Parallel Preconditioned Eigenvalue and Singular Value Solvers

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This award, from the computational mathematics program of the Division of Mathematical Sciences, supports the research of the principal investigator, which covers a balanced mix of theoretical investigations, software support and development, and advances for modern applications. The research focuses on design of new efficient robust parallel preconditioned methods for interior eigenvalue and singular value computations. The main emphasis is on the following challenging issues: investigating availability and efficiency of preconditioning; avoiding the folded spectrum approach, analyzing the possibility of computing the singular values of an operator using preconditioning; and establishing convergence theory for novel iterative solvers. For the latter, a novel application of classical majorization theory permits analysis of the convergence behavior of block eigenvalue and singular value solvers. This work is also relevant for applications; the singular vectors, corresponding to the largest singular values of a matrix are used for performing the principal component analysis of data described by a data matrix. Here the focus is on multi-dimensional image segmentation.The topic of this research is motivated by the fact that the class of problems under consideration describes many phenomena in physics and mechanics, and for practical calculations may require extreme computational power. Advances in the computational approaches can provide noticeable improvements in the accuracy and efficiency of calculations. Implementing the developments in publicly available software enhances the potential for significant broader impact, due to the relevance of the software for a large range of applications. The application specifically targeted by the PI is image segmentation, such as occurs in tracking moving objects in videos. Additional impact of the project is on graduate student training. The research of the PI provides an opportunity to train students in important computational research motivated by real practical applications.