Mathematical Sciences: Preconditioned Parallel Methods for Large Symmetric Eigenproblems

小行星9501507 本项目将研究计算的预条件迭代方法 广义特征值问题的几个极值特征对， 大型对称矩阵 这些问题在许多国家都是非常重要的。 科学与工程应用：结构动力学 屈曲，海洋建模，量子化学， 磁流体力学它们通常来自于 连续模型，由偏微分方程系统描述。 矩阵可能是如此之大，以至于标准的数值方法变得 即使在最强大的国家也无法实施， 现代超级计算机，因为它们太慢或需要太多的内存。 预条件迭代法是专门为求解 现在他们被很好地理解为解决大型病态 线性代数方程组对于特征值问题， 这些方法很少使用。理想情况下，这些方法将 计算特征值和相应的 本征空间的计算成本相同的顺序为解决 对应的线性代数系统。这些过程可以是有效的 可并行化。区域分解和迭代子结构 方法特别有前途。 研究的最终目标是发展一个完整的理论， 预处理方法现在已知的对称特征值问题， 寻找新的快速、准确和鲁棒的迭代方法。这将提供 为高级软件和算法的开发奠定了基础 性能并行计算系统。预计一项研究 大型对称问题的预条件迭代法 本征值问题将准备总结的结果， 项目

9501507 Knyazev This project will study preconditioned iterative methods for computing several extreme eigenpairs of generalized eigenvalue problems, with very large symmetric matrices. Such problems are of major importance in many grand challenge scientific and engineering applications: structural dynamics and buckling, ocean modeling, quantum chemistry, and magnetohydrodynamics. They arise typically from the discretization of continuous models, described by systems of partial differential equations. Matrices may be so large that the standard numerical methods become unsatisfactory and cannot be implemented even on the most powerful modern supercomputers as they are too slow or require too much memory. Preconditioned iterative methods were specially designed for problems of that kind and now they are well understood for solving large ill-conditioned linear algebraic systems of equations. For eigenproblems the theory is still poor, and these methods are rarely used. Ideally, the methods would compute well-separated clusters of eigenvalues and corresponding eigenspaces at the same order of computational cost as that for solution of the corresponding linear algebraic system. Such processes can be effective and parallelizable. Domain decomposition and iterative substructuring methods are particularly promising. The ultimate goal of the research is to develop a complete theory of the preconditioned methods now known for symmetric eigenproblems and to find new fast, accurate, and robust iterative methods. This would provide a basis for the development of advanced software and algorithms on high performance parallel computing systems. It is expected that a research monograph on preconditioned iterative methods for large symmetric eigenvalue problems will be prepared to summarize the results of the project.