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

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 alpha and gamma to be determined. When the weighting parameter alpha is optimized by minimizing a properly defined objective function, the relaxation parameter gamma can be derived to accelerate the convergence speed under a switching criterion. When the switch is turned-on, by using the derived value of gamma 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 gamma 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.