Design and Implementation of New Scalable Algorithms in Nano-Scale Materials Science

大规模特征值相关问题构成了广泛的科学和工程学科的计算瓶颈。本项目的中心目标是为这些瓶颈问题之一设计和实现新颖的、可扩展的算法：来自第一性原理密度泛函理论（DFT）计算的非线性特征值问题。DFT是最重要的科学成果之一，它彻底改变了研究原子和分子电子结构的方法，并已成为凝聚态物理和材料科学等领域不可或缺的工具。DFT的应用和计算已经取得了巨大的进步。然而，DFT的全部能力仍然受到计算约束的严重限制。一个主要的瓶颈是大规模特征值问题的重复求解，这些问题在自洽场（SCF）环路中可能具有前所未有的维度。利用自适应多项式滤波器，提出了一种几乎不含特征向量的非线性切比雪夫滤波子空间迭代（CheFSI）方法。该方法显著减轻了SCF计算的计算负担。本研究将：(1)进一步提高CheFSI方法的效率，特别是用于SCF第一步初始对角化的算法；(2)继续改进和利用我们自己的DFT软件包parsec，该软件包使用CheFSI，对具有科学和技术意义的材料进行具有挑战性的DFT模拟；(3)扩展了CheFSI中使用的滤波思想，显著改进了平面波包（PWscf）中的对角化方法；(4)探索和发展新的DFT计算中非线性特征值的滤波和预处理技术。DFT具有根本性的重要性，因为它描述了材料基本组成部分的规律。使DFT在复杂材料中更具计算可行性的进展将对智力产生深远的影响。两个目标包PARSEC和PWscf都是开源软件，拥有庞大的用户群，能够提高瓶颈特征问题求解器的效率将使大量研究人员受益，并有助于应用DFT探索以前未研究过的更大和更复杂的材料的性质。此外，所探索的滤波和预处理技术与原理子空间的计算密切相关，它们在模型约简、数据挖掘和海量数据集的信息检索等领域具有潜在的广泛应用前景。

Large scale eigenvalue-related problems constitute the computational bottlenecks in a wide range of science and engineering disciplines. The central goal of this project is to design and implement novel,scalable algorithms for one of these bottleneck problems: the nonlinear eigenvalue problems from first principles density functional theory (DFT) calculations. DFT is one of the most significantscientific achievements, it has revolutionized approaches used in studying electronic structures of atoms and molecules and has been an indispensable tool in fields such as condensed matter physics andmaterials science. Tremendous progress has been made in DFT applications/calculations. However, the full power of DFT is still severely limited by computational constraints. One major bottleneck isthe repeated solution of large scale eigenvalue problems that can be of unprecedented dimension in the self-consistent-field (SCF) loop. By exploiting adaptive polynomial filters, we recently developed thenonlinear Chebyshev filtered subspace iteration (CheFSI) method which is almost eigenvector-free. This method significantly alleviates the computational burden for SCF calculations. The proposed research will(1) further improve the efficiency of the CheFSI method, especially the algorithm used for the initial diagonalization at the first SCF step; (2) continue improving and utilizing our own DFT package calledPARSEC, which uses CheFSI, to perform challenging DFT simulations on materials of scientific and technological significance; (3) extend the filtering ideas used in CheFSI to significantly improve thediagonalization methods in the plane-wave package called PWscf; (4) explore and develop novel filtering and preconditioning techniques for the nonlinear eigenvalues in DFT calculations.DFT is of fundamental importance because it describes the law of the fundamental building blocks of materials. Advances in making DFT more computationally feasible for complex materials will have far-reaching intellectual impact. The two targeted packages PARSEC and PWscf areboth open source software with large user group, being able to improve the solver efficiency for the bottleneck eigenproblems will benefit a large number of researchers and facilitate applying DFT to exploreproperties of larger and more complex materials that have not been studied before. Furthermore, the filtering and preconditioning techniques to be explored are closely related to calculating principaleigensubspaces, they have potentially broad range of applications in areas such as model reduction, data mining and information retrieval of massive data sets.