@Article{cmes.2013.092.241,
AUTHOR = {Chein-Shan Liu},
TITLE = {An Optimal Preconditioner with an Alternate Relaxation Parameter Used to Solve Ill-Posed Linear Problems},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {92},
YEAR = {2013},
NUMBER = {3},
PAGES = {241--269},
URL = {http://www.techscience.com/CMES/v92n3/26940},
ISSN = {1526-1506},
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 <i>it can pull back the iterative orbit to the</i> <b>fast manifold</b>. 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.},
DOI = {10.32604/cmes.2013.092.241}
}