An Inverse-Free Projected Gradient Descent Method for the Generalized Eigenvalue Problem
| dc.contributor.advisor | Zhang, Yin | en_US |
| dc.creator | Camacho, Frankie | en_US |
| dc.date.accessioned | 2017-08-01T16:15:46Z | en_US |
| dc.date.available | 2017-08-01T16:15:46Z | en_US |
| dc.date.created | 2017-05 | en_US |
| dc.date.issued | 2017-05-26 | en_US |
| dc.date.submitted | May 2017 | en_US |
| dc.date.updated | 2017-08-01T16:15:46Z | en_US |
| 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. | en_US |
| dc.format.mimetype | application/pdf | en_US |
| dc.identifier.citation | Camacho, Frankie. "An Inverse-Free Projected Gradient Descent Method for the Generalized Eigenvalue Problem." (2017) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/96001">https://hdl.handle.net/1911/96001</a>. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1911/96001 | en_US |
| dc.language.iso | eng | en_US |
| dc.rights | Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder. | en_US |
| dc.subject | generalized eigenvalue problem | en_US |
| dc.subject | linear algebra | en_US |
| dc.subject | unconstrained optimization | en_US |
| dc.title | An Inverse-Free Projected Gradient Descent Method for the Generalized Eigenvalue Problem | en_US |
| dc.type | Thesis | en_US |
| dc.type.material | Text | en_US |
| thesis.degree.department | Computational and Applied Mathematics | en_US |
| thesis.degree.discipline | Engineering | en_US |
| thesis.degree.grantor | Rice University | en_US |
| thesis.degree.level | Masters | en_US |
| thesis.degree.name | Master of Arts | en_US |
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