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方案。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提供了例子。成功的计算机模拟这样的问题需要仔细的建模以及有效的方法，以获得往往非常大的方程组，将出现在任何试图提供可靠的支持，复杂的工程结构的设计和优化的及时解决方案。已经安装或正在建造的石油平台发生的代价高昂的故障可以说明这种工作的必要性。这项工作涉及开发精确的数学模型以及设计快速求解器;目前的项目将侧重于开发此类求解器。大规模计算模型需要使用现代计算机技术，特别是具有许多处理器的计算机系统。首席研究员将继续积极与软件工程师合作，为各种问题类开发改进的解决方案。在这个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内存层次结构，实验清楚地表明，它们可以很好地扩展到完整的处理器和数十亿的自由度。区域分解方法提供了基于共轭梯度算法结合预处理器的迭代求解器。预条件提供了一个近似逆的偏微分方程的刚度矩阵制定变分和近似使用伽辽金方法。任何成功的区域分解算法都可以在给定偏微分方程的域被细分成的许多子域上使用求解器。此外，为了获得可缩放算法，即，当子域和处理器的数量增加时，收敛速度不会恶化的算法，必须引入预处理器的粗略全局部分;对于大问题，还引入第三个甚至更粗糙的级别。这些算法牢牢扎根于数学理论，现在正以几年前无法预见的方式迅速发展。强大的想法，现在正在开发，提供了更好的设计的粗糙组件的代价，解决相对较小的广义本征值问题，在设置阶段的计算。最近的实验表明，这些设备大大提高了算法的鲁棒性，即使是材料特性变化很大的问题。 本项目旨在促进这些发展。

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.