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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目涉及解决大型稀疏线性方程组鞍点型的新技术的开发，分析和实施。尽管最近取得了很大的进展，大型方程组的解决方案仍然是许多数值模拟的主要瓶颈之一。应用包括流体动力学，磁流体动力学，图像处理，以及更多。这个项目涉及层次（H-）矩阵技术的进一步发展。H-矩阵提供了一种有效的计算技术，它涉及到对完全填充矩阵的近似。类似的generalizationof几何代数多重网格方法，PI提出开发一个算法的代数构造的H-矩阵预处理鞍点问题。一系列三维Oseen问题的数值解的Navier-Stokes方程将作为新技术的主要应用，在这个项目中要考虑的另一个主题是H-技术在最近开发的Kronecker乘积预处理器中的应用。该提议的技术具有很高的潜力，H-矩阵是在科学和技术问题的规模和复杂性不断增加的模拟中出现的线性系统。H-矩阵于1998年首次引入，并从那时起进入了广泛的应用。H-矩阵的方法矩阵在其自身的数值分析领域以及科学家目前面临的实际大规模计算挑战方面具有重要意义。应用的例子包括磁聚变、加速器设计、电化学过程或陶瓷纳米结构生长的模型。该提案所针对的领域的进展将通过允许更快和更准确的计算机模拟对科学和工程产生积极影响。

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.