CS&E Postdoctoral Associate: Parallel Iterative Methods and Preconditioners for the Large, Sparse, Symmetric Eigenvalue Problem

大规模、稀疏、对称 许多科学和工程都需要解决这个问题 应用，如电子结构计算。 迭代方法 只有满足这些要求的手段，面临着两个挑战。 第一 涉及退化或病态接近的特征值，第二个涉及 可能无法存储的非常大的矩阵。 并行计算和 预处理是提高迭代算法性能的有效途径。 这项建议有三个主要目标。 第一，延伸平行线 从原子结构计算中使用的预处理本征值方法到固态物理中复杂的分子结构计算。 第二，扩展现有的特征值编码，以解决非常大的矩阵的大量特征值。 第三，进一步探索 本征值预处理，并应用于分子结构矩阵 从有限差分/单元离散化技术。 对于无法存储的大型矩阵，也正在研究预处理。 最后，区域分解预条件子的工作继续在两个 应用程序和任意矩阵。 域分区 想法正在测试结合一个粗略解决正确的修正 选择小系统 聚合/分解技术正在 为创建小型系统而进行的调查。

The numerical solution of the large, sparse, symmetric problem is required by many scientific and engineering applications, such as electronic structure calculations. Iterative methods, the only means of meeting these demands, are faced with two challenges. The first involves degenerate or pathologically close eigenvalues, and the second involves matrices of very large size that may not stored. Parallel computing and preconditioning are powerful ways for improving the performance of iterative methods. This proposal has three major goals. First, to extend the parallel preconditioned eigenvalue methods that are used in atomic structure calculations to the complex molecular structure calculations in solid state physics. Second, to extend the existing eigenvalue codes to solve for large number of eigenvalues of very large matrices. Third, to explore further the eigenvalue preconditioning, with application to molecular structure matrices resulting from finite difference/element discretization techniques. Preconditioning for large matrices that cannot be stored are also being examined. Finally, work on domain decomposition preconditioners is continued on both the application and on arbitrary matrices. Domain partitioning ideas is being tested in combination with a coarse solve correction on a properly chosen small system. The aggregation/disaggregation technique is being investigated for creating the small system.