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.