Open Access

ARTICLE

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

Chein-Shan Liu¹

1 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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Citations