We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval |$(0, \, 2)$| when the iteration parameter is close to |$0$| and, furthermore, they can be clustered near |$0$| and |$2$| when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

中文翻译：

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稳定鞍点问题的正则HSS迭代方法

我们将标准鞍点问题的正则化Hermitian和skew-Hermitian分裂（RHSS）迭代方法扩展到稳定的鞍点问题，并为所得方法建立了相应的无条件收敛理论。除了用作平稳迭代求解器外，此类RHSS方法还可以用作Krylov子空间方法的前置条件。结果表明，相应预处理矩阵的特征值聚集在| $（0，\，2）$ |区间中的少数点上 当迭代参数接近| $ 0 $ |时 而且，它们可以聚集在| $ 0 $ |附近 和| $ 2 $ |当正则矩阵适当地选择。最优边界控制问题和Cahn-Hilliard图像修复问题的有限元离散化以及非线性图像恢复问题的高斯-牛顿线性化引起的稳定鞍点问题的数值结果表明，RHSS迭代当方法用作线性迭代求解器或用作Krylov子空间方法的矩阵拆分预处理器时，该方法在迭代计数和计算时间方面显着优于Hermitian和skew-Hermitian拆分迭代方法，并且当使用不精确的建议的RHSS预处理器。