Advanced Eigensolvers for Science and Engineering Applications

University Of California-Davis, Davis CA

Investigators

Abstract

Large-scale eigenvalue computation is a long-standing problem in computational mathematics and computational science and engineering. It is frequently encountered as a critical kernel in simulations and data analysis. Significant progress has been made both in general-purpose eigensolvers and also in specialized eigensolvers that exploit underlying particular mathematical properties and data structure. However, new needs and challenges continue to emerge from science and engineering applications. This project involves the development of advanced mathematical analysis and robust, efficient algorithms for two emerging classes of eigenvalue problems: sparse plus low rank linear eigenvalue problems and eigenvalue problems with eigenvalue nonlinearity. In addition, this project has broader impacts in training graduate students in interdisciplinary research. While much of the work involves significant technical expertise, other areas can be successfully understood and tackled by advanced undergraduates. The computational stability, efficiency and reliability of the new solvers for the two classes of eigenvalue problems will be greatly enhanced by skillful exploitation of underlying mathematical properties and matrix structure. In particular, for the eigenvalue problems with eigenvalue nonlinearity, new solver will combine rational approximations of nonlinearity for high accuracy, trimmed linearizations for low dimensionality, and compact representations of the projection subspace bases for memory-saving and communication efficiency. The outcomes of this project will be the publication of new theory and algorithms and open-source software.