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。DFT的根本重要性源于它在薛定谔方程上对物质基本组成部分定律的简化描述。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，复杂材料的第一主元DFT计算往往会导致巨大的计算问题。如果没有有效的可扩展算法，即使是最强大的超级计算机，也很容易被问题的规模所淹没。DFT计算中的一个主要计算瓶颈是自洽环内大规模特征问题的重复求解。为了在不影响精度的情况下显著降低计算成本，需要新颖的算法。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Chebyshev多项式进行自适应子空间滤波，除了自洽环路的第一步需要对角化外，它与无特征向量的方法接近。该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型无特征向量算法，用于DFT计算中的非线性特征值问题。重点还将放在扩展非线性子空间滤波技术发展在实空间设置到广泛使用的平面波设置。目标是使第一主DFT计算更有效和可行，以研究越来越复杂的材料。大规模特征值相关问题在科学和工程中普遍存在，往往构成计算瓶颈。可扩展特征算法的进步可以显著降低计算成本，同时保持准确性，这将对智力产生广泛而深远的影响。除材料科学外，所研究的滤波和预处理技术在模式识别、信息检索和数据挖掘以及其他大规模数据集的降维方法等广泛领域也有潜在的应用。

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.