Spectral Analysis of Large Finite Element Problems by Optimization Methods
Recently an efficient method for the solution of the partial symmetric eigenproblem (DACG, deflated-accelerated conjugate gradient) was developed, based on the conjugate gradient (CG) minimization of successive Rayleigh quotients over deflated subspaces of decreasing size. In this article four diffe...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Wiley
1994-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.3233/SAV-1994-1603 |
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| _version_ | 1832554826710384640 |
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| author | Luca Bergamaschi Giuseppe Gambolati Giorgio Pini |
| author_facet | Luca Bergamaschi Giuseppe Gambolati Giorgio Pini |
| author_sort | Luca Bergamaschi |
| collection | DOAJ |
| description | Recently an efficient method for the solution of the partial symmetric eigenproblem (DACG, deflated-accelerated conjugate gradient) was developed, based on the conjugate gradient (CG) minimization of successive Rayleigh quotients over deflated subspaces of decreasing size. In this article four different choices of the coefficient βk required at each DACG iteration for the computation of the new search direction Pk are discussed. The “optimal” choice is the one that yields the same asymptotic convergence rate as the CG scheme applied to the solution of linear systems. Numerical results point out that the optimal βk leads to a very cost effective algorithm in terms of CPU time in all the sample problems presented. Various preconditioners are also analyzed. It is found that DACG using the optimal βk and (LLT)−1 as a preconditioner, L being the incomplete Cholesky factor of A, proves a very promising method for the partial eigensolution. It appears to be superior to the Lanczos method in the evaluation of the 40 leftmost eigenpairs of five finite element problems, and particularly for the largest problem, with size equal to 4560, for which the speed gain turns out to fall between 2.5 and 6.0, depending on the eigenpair level. |
| format | Article |
| id | doaj-art-2ddf3ba261714ab0b0e6a6080bd3f3a6 |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 1994-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-2ddf3ba261714ab0b0e6a6080bd3f3a62025-02-03T05:50:19ZengWileyShock and Vibration1070-96221875-92031994-01-011652954010.3233/SAV-1994-1603Spectral Analysis of Large Finite Element Problems by Optimization MethodsLuca Bergamaschi0Giuseppe Gambolati1Giorgio Pini2Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, ItalyDipartimento di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, ItalyDipartimento di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, ItalyRecently an efficient method for the solution of the partial symmetric eigenproblem (DACG, deflated-accelerated conjugate gradient) was developed, based on the conjugate gradient (CG) minimization of successive Rayleigh quotients over deflated subspaces of decreasing size. In this article four different choices of the coefficient βk required at each DACG iteration for the computation of the new search direction Pk are discussed. The “optimal” choice is the one that yields the same asymptotic convergence rate as the CG scheme applied to the solution of linear systems. Numerical results point out that the optimal βk leads to a very cost effective algorithm in terms of CPU time in all the sample problems presented. Various preconditioners are also analyzed. It is found that DACG using the optimal βk and (LLT)−1 as a preconditioner, L being the incomplete Cholesky factor of A, proves a very promising method for the partial eigensolution. It appears to be superior to the Lanczos method in the evaluation of the 40 leftmost eigenpairs of five finite element problems, and particularly for the largest problem, with size equal to 4560, for which the speed gain turns out to fall between 2.5 and 6.0, depending on the eigenpair level.http://dx.doi.org/10.3233/SAV-1994-1603 |
| spellingShingle | Luca Bergamaschi Giuseppe Gambolati Giorgio Pini Spectral Analysis of Large Finite Element Problems by Optimization Methods Shock and Vibration |
| title | Spectral Analysis of Large Finite Element Problems by Optimization Methods |
| title_full | Spectral Analysis of Large Finite Element Problems by Optimization Methods |
| title_fullStr | Spectral Analysis of Large Finite Element Problems by Optimization Methods |
| title_full_unstemmed | Spectral Analysis of Large Finite Element Problems by Optimization Methods |
| title_short | Spectral Analysis of Large Finite Element Problems by Optimization Methods |
| title_sort | spectral analysis of large finite element problems by optimization methods |
| url | http://dx.doi.org/10.3233/SAV-1994-1603 |
| work_keys_str_mv | AT lucabergamaschi spectralanalysisoflargefiniteelementproblemsbyoptimizationmethods AT giuseppegambolati spectralanalysisoflargefiniteelementproblemsbyoptimizationmethods AT giorgiopini spectralanalysisoflargefiniteelementproblemsbyoptimizationmethods |