Innovative Algorithms and Techniques for Large Scale Simulations

该项目旨在为稀疏线性系统和稠密线性系统开发各种健壮、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，仔细地交叉培育这两个领域的技术能够在设计有效的并行预处理器和解算器方面产生所需的突破。所提出的稀疏线性系统的技术是基于使用块低秩阵和逆矩阵的多极近似的表示的层次结构。该项目还将研究这些技术对用于近似稠密线性系统解算器的压缩方案的影响。该项目还将研究稀疏线性系统的代数多水平技术，在该技术中，使用历史表示法来获得另一个预条件系统。所有这些技术的并行公式将与收敛和并行性能的理论估计一起开发。除此之外，这项研究还将研究基于投影的技术，这些技术通过一组同时投射到适当的子空间来隐含地影响预适应。这种技术包括行投影法，是近似逆预条件子的稳健形式。

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.