Intrinsically Parallel Spectrum Decomposition Algorithm for Large Eigenvalue Problems

本征值问题在许多科学和工程学科中都是非常重要的，包括材料科学中的电子结构计算，以及数据科学中的聚类、排序和匹配。在这些材料科学和数据科学计算中，大规模的特征值问题自然产生于重要的现代应用。然而，随之而来的特征值问题维度的快速增加很容易淹没现有的算法。这就产生了对更高效算法的迫切需求，特别是那些可以在拥有数千个内核的现代超级计算机上很好地扩展的算法。本研究以谱滤波(多项式滤波和有理函数“预条件”)为代表，重点研究了特征值问题的算法加速问题，并将其应用于稳健的软件系统中。研究人员将同时研究标准和广义特征值问题，这两个问题通常来自第一原理计算。该项目将使用定制的滤波方法作为频谱估计的第一步，并开发出一种本质上并行的实用频谱分解算法。频谱分解方法是为了克服目前频谱切片算法中遇到的主要困难而设计的。研究人员将开发新的方法，如自适应Ritz迭代和(局部)最优移位的自适应选择；这两种技术对于实现本质上可扩展的方法的效率和稳健性都很重要。该研究将极大地拓展解决大规模特征值问题的实用方法的前沿。该项目将提供有用的算法和软件，以促进需要解决越来越大的特征值问题的尖端研究。

Eigenvalue problems are of fundamental importance in a wide range of science and engineering disciplines, including electronic structure calculations in materials science, and clustering, ranking, and matching in data science. Large-scale eigenvalue problems arise naturally from significant modern applications in these materials science and data science computations. However, the associated rapid increase in the dimension of the eigenvalue problems can easily overwhelm existing algorithms. This generates pressing demand for more efficient algorithms, especially those that can scale well on modern supercomputers with many thousands of cores. This research project centers on designing highly efficient algorithms for solving large-scale eigenvalue problems and implementing them in robust software that can be effectively utilized by other researchers.The project focuses on the essence of algorithm acceleration for eigenvalue problems, represented by spectrum filtering (both by polynomial filtering and by rational function "preconditioning"). The investigator will study both standard and generalized eigenvalue problems, which often arise from first-principles calculations. The project will use a tailored filtering method as the first step for spectrum estimation and develop a practical spectrum decomposition algorithm that is intrinsically parallel. The spectrum decomposition method is designed to overcome the main difficulties encountered in state-of-the-art spectrum slicing algorithms. The investigator will develop novel approaches such as adaptive Ritz iteration and adaptive selection of (locally) optimal shifts; both techniques are important for achieving efficiency as well as robustness for intrinsically scalable methods. The research will significantly extend the forefront of practical methods for solving large-scale eigenvalue problems. The project will provide useful algorithms and software to facilitate cutting-edge research that requires solving increasingly larger eigenvalue problems.