Preconditioned Iteration Methods For Two Kinds Of Linear Saddle Point Problems And Low-rank Parallelization Methods For Two Kinds Of Linear Matrix Equations

This dissertation is mainly focused on the numerical solutions for two different kinds of block three-by-three linear systems and linear matrix equations.Due to their widely application settings,it is of great theoretical value and practical meaning to construct fast and efficient numerical methods for solving these problems.In Chapter 2,for a three-by-three block linear system with saddle-point structure,we develop a parameterized relaxed dimensional factorization（PRDF）preconditioner to improve the flexibility and the preconditioning efficiency of the relaxed dimen-sional factorization（RDF）preconditioner.Similar to the RDF preconditioner,the preconditioning effect of the PRDF preconditioner also depends on the choice of its parameter-values.In order to achieve the best efficiency of PRDF preconditioner,we further derive a rapid and effective method to compute the optimal values of its pa-rameters.Numerical examples arisen from the Navier-Stokes equations and the partial differential equation constraint optimization problems are employed to illustrate the robustness and the efficiency of the PRDF preconditioner when its parameter-values are computed by the proposed method.In Chapter 3,for a class of double saddle point linear system arising from liq-uid crystal directors modeling,to improve the performance of block triangular（BT）preconditioner,we develop a two-parameter BT（TPBT）preconditioner by analyz-ing the the properties of the BT preconditioner.Theoretical analysis shows that all the eigenvalues of the TPBT preconditioned coefficient matrix are real and locate in an interval（0,2）no matter which value the spectral radius of matrix^-1-1is chosen.Moreover,an upper bound of the degree of the minimal polynomial of the TPBT preconditioned coefficient matrix and the dimension of its corresponding Krylov subspace is also obtained.Inasmuch as the efficiency of the TPBT precondi-tioner depends on the values of its parameters,we further derive a class of fast and effective formulas to compute the quasi-optimal values of the parameters involved in the TPBT preconditioner.Finally,numerical results are reported to illustrate the feasibility and the efficiency of the TPBT preconditioner.In chapter 4,for solving two kinds of linear matrix equations whose coefficients are banded and Topelitz,namely Sylvester and generalized Sylvester equations,we propose a low-rank parallelization method by combining Topelitz structures with cyclic matrices.Meanwhile,a rational variant of extended Krylov subspace method（REKSM）is proposed to solve the corresponding low-rank correction equation,and a practical method is also provided to compute the optimal value of shift parameters involved in the REKSM via analyzing the solution properties of correction equation.In addition,for a class of generalized Sylvester equations arising from the linear wave equations,by combining one-sided extended Krylov subspace method（EKSM）with Sherman-Morrison-Woodbury（SMW）formula,we propose a new algorithm,named EKSM-SMW,to solve the corresponding low-rank correction equation.Finally,nu-merical examples are given to verify the effectiveness of the proposed low-rank paral-lelization method for solving the above linear matrix equations.In chapter 5,we concerned the linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations（PDEs）.Such matrix equations have been considered,for example,in the context of parallel-in-time integration leading to a class of algorithms called Para Diag.We develop two novel approaches for the numerical solution of such equations.Our first approach is based on the observation that the modification of these equations performed by Para-Diag in order to solve them in parallel has low rank.Building upon previous work on low-rank updates of matrix equations,this allows us to make use of tensorized Krylov subspace methods to account for the modification.Our second approach is based on interpolating the solution of matrix equation from the solutions of several modi-fications.Both approaches avoid the use of iterative refinement needed by Para Diag and related space-time approaches in order to attain good accuracy.In turn,our new approaches have the potential to outperform,sometimes significantly,existing approaches.This potential is demonstrated for several different types of PDEs.