Mathematical Sciences: Preconditioned Parallel Methods for Large Symmetric Eigenproblems

这个项目将研究计算具有非常大对称矩阵的广义特征值问题的几个极端特征对的预条件迭代方法。这些问题在许多具有重大挑战性的科学和工程应用中具有重要意义：结构动力学和屈曲，海洋建模，量子化学和磁流体动力学。它们通常是由偏微分方程组描述的连续模型的离散化引起的。矩阵可能太大，以至于标准的数值方法变得不令人满意，甚至不能在最强大的现代超级计算机上实现，因为它们太慢或需要太多的内存。预条件迭代法是专门为这类问题设计的，现在人们很好地理解了它用于求解大型病态线性代数方程组。对于特征问题，理论仍然很差，这些方法很少使用。理想情况下，该方法将计算分离良好的特征值簇和相应的特征空间，其计算成本与求解相应的线性代数系统的计算成本相同。这样的过程可以是有效的和并行的。区域分解和迭代子结构方法尤其有前途。研究的最终目标是建立一个完整的理论，现在已知的对称特征问题的预条件方法，并找到新的快速，准确和鲁棒的迭代方法。这将为在高性能并行计算系统上开发先进的软件和算法提供基础。预计将编写一本关于大型对称特征值问题的预条件迭代方法的研究专著，以总结该项目的结果。

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.