Improving Numerical Methods for Large Eigenvalue Problems

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Eigenvalue problems are the cornerstones of computational science and engineering. They arise in applications ranging from electronic structure calculations in physics and chemistry, dynamic of supramolecular systems in biology to structural and vibration in civil and mechanical engineering. Emerging applications include the investigation and design of new materials such as Lithium-ion electrolyte and graphene and the study of dynamics of viral capsids of supramolecular systems such as Zika and West Nile viruses. These applications require large number of eigenvalues. The capability of being able to efficiently compute large number of eigenvalues will not just be appealing, but also mandatory for the next generation of eigensolvers. In this project, we will undertake synergistic efforts to develop mathematical theory and numerical methods for large eigenvalue problems. The outcome of this project will provide mathematical theory and computational tools for scientists and engineers to obtain more precise simulation outputs in much less time, and to allow them to pursue more productive simulation strategies. This project will integrate research activities into interdisciplinary teaching, education and training of graduate students in the forefront of computational mathematics. One graduate student will be funded by this award.To address challenging issues of existing algorithms and software for large linear eigenvalue problems, we will focus on two core techniques. One is an explicit external deflation for reliably moving away the computed eigenpairs to prevent the algorithm from computing over again those quantities. The second technique is a communication-avoiding matrix powers kernel for fast sparse-plus-low-rank matrix-vector products in Krylov subspace solvers with the explicit external deflation. For large nonlinear eigenvalue problems, we develop mathematical theory and algorithm templates for guiding the design and implementation of approximation based nonlinear eigensolvers. In addition, we will explore an emerging formulation of large nonlinear eigenvalue problems where underlying nonlinear matrix-valued functions are not explicitly available.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.