Innovative Algorithms and Techniques for Large Scale Simulations

该项目旨在为有限元建模产生的稀疏线性系统和边界元建模方法产生的密集线性系统开发各种鲁棒，有效和可并行的求解器和预处理器。正如我们的初步结果所证明的那样，从这两个领域的技术仔细杂交能够产生设计有效的并行预处理器和求解器所需的突破。稀疏线性系统提出的技术是基于一个层次的表示使用块低秩矩阵和多极近似的逆。该项目还将研究这些技术对近似密集线性系统求解器中使用的压缩方案的影响。该项目还将研究稀疏线性系统的代数多级技术，其中层次表示用于获得替代的预处理系统。所有这些技术的并行配方将开发沿着与收敛性和并行性能的理论估计。除了这些，本研究将调查基于投影的技术，影响预处理隐含通过一组并发投影到适当的子空间。这些技术包括行投影方法，并且是近似逆预处理器的鲁棒形式。

This project aims to develop a variety of robust, effective, and parallelizable solvers and preconditioners for sparse linear systems arising from finite element modeling and dense linear systems arising from boundary element modeling approaches. As is evidenced from our preliminary results, a careful cross-breeding of techniques from these two areas is capable of yielding the breakthrough needed in designing effective parallel preconditioners and solvers. The techniques proposed for sparse linear systems are based on a hierarchy of representations using block low-rank matrices and multipole approximations of the inverse. The project will study the implications of these techniques for compression schemes used in approximate dense linear system solvers as well. The project will also investigate algebraic multilevel techniques for sparse linear systems in which hiearchical representations are used to obtain an alternate preconditioned system. Parallel formulations of all these techniques will be developed along with theoretical estimates for convergence and parallel performance. In addition to these, this study will investigate projection-based techniques that affect preconditioning implicitly through a set of concurrent projections onto appropriate subspaces. Such techniques include row-projection methods and are robust forms of approximate inverse preconditioners.