Mathematical Sciences: Iterative Methods for Large-Scale NonLinear and Linear Systems

研究人员开发了更有效和高效的牛顿迭代(截断牛顿)方法，即使用迭代线性代数方法的牛顿方法的实现。随之而来的一个目标是开发用于线性问题的改进的Krylov子空间方法和相关技术；这项工作支持主要目标，本身也是重要的。该项目围绕最近开发的全局收敛的不精确牛顿算法、GMRES的改进实现以及基于Lanczos的方法和“残差平滑”技术的新应用展开。一个具体的目标是构造适用于公开分发的牛顿迭代法和GMRES的高质量代码。牛顿迭代方法和Krylov子空间方法有许多重要的科学和工业应用，例如飞机动力学建模、半导体建模、气候建模、油藏模拟和含水层中的污染物传输。它们通常应用于大规模问题，通常需要高性能的向量计算机和并行计算机，并且确实为它们的发展提供了巨大的动力。这项研究有助于改进科学和工业应用的计算建模技术，并更有效地利用高性能计算机。

The investigator develops more effective and efficient Newton iterative (truncated Newton) methods, i.e., implementations of Newton's method that use iterative linear algebra methods. A concomitant objective is to develop improved Krylov subspace methods and associated techniques for linear problems; this work supports the primary objective and is important in its own right. The project centers around recently developed globally convergent inexact Newton algorithms, improved implementations of GMRES, and new applications of Lanczos-based methods and "residual smoothing" techniques. A specific goal is to construct high-quality codes for Newton iterative methods and for GMRES that are suitable for public distribution. Newton iterative methods and Krylov subspace methods have many important scientific and industrial applications; examples include aircraft dynamics modeling, semiconductor modeling, climate modeling, oil reservoir simulation, and contaminant transport in aquifers. The large-scale problems to which they are usually applied often require high-performance vector and parallel computers and, indeed, have provided significant incentive for their development. This research contributes to improved computational modeling techniques for scientific and industrial applications and more effective utilization of high-performance computers.