Semi-regularized Hermitian and Skew-Hermitian Splitting Preconditioning for Saddle-Point Linear Systems

In this paper, a two-step semi-regularized Hermitian and skew-Hermitian splitting (SHSS) iteration method is constructed by introducing a regularization matrix in the (1,1)-block of the first iteration step, to solve the saddle-point linear system. By carefully selecting two different regularization matrices, two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods. Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0. The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic. In addition, the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods, and their convergence rates are independent of the discrete mesh size.