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

大型、稀疏、对称问题的数值解是许多科学和工程应用的需要，如电子结构计算。迭代方法是满足这些需求的唯一手段，它面临着两个挑战。第一个涉及退化的或病理上接近的特征值，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。这项提议有三个主要目标。首先，将用于原子结构计算的并行预条件本征值方法推广到固体物理中的复杂分子结构计算。其次，对已有的特征值码进行扩展，使之适用于求解超大型矩阵的大量特征值。第三，进一步探讨了特征值预处理方法，并将其应用于有限差分/单元离散化技术得到的分子结构矩阵。此外，还在研究无法存储的大型矩阵的预处理问题。最后，在应用程序和任意矩阵上继续研究区域分解预条件子。在适当选择的小系统上，域划分思想正在与粗略的求解校正相结合进行测试。正在研究聚集/解聚技术来创建小系统。

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.