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dc.contributor.advisor Zhang, Yin
dc.creatorCamacho, Frankie
dc.date.accessioned 2017-08-01T16:15:46Z
dc.date.available 2017-08-01T16:15:46Z
dc.date.created 2017-05
dc.date.issued 2017-05-26
dc.date.submitted May 2017
dc.identifier.citation Camacho, Frankie. "An Inverse-Free Projected Gradient Descent Method for the Generalized Eigenvalue Problem." (2017) Master’s Thesis, Rice University. https://hdl.handle.net/1911/96001.
dc.identifier.urihttps://hdl.handle.net/1911/96001
dc.description.abstract The generalized eigenvalue problem is a fundamental numerical linear algebra problem whose applications are wide ranging. For truly large-scale problems, matrices themselves are often not directly accessible, but their actions as linear operators can be probed through matrix-vector multiplications. To solve such problems, matrix-free algorithms are the only viable option. In addition, algorithms that do multiple matrix-vector multiplications simultaneously (instead of sequentially), or so-called block algorithms, generally have greater parallel scalability that can prove advantageous on highly parallel, modern computer architectures. In this work, we propose and study a new inverse-free, block algorithmic framework for generalized eigenvalue problems that is based on an extension of a recent framework called eigpen -- an unconstrained optimization formulation utilizing the Courant Penalty function. We construct a method that borrows several key ideas, including projected gradient descent, back-tracking line search, and Rayleigh-Ritz (RR) projection. We establish a convergence theory for this framework. We conduct numerical experiments to assess the performance of the proposed method in comparison to two well-known existing matrix-free algorithms, as well as to the popular solver ARPACK as a benchmark (even though it is not matrix-free). Our numerical results suggest that the new method is highly promising and worthy of further study and development.
dc.format.mimetype application/pdf
dc.language.iso eng
dc.subjectgeneralized eigenvalue problem
linear algebra
unconstrained optimization
dc.title An Inverse-Free Projected Gradient Descent Method for the Generalized Eigenvalue Problem
dc.date.updated 2017-08-01T16:15:46Z
dc.type.genre Thesis
dc.type.material Text
thesis.degree.department Computational and Applied Mathematics
thesis.degree.discipline Engineering
thesis.degree.grantor Rice University
thesis.degree.level Masters
thesis.degree.name Master of Arts


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