Solving large-scale eigen-related problems: Efficient and scalable algorithms

这个项目的主要目标集中在解决大规模特征相关问题的新的有效算法。术语“本征相关”是指可以使用某些本征值和/或本征向量来表达解的任何问题。这些与本征值相关的问题是许多科学和工程学科的核心，包括材料科学、统计计算、数据挖掘和机器学习。现代科学问题的高维性和复杂性给特征值计算带来了巨大的计算负担，使得现有的特征值算法失效。对于能够考虑现代科学问题的动态和非线性性质的特征算法有很大的需求，从而导致能够更有效地处理大量数据集的算法。 研究者建议系统地研究和开发新的可扩展的方法，以减轻与解决大规模特征相关问题有关的计算瓶颈。研究的方法包括全局频谱滤波以及多级采样和聚类。具体来说，研究者将开发方法（a）以减少密度泛函理论计算的对角化成本;（B）以加速在广泛应用中使用的部分SVD计算;以及（c）通过Nystrom型方法加速低秩矩阵近似。这个项目将开发有效的可扩展算法来解决大型特征相关问题。 拟议工作的一个重要的更广泛的影响是，由此产生的方法将加速在广泛的研究领域的发现，包括材料科学和数据挖掘，这对国家能源安全和经济竞争力至关重要。该项目还将为博士提供有趣和具有挑战性的主题。论文研究，本科生研究，和推广活动。

The main objectives of this project focus on novel efficient algorithms for solving large-scale eigen-related problems. The term "eigen-related" refers to any problem for which a solution can be expressed using certain eigenvalues and/or eigenvectors. Such eigen-related problems are central to a wide range of scientific and engineering disciplines, including materials science, statistical computing, data mining and machine learning. The ever-increasing high dimensions and complexity of modern scientific problems bring overwhelming computational burden for eigen-related computations and render existing eigen-algorithms ineffective. There is great demand for eigen-algorithms that can take into account the dynamic and nonlinear nature of modern scientific problems, leading to algorithms that can handle massive datasets more efficiently. The investigator proposes to systematically study and develop novel scalable approaches for alleviating the computational bottlenecks related to solving large-scale eigen-related problems. Approaches to be investigated include global spectrum filtering as well as multilevel sampling and clustering. Specifically, the investigator will develop methods (a) to reduce the diagonalization cost for density functional theory calculations; (b) to accelerate partial SVD calculations that are used in a significantly broad range of applications; and (c)to accelerate the low-rank matrix approximations via Nystrom-type methods.Large eigen-related problems are ubiquitous in modern science, engineering, and economics applications. This project will develop efficient scalable algorithms for solving large eigen-related problems. One important broader impact of the proposed work is that the resulting methods will accelerate discoveries in a wide range of research fields, including material science and data mining, which are critical to national energy security and economic competitiveness. The project will also provide interesting and challenging topics for Ph.D. dissertation research, for undergraduate research, and for outreach activities.