Intrinsically Parallel Spectrum Decomposition Algorithm for Large Eigenvalue Problems

Southern Methodist University, Dallas TX

Investigators

Abstract

Eigenvalue problems are of fundamental importance in a wide range of science and engineering disciplines, including electronic structure calculations in materials science, and clustering, ranking, and matching in data science. Large-scale eigenvalue problems arise naturally from significant modern applications in these materials science and data science computations. However, the associated rapid increase in the dimension of the eigenvalue problems can easily overwhelm existing algorithms. This generates pressing demand for more efficient algorithms, especially those that can scale well on modern supercomputers with many thousands of cores. This research project centers on designing highly efficient algorithms for solving large-scale eigenvalue problems and implementing them in robust software that can be effectively utilized by other researchers. The project focuses on the essence of algorithm acceleration for eigenvalue problems, represented by spectrum filtering (both by polynomial filtering and by rational function "preconditioning"). The investigator will study both standard and generalized eigenvalue problems, which often arise from first-principles calculations. The project will use a tailored filtering method as the first step for spectrum estimation and develop a practical spectrum decomposition algorithm that is intrinsically parallel. The spectrum decomposition method is designed to overcome the main difficulties encountered in state-of-the-art spectrum slicing algorithms. The investigator will develop novel approaches such as adaptive Ritz iteration and adaptive selection of (locally) optimal shifts; both techniques are important for achieving efficiency as well as robustness for intrinsically scalable methods. The research will significantly extend the forefront of practical methods for solving large-scale eigenvalue problems. The project will provide useful algorithms and software to facilitate cutting-edge research that requires solving increasingly larger eigenvalue problems.