Algebraic hierarchical matrix preconditioners for two- and three-dimensional saddle point problems

用于二维和三维鞍点问题的代数分层矩阵预处理器

• 批准号: 0913017
• 负责人: Allan Mills
• 金额: $ 13.71万
• 依托单位: Tennessee Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0913017&HistoricalAwards=false
• 关键词: Algebraic; hierarchical; matrix; preconditioners; two

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project deals with the development, analysis and implementationof novel techniques for the solution of large, sparse linear systemsof equations of saddle point type. Despite much recent progress,the solution of large systems of equations remains one of the mainbottlenecks in many numerical simulations. Applications includefluid dynamics, magnetohydrodynamics, image processing, and many more.This project deals with the further development of the technique of hierarchical (H-)matrices. H-matrices provide an efficient technique for computationsinvolving approximations to fully populated matrices.The standard construction of an H-matrix is basedon the underlying geometry of the application. Similar to the generalizationof geometric to algebraic multigrid methods, the PI proposes to develop an algorithm for the algebraic construction of H-matrix preconditioners for saddle point problems. Sequences of three-dimensional Oseen problems that result in the numerical solution of the Navier-Stokes equations will serve as the major application of the novel techniques.An additional topic to be considered in this project isthe application of H-techniques in only recently developedKronecker product preconditioners. The techniques of this proposal have a high potential to lead to robust and scalable blackbox solvers for large, linear systems arising in the simulation of scientific and technical problems of increasing size and complexity.H-matrices have first been introduced in 1998 and since then entered into a wide range of applications.The approach of H-matrices is of significant importance within its own field of numerical analysis and also with respect topractical large-scale computing challenges that scientists are currently facing. Examples for applications include models for magnetic fusion, accelerator design, electrochemical processes or the growth of ceramic nanostructures.Progress in the areas targeted in this proposal will have a positive impact on science and engineering by allowing for fasterand more accurate computer simulations.

该奖项是根据2009年《美国回收与再投资法》（公法111-5）资助的。本项目介绍了用于解决鞍点类型的大型稀疏线性线性系统的新技术的开发，分析和实施。尽管最近取得了很多进展，但在许多数值模拟中，大型方程式的解决方案仍然是主链带之一。应用程序包括流体动力学，磁性流动力学，图像处理等等。该项目涉及层次（H-）矩阵技术的进一步发展。 H-膜为计算近似值的计算提供了一种有效的技术。H-Matrix的标准结构是基于应用程序的基本几何形状。类似于几何形状对代数多族方法的概括，PI提议开发一种用于用于鞍点问题的H-马trix预处理器代数构建的算法。导致Navier-Stokes方程的数值解决方案的三维OSEEN问题的序列将作为新技术的主要应用。在此项目中要考虑的其他主题是H-Techniques在最近的ReveartionKronecker产品预处理中的应用。该提案的技术具有很高的潜力，可以导致可靠的大型，线性系统在模拟型号和技术问题的模拟中产生的大型和技术问题的强大和可扩展的求解器。H-H-ATRICES于1998年首先引入了1998年引入，自那时以来，当时进入了广泛的应用范围内，h-matrices在其自身的范围内也具有重要的范围，并且在数字分析中也具有重要的重要性。目前面对。应用程序的示例包括用于磁融合的模型，加速器设计，电化学过程或陶瓷纳米结构的生长。在该提案中针对的领域中的过程将对科学和工程产生积极影响，通过允许更快的和更准确的计算机模拟。