Domain Decomposition Methods: Algorithms and Theory

该研究项目的主要目的是进一步开发快速可靠的大规模计算方法，以支持复杂工程问题的解决。石油平台、复杂的天线系统以及石油、天然气和污染物在多孔介质中的流动都是例子。对这类问题的成功的计算机模拟需要仔细的建模和有效的方法，以获得在任何尝试为复杂工程结构的设计和优化提供可靠支持时经常会出现的非常大的方程组的及时解。已经安装或正在建造的石油平台发生了代价非常高昂的故障，这说明需要进行这种工作。这项工作涉及开发精确的数学模型以及设计快速求解器；目前的项目将专注于开发这种求解器。大规模计算模型需要接触到现代计算机技术，特别是具有许多处理器的计算机系统。首席调查员将继续与软件工程师积极合作，为各种问题类别开发改进的解算器。本项目中开发的算法都将基于区域分解。区域分解算法尊重现代并行计算系统的存储层次结构，实验清楚地表明，它们可以很好地扩展到整个处理器集和数十亿个自由度。区域分解方法提供了基于共轭梯度算法和预条件的迭代求解器。预处理器提供偏微分方程的刚度矩阵的近似逆，该偏微分方程用Galerkin方法进行变分表示和近似。任何成功的区域分解算法都需要在给定偏微分方程域被细分的许多子域上使用求解器。此外，为了获得一个可伸缩的算法，即一个算法的收敛速度不会随着子域和处理器的数量的增加而恶化，必须引入预条件算子的粗全局部分；对于大的问题，还引入了第三个更粗的水平。这些算法牢牢植根于数学理论，现在正以几年前还没有预见到的方式迅速发展。现在正在发展强大的想法，这些想法以在计算的建立阶段解决相对较小的广义特征值问题为代价，提供了大大改进的粗略组件的设计。最近的实验表明，这些设备极大地提高了算法的稳健性，即使是在材料性质变化很大的问题上也是如此。该项目旨在为这些发展做出贡献。

This research project aims primarily to further develop fast and reliable methods for large scale computations to support the solution of complicated engineering problems. Examples are provided by oil platforms, complicated antenna systems, and flow of oil, gas, and contaminants in porous media. Successful computer simulations of such problems require careful modeling as well as efficient methods to obtain timely solutions of the often very large systems of equations that will arise in any attempt to provide reliable support for the design and optimization of complicated engineering structures. The need for such work can be illustrated by the very costly failures that have happened to oil platforms already installed or in the process of being built. This work involves the development of accurate mathematical models as well as the design of fast solvers; the current project will focus on developing such solvers. Large scale computational models require access to modern computer technology, in particular to computer systems with many processors. The principal investigator will continue to work actively with software engineers to develop improved solvers for a variety of problem classes. The algorithms developed in this project will all be based on domain decomposition. Domain decomposition algorithms respect the memory hierarchies of modern parallel computing systems, and experiments clearly illustrate that they scale very well up to the full set of processors and billions of degrees of freedom. Domain decomposition methods provide iterative solvers based on a conjugate gradient algorithm combined with a preconditioner. A preconditioner provides an approximate inverse of the stiffness matrix of the partial differential equation formulated variationally and approximated using a Galerkin method. Any successful domain decomposition algorithm works with solvers on often very many subdomains into which the domain of the given partial differential equation has been subdivided. In addition, to obtain a scalable algorithm, i.e., an algorithm with a convergence rate that does not deteriorate when the number of subdomains and processors are increased, a coarse global part of the preconditioner must be introduced; for large problems a third even coarser level is also introduced. Firmly rooted in mathematical theory, these algorithms are now developing rapidly in a way not foreseen just a few years ago. Powerful ideas are now developing that provide much improved design of the coarse components at the expense of solving relatively small generalized eigenvalue problems in the set-up phase of the computation. Recent experiments show that these devices greatly improve the robustness of the algorithms even of problems with greatly varying material properties. This project aims to contribute to these developments.