An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems
It is known that the steepest-descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and descent direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm w...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/154358 |
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| author | Chein-Shan Liu |
| author_facet | Chein-Shan Liu |
| author_sort | Chein-Shan Liu |
| collection | DOAJ |
| description | It is known that the steepest-descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and descent direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm with m-vector descent direction in a Krylov subspace is constructed, of which the m optimal weighting parameters are solved in closed-form to accelerate the convergence speed in solving ill-posed linear problems. The optimally generalized steepest-descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems. |
| format | Article |
| id | doaj-art-a56c624d758f484d850e49dfddd3be0d |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-a56c624d758f484d850e49dfddd3be0d2025-02-03T01:26:14ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/154358154358An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear SystemsChein-Shan Liu0Department of Civil Engineering, National Taiwan University, Taipei 106-17, TaiwanIt is known that the steepest-descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and descent direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm with m-vector descent direction in a Krylov subspace is constructed, of which the m optimal weighting parameters are solved in closed-form to accelerate the convergence speed in solving ill-posed linear problems. The optimally generalized steepest-descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems.http://dx.doi.org/10.1155/2013/154358 |
| spellingShingle | Chein-Shan Liu An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems Journal of Applied Mathematics |
| title | An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems |
| title_full | An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems |
| title_fullStr | An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems |
| title_full_unstemmed | An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems |
| title_short | An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems |
| title_sort | optimally generalized steepest descent algorithm for solving ill posed linear systems |
| url | http://dx.doi.org/10.1155/2013/154358 |
| work_keys_str_mv | AT cheinshanliu anoptimallygeneralizedsteepestdescentalgorithmforsolvingillposedlinearsystems AT cheinshanliu optimallygeneralizedsteepestdescentalgorithmforsolvingillposedlinearsystems |