Presidential Young Investigator Award: Research in Sparse Matrix Methods

总统青年研究员奖：稀疏矩阵方法研究

• 批准号: 8958544
• 负责人: Howard Elman
• 金额: $ 30.22万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8958544&HistoricalAwards=false
• 关键词: Presidential; Young; Investigator; Award; Research

Methods for solving sparse linear systems of equations, especially of the type arising from discretized elliptic and parabolic partial differential equations, will be studied. The solution of such systems is often the most costly computation in scientific codes. Three general areas will be emphasized. Iterative solution of sparse linear systems of equations, including parallel techniques and nonsymmetric systems: "Multicolor" variants of incomplete factorization preconditioners display slower convergence than standard preconditioners on serial architectures, but theoretical and computational studies with model problems show them to be superior on parallel machines. These methods will be tested on problems arising in scientific application codes, where it is not certain that they will be robust. Alternatives, such as block ordering methods which have less efficient parallel implementations but display faster convergence on two-dimensional problems, will be examined. In addition, new analytic and experimental results suggest that iterative methods for nonsymmetric linear systems based on partial elimination and "line" preconditioners are very effective for solving the convection-diffusion equation. Further studies of these techniques will be made, including implementation on parallel architectures. Numerical methods for three-dimensional problems: Linear systems arising from three-dimensional elliptic problems cannot be solved at reasonable cost at the present time. Successful development of parallel and/or faster converging methods will expand the domain of solvable problems, but special attention must be paid to the specific issues associated with three-dimensionality. Two ideas will be examined: three-dimensional multicolor schemes, and "plane" preconditioners (which would generalize line methods). The latter idea build upon effective parallel two- dimensional solvers. Parallel implementation of finite element methods: Finite element methods comprise a widely used solution technique for elliptic problems that present special difficulties for parallel computing, including the presence of irregular grids and three- dimensional problems. High order finite element methods appear to offer some advantages for parallel solution of two-dimensional problems on uniform grids. Starting from this preliminary observation, the parallel solution of finite element models on irregular grids and in three dimensions will be studied. Two methodologies that will be considered are the combination of local direct solution with global iterative solution methods, and parallel solution using hierarchical basis functions.

将研究求解稀疏线性方程组的方法，特别是由离散椭圆和抛物型偏微分方程产生的类型。 此类系统的解决方案通常是科学代码中成本最高的计算。 将强调三个一般领域。 稀疏线性方程组的迭代求解，包括并行技术和非对称系统：不完全因式分解预处理器的“多色”变体在串行架构上显示出比标准预处理器更慢的收敛速度，但对模型问题的理论和计算研究表明它们在并行机上更优越。 这些方法将针对科学应用代码中出现的问题进行测试，但不确定它们是否稳健。 将研究替代方案，例如并行实现效率较低但在二维问题上表现出更快收敛的块排序方法。 此外，新的分析和实验结果表明，基于部分消除和“线”预处理器的非对称线性系统的迭代方法对于求解对流扩散方程非常有效。 将进一步研究这些技术，包括并行架构上的实现。 三维问题的数值方法：目前无法以合理的成本解决由三维椭圆问题产生的线性系统。 并行和/或更快收敛方法的成功开发将扩大可解决问题的范围，但必须特别注意与三维相关的具体问题。 将研究两个想法：三维多色方案和“平面”预处理器（这将推广线方法）。 后一个想法建立在有效的并行二维求解器的基础上。 有限元方法的并行实现：有限元方法是一种广泛使用的椭圆问题求解技术，椭圆问题给并行计算带来了特殊的困难，包括不规则网格和三维问题的存在。 高阶有限元方法似乎为均匀网格上二维问题的并行求解提供了一些优势。 从这个初步观察开始，将研究不规则网格和三维有限元模型的并行求解。 将考虑的两种方法是局部直接求解与全局迭代求解方法的组合，以及使用分层基函数的并行求解。