large Scale Nonlinear and Linear Systems

通过偏微分方程、积分方程等的离散化，在物理现象的计算建模中，线性和非线性方程组的大规模系统几乎无处不在。Krylov子空间方法是一类迭代线性代数方法，已广泛应用于这些大规模线性系统。这些方法已经通过Newton- krylov实现适应于大规模非线性系统，其中它们用于求解具有牛顿步骤特征的线性系统。本研究是一个正在进行的项目的延续，旨在为非线性问题开发更鲁棒和有效的牛顿-克雷洛夫方法，同时为线性问题开发更有效的克雷洛夫子空间方法。目前的研究主要集中在大规模非线性问题的Newton-Krylov方法及相关方法。具体目标是开发更强大的全球化，实现和测试额外的和更有效的Krylov解算器，改进预处理策略和选项，并探索自动区分等新技术。对先前计划目标的一个主要扩展是发展大规模延拓问题的有效方法。该方法基于将Krylov子空间方法和牛顿-Krylov方法适应于允许使用现有预调节器和其他先前开发的问题技术的延续设置的某些方法。这项工作的一个特殊目的是开发改进的方法，用于在高性能计算机上对复杂物理现象进行大规模计算建模。特定的目标应用包括反应流动（燃烧和其他化学反应流动）的全物理建模以及计算流体动力学中的相关问题。这些应用的成功将为工业设施的设计和运行带来巨大的潜在好处，例如用于供热和发电的炉以及用于半导体制造的化学气相沉积反应器。这项工作将建立在先前与桑迪亚国家实验室、劳伦斯利弗莫尔国家实验室和犹他大学高性能计算中心的研究人员建立的合作活动的基础上，在这些合作活动中，正在开发的方法已经成功地应用于大量并行计算机上涉及热量和质量传输的许多现实流动问题。未来可能的合作是与犹他大学的研究人员模拟意外火灾和爆炸，以及劳伦斯利弗莫尔国家实验室的研究人员模拟含水层中的污染物运输，并将其应用于环境管理和生物修复。

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.