Abstract
Two new preconditioners, which can be viewed as variants of the deteriorated positive definite and skew-Hermitian splitting preconditioner, are proposed for solving saddle point problems. The corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks. The choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated Cayley transform, which result in faster convergence rate and more clustered spectrum. Compared with some preconditioners of similar structures, the new preconditioners have better convergence properties and spectrum distributions. In addition, more practical preconditioning variants of the new preconditioners are considered. Numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate GMRES.
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