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An Optimal Preconditioner with an Alternate Relaxation Parameter Used to Solve Ill-Posed Linear Problems

Chein-Shan Liu1

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw

Computer Modeling in Engineering & Sciences 2013, 92(3), 241-269. https://doi.org/10.32604/cmes.2013.092.241

Abstract

In order to solve an ill-posed linear problem, we propose an innovative Jacobian type iterative method by presetting a conditioner before the steepest descent direction. The preconditioner is derived from an invariant manifold approach, which includes two parameters α and γ to be determined. When the weighting parameter α is optimized by minimizing a properly defined objective function, the relaxation parameter γ can be derived to accelerate the convergence speed under a switching criterion. When the switch is turned-on, by using the derived value of γ it can pull back the iterative orbit to the fast manifold. It is the first time that we have a formula for the relaxation parameter, by recognizing that γ is specified case by case, previously. The presently developed optimal and generalized steepest descent method with an alternate value of the relaxation parameter is able to overcome the ill-posedness of linear inverse problem, and provides a rather accurate numerical solution.

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APA Style
Liu, C. (2013). An optimal preconditioner with an alternate relaxation parameter used to solve ill-posed linear problems. Computer Modeling in Engineering & Sciences, 92(3), 241-269. https://doi.org/10.32604/cmes.2013.092.241
Vancouver Style
Liu C. An optimal preconditioner with an alternate relaxation parameter used to solve ill-posed linear problems. Comput Model Eng Sci. 2013;92(3):241-269 https://doi.org/10.32604/cmes.2013.092.241
IEEE Style
C. Liu, "An Optimal Preconditioner with an Alternate Relaxation Parameter Used to Solve Ill-Posed Linear Problems," Comput. Model. Eng. Sci., vol. 92, no. 3, pp. 241-269. 2013. https://doi.org/10.32604/cmes.2013.092.241

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cc Copyright © 2013 The Author(s). Published by Tech Science Press.
This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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