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

许多科学和工程应用，如电子结构计算，都需要大型、稀疏、对称问题的数值解。作为满足这些需求的唯一手段，迭代方法面临着两方面的挑战。第一种涉及退化或病态接近的特征值，第二种涉及可能不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力手段。这项提议有三个主要目标。首先，将原子结构计算中使用的平行预条件特征值方法推广到固体物理中复杂分子结构的计算中。第二，扩展现有的特征值编码，求解超大矩阵的大量特征值。第三，进一步探索特征值预处理，并将其应用于由有限差分/单元离散化技术产生的分子结构矩阵。对于不能存储的大矩阵的预处理也正在研究中。最后，从应用和任意矩阵两个方面继续研究了域分解预条件。在一个适当选择的小系统上，结合粗解校正对领域划分思想进行了测试。正在研究用于创建小系统的聚合/分解技术。

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.