Accelerated Residual Methods for the Iterative Solution of Systems of Equations
© 2018 Society for Industrial and Applied Mathematics. We present accelerated residual methods for the iterative solution of systems of equations by leveraging recent developments in accelerated gradient methods for convex optimization. The stability properties of the proposed method are analyzed fo...
| Main Authors: | , , , |
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| Format: | Article |
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2021
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| Online Access: | https://hdl.handle.net/1721.1/132158 |
| _version_ | 1826203074781249536 |
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| author | Nguyen, N. C. Fernandez, P. Freund, R. M. Peraire, J. |
| author_facet | Nguyen, N. C. Fernandez, P. Freund, R. M. Peraire, J. |
| author_sort | Nguyen, N. C. |
| collection | MIT |
| description | © 2018 Society for Industrial and Applied Mathematics. We present accelerated residual methods for the iterative solution of systems of equations by leveraging recent developments in accelerated gradient methods for convex optimization. The stability properties of the proposed method are analyzed for linear systems of equations by using the finite difference equation theory. Next, we introduce a residual descent restarting strategy and an adaptive computation of the acceleration parameter to enhance the robustness and efficiency of our method. Furthermore, we incorporate preconditioning techniques into the proposed method to accelerate its convergence. We demonstrate the performance of our method on systems of equations resulting from the finite element approximation of linear and nonlinear partial differential equations. In a variety of test cases, the numerical results show that the proposed method is competitive with the pseudo-time-marching method, Nesterov's method, and Newton-Krylov methods. Finally, we discuss some open issues that should be addressed in future research. |
| first_indexed | 2024-09-23T12:30:56Z |
| format | Article |
| id | mit-1721.1/132158 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T12:30:56Z |
| publishDate | 2021 |
| record_format | dspace |
| spelling | mit-1721.1/1321582021-09-21T03:35:56Z Accelerated Residual Methods for the Iterative Solution of Systems of Equations Nguyen, N. C. Fernandez, P. Freund, R. M. Peraire, J. © 2018 Society for Industrial and Applied Mathematics. We present accelerated residual methods for the iterative solution of systems of equations by leveraging recent developments in accelerated gradient methods for convex optimization. The stability properties of the proposed method are analyzed for linear systems of equations by using the finite difference equation theory. Next, we introduce a residual descent restarting strategy and an adaptive computation of the acceleration parameter to enhance the robustness and efficiency of our method. Furthermore, we incorporate preconditioning techniques into the proposed method to accelerate its convergence. We demonstrate the performance of our method on systems of equations resulting from the finite element approximation of linear and nonlinear partial differential equations. In a variety of test cases, the numerical results show that the proposed method is competitive with the pseudo-time-marching method, Nesterov's method, and Newton-Krylov methods. Finally, we discuss some open issues that should be addressed in future research. 2021-09-20T18:21:12Z 2021-09-20T18:21:12Z 2019-02-13T16:11:52Z Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 https://hdl.handle.net/1721.1/132158 Nguyen, N. C., P. Fernandez, R. M. Freund, and J. Peraire. “Accelerated Residual Methods for the Iterative Solution of Systems of Equations.” SIAM Journal on Scientific Computing 40, no. 5 (January 2018): A3157–A3179. doi:10.1137/17m1141369. http://dx.doi.org/10.1137/17M1141369 SIAM Journal on Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf SIAM |
| spellingShingle | Nguyen, N. C. Fernandez, P. Freund, R. M. Peraire, J. Accelerated Residual Methods for the Iterative Solution of Systems of Equations |
| title | Accelerated Residual Methods for the Iterative Solution of Systems of Equations |
| title_full | Accelerated Residual Methods for the Iterative Solution of Systems of Equations |
| title_fullStr | Accelerated Residual Methods for the Iterative Solution of Systems of Equations |
| title_full_unstemmed | Accelerated Residual Methods for the Iterative Solution of Systems of Equations |
| title_short | Accelerated Residual Methods for the Iterative Solution of Systems of Equations |
| title_sort | accelerated residual methods for the iterative solution of systems of equations |
| url | https://hdl.handle.net/1721.1/132158 |
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