In this paper,the efficient preconditioned modified Hermitian and skew-Hermitian splitting(PMHSS)iteration method is further explored and it is extended to solve more general block two-by-two linear systems with diffe...In this paper,the efficient preconditioned modified Hermitian and skew-Hermitian splitting(PMHSS)iteration method is further explored and it is extended to solve more general block two-by-two linear systems with different and nonsymmetric off-diagonal blocks.With the aid of the singular value decomposition technique,the detailed analysis of the algebraic and convergence properties of the PMHSS iteration method demonstrates that it is still convergent unconditionally as when it is used to solve the well-studied case of block two-by-two linear systems with same and symmetric off-diagonal blocks.Moreover,the PMHSS preconditioned matrix is almost unitary diagonalizable with clustered eigenvalue distributions for this more general case.On account of the favorable spectral properties of the PMHSS preconditioned matrix,a parameter free Chebyshev accelerated PMHSS(CAPMHSS)method is established to further improve its convergence rate.Numerical experiments about Kroncker structured block two-by-two linear systems arising from a time-dependent PDE-constrained optimal control problem demonstrate quite satisfactory and competitive performance of the CAPMHSS method compared with some existing preconditioned Krylov subspace methods.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11801242,11771193,and 11901267)the Fundamental Research Funds for the Central Universities(No.lzujbky-2022-05)the Natural Science Foundation of Gansu Province of China(Grant No.23JRRA1104).
文摘In this paper,the efficient preconditioned modified Hermitian and skew-Hermitian splitting(PMHSS)iteration method is further explored and it is extended to solve more general block two-by-two linear systems with different and nonsymmetric off-diagonal blocks.With the aid of the singular value decomposition technique,the detailed analysis of the algebraic and convergence properties of the PMHSS iteration method demonstrates that it is still convergent unconditionally as when it is used to solve the well-studied case of block two-by-two linear systems with same and symmetric off-diagonal blocks.Moreover,the PMHSS preconditioned matrix is almost unitary diagonalizable with clustered eigenvalue distributions for this more general case.On account of the favorable spectral properties of the PMHSS preconditioned matrix,a parameter free Chebyshev accelerated PMHSS(CAPMHSS)method is established to further improve its convergence rate.Numerical experiments about Kroncker structured block two-by-two linear systems arising from a time-dependent PDE-constrained optimal control problem demonstrate quite satisfactory and competitive performance of the CAPMHSS method compared with some existing preconditioned Krylov subspace methods.
