Iterative Solution Of Linear Systems In Saddle Point Form

Within plenty of application backgrounds,linear systems of saddle point form arise widely from numerical solution procedure of related problems in scientific com-puting and engineering applications.As a class of special block linear systems,such linear systems have large and sparse structural features,which are suitable for itera-tion solution methods.Due to the indefiniteness and poor spectral properties of their coefficient matrices,it is difficult to solve them efficiently.The primary focus of this thesis is the numerical solution of saddle point linear systems arising from problems such as Navier-Stokes equations,equivalent transformation of complex linear system and PDE-constrained optimization.Based on the specific problems,we construct efficient iteration methods and robust preconditioners.Iterative solution methods for nonsingular saddle point problems are studied.A class of SIMPLE-like?SL?preconditioners are constructed for solving saddle point problems arising form the Navier-Stokes equations.Compared with some similar preconditioners,the SL preconditioners can result in much better convergence and spectral distribution properties.Two modified HSS?MHSS?preconditioners are con-structed for solving generalized saddle point problems.The corresponding MHSS iteration methods are convergent unconditionally.Moreover,the MHSS precondi-tioned matrices display much better spectral properties than the HSS and some other preconditioners.By technically using Sherman-Morrison-Woodbury formula,a fur-ther improved convergence result of the NIU algorithm for generalized saddle point problems is presented.Some adaptive versions of the NIU algorithm with variable iteration parameters are constructed,which exhibit much better numerical perfor-mance than the original NIU algorithms to solve generalized saddle point problems.Combining the classical matrix splitting of sub-matrix with Uzawa-type methods,we construct two class of iteration methods which are suitable for solving saddle point problems of coefficient matrix with symmetric positive definite?1,1?block and non-symmetric positive definite but nonsymmetric dominate?1,1?block,respectively.The proposed methods improve the solving efficiency of the similar methods.Iterative solution methods for saddle point problems of coefficient matrix with square blocks are studied.The SSOR method is analyzed for solving block two-by-two linear systems arising from the equivalent transformation of the complex linear systems and the corresponding optimal iteration parameters are presented.An accel-erated variant of the SSOR method,called ASSOR,is given,which has much better convergence properties.Moreover,a practical way to choose iteration parameters for the ASSOR method is proposed.A robust structured preconditioner is constructed for solving block two-by-two linear systems arising from optimal control problems con-strained by the time-harmonic parabolic equations.The algorithmic implementation of the preconditioner is simple and the eigenvalues of the corresponding preconditioned matrix are located in the interval[₂¹,1].When using to accelerate Krylov subspace methods,it outperforms some efficient known preconditioners.Iterative solution methods for singular saddle point problems are studied.Based on suitable parameterizing and preconditioning techniques,the DPSS iteration method is generalized to a PDPSS iteration method which is semi-convergent unconditional-ly for solving singular saddle point problems.Moreover,spectral properties of the corresponding DPSS preconditioned matrix are studied.A relaxed variant,called R-PDPSS,is given,which results in much better convergence and spectral properties.The PDPSS and RPDPSS preconditioned GMRES method perform much better than the HSS preconditioned GMRES method for solving singular saddle point problems.By using a singular preconditioning technique,the PU splitting resulting from the PU iteration method is generalized to a GPU splitting which can result in proper splitting.Based on this splitting,a GPU iteration method is constructed for solving singular saddle point problems.The GPU iteration method and the GPU precondi-tioned GMRES method can both converge to the least squares solution.The obtained convergence results improve those for the PU method for solving singular saddle-point problems.