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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:

Computer Modeling in Engineering & Sciences 2013, 92(3), 241-269.


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.


Linear inverse problem, Ill-posed linear problem, Generalized relaxed steepest descent method (GRSDM), Optimal GRSDM (OGRSDM), Relaxation parameter, Optimal GRSDM with an Alternate Relaxation Parameter (OGRSDM- γ), Fast manifold

Cite This Article

Liu, C. (2013). An Optimal Preconditioner with an Alternate Relaxation Parameter Used to Solve Ill-Posed Linear Problems. CMES-Computer Modeling in Engineering & Sciences, 92(3), 241–269.


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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