Novel Scalable Algorithms in Density Functional Theory Calculations

摘要：密度泛函理论(DFT)是20世纪最伟大的科学成就之一。密度泛函理论的根本重要性源于它对薛定谔方程上的基本材料积木规律的简化描述。DFT在包括凝聚态物理、量子化学和材料科学在内的广泛领域产生了重大影响。然而，首先对复杂材料进行密度泛函理论计算，往往会导致大量的计算问题。由于没有高效的可扩展算法，即使是最强大的超级计算机，问题的巨大规模也很容易被淹没。DFT计算中的一个主要计算瓶颈是在自洽循环内重复求解大规模特征问题。为了在不影响精度的情况下显著降低计算成本，需要新的算法。我们将研究基于最近发展的切比雪夫滤波子空间迭代(CheFSI)方法的滤波方法。CheFSI利用切比雪夫多项式进行自适应子空间滤波，除了自洽循环的第一步需要对角化之外，它接近于无特征向量的方法。提出的研究将进一步研究和开发基于多项式滤波和预处理技术的新的无特征向量算法来解决DFT计算中的非线性特征值问题。重点还将强调将在真实空间环境中发展的非线性子空间滤波技术扩展到广泛使用的平面波环境。其目的是使第一性原理的DFT计算更加高效和可行，以研究日益复杂的材料。大规模的与特征值相关的问题在科学和工程中普遍存在，它们往往构成计算瓶颈。在可伸缩特征算法方面的进展可以显著降低计算成本，同时保持准确性，将产生广泛而深远的智力影响。除材料科学外，待研究的过滤和预处理技术在模式识别、信息检索和数据挖掘等领域也有潜在的应用前景，以及其他针对海量数据集的降维方法。

Title: Novel Scalable Algorithms in Density Functional Theory CalculationsAbstract:Density functional theory (DFT) is one of the great scientific achievements of the 20th century. The fundamental importance of DFT originates from its much simplified description of the law of the fundamental building blocks of materials over the Schrodinger equation. DFT has had significant impact in a broad range of fields including condensed matter physics, quantum chemistry and materials science. However, first principals DFT calculations for complex materials often lead to computational problems of enormous dimension. Without efficient scalable algorithms, the sheer size of the problems can easily overwhelm even the most powerful supercomputers.One major computational bottleneck in DFT calculations is the repeated solution of large scale eigenproblems inside a self-consistent loop.Novel algorithms are needed in order to significantly reduce the computational cost without scarifying accuracy. We will investigate filtering approaches based on the recently developed Chebyshev filtered subspace iteration (CheFSI) method.CheFSI utilizes Chebyshev polynomials for adaptive subspace filtering, it is close to an eigenvector-free approach except that the first step of the self-consistent loop requires a diagonalization. The proposed research will further investigate and develop novel eigenvector-free algorithms based on polynomial filtering and preconditioning techniques for the nonlinear eigenvalue problems in DFT calculations. Emphasis will also be placed on extending the nonlinear subspace filtering techniques developed in the real-space setting to the widely used plane-wave setting. The goal is to make first principals DFT calculations more efficient and feasible for the study of increasingly more complex materials.Large scale eigenvalue related problems are ubiquitous in science and engineering and they often constitute the computational bottlenecks. Advances in scalable eigen-algorithms that can significantly reduce computational cost while maintaining accuracy will have broad and far-reaching intellectual impact. Besides materials science, the filtering and preconditioning techniques to be investigated also have potential applications in a broad range of areas such as pattern recognition, information retrieval and data mining, and other dimensionality reduction approaches for massive data sets.