Efficient Iterative Methods And Preconditioning Techniques For Three Classes Of Structured Discretized Systems

Numerous large-scale scientific computing problems,such as PDE-constraint dis-tributed control problems,Navier-Stokes equations,least-square problems,fractional differential equations,etc.,will lead to special structured linear systems or matrix equations after being discretized.In this thesis,we consider some new iterative meth-ods and preconditioning techniques for three classes of structured discretized systems,i.e.,the saddle-point-structured linear systems,the Toeplitz-like-structured linear sys-tems and the Sylvester-structured matrix equations.In Chapter 1,we describe the background,the research significance and the research status of the three classes of structured discretized systems in detail.The main work and the main innovations of the researches in this thesis are proposed at the end of Chapter 1.In the first part of Chapter 2,we concentrate on the efficient solvers and pre-conditioners for saddle point structured systems arising from the discretization of the PDE-optimization problem.Firstly,by exploiting the special block and sparse structure of the coefficient matrix of the linear system arising from the discretiza-tion of the heat equations constraint,a natural order-reduction is performed.Then a class of new preconditioning techniques are proposed for the reduced-order linear systems.For the saddle point structured systems arising from the discretized op-timization problem with the unsteady Burgers equation constraint,we propose an efficient method by introducing a new preconditioning technique with an approxima-tion for the Schur complement in nonstandard inner product.Further,we establish a relaxed splitting iterative method and a bi-parameter relaxed splitting preconditioner for the discretized system arising from the optimized velocity tracking problem with the Stokes equation constraint.The spectral properties for the preconditioned ma-trices corresponding to the three preconditioners are analyzed.In the second part of Chapter 2,we propose three classes of preconditioners for the two-by-two block linear systems,i.e.,a generalized additive block diagonal preconditioner,a generalized shift splitting preconditioner and a class of parameterized rotated block preconditioners.The convergence conditions for the new iterative method are given.The spectral properties for the preconditioned matrices corresponding to the new preconditioners are analyzed.The numerical experiments are used to test the validity of the new preconditioners and the new iterative methods in this chapter.In Chapter 3,we focus on the Toeplitz-like discretized systems arising from space-fractional diffusion equations with initial-boundary value.Based on the incomplete circulant and skew-circulant splitting of the system matrix,a three-step alternating iterative method is presented.Besides,the convergence conditions for the new itera-tive method are studied in detail.The effectiveness of the new method is verified by the experimental examples.The main advantage of the new method is that one only have to carry out the iterations by two fast Fourier transforms and a product of a diagonal matrix and a vector at each iterative step.Therefore,the workload for this new method is light.In Chapter 4,firstly,a preconditioned asymmetric Hermitian and skew-Hermitian(PAHSS)iterative method and an inexact PAHSS(IPAHSS)iterative method are es-tablished for solving large sparse Sylvester equations.The convergence condition and the optimal iterative parameters are derived.Then a preconditioned backward substituting method by using a banded preconditioner is introduced for solving the Sylvester equation from the discretization of the time-space fractional advection-diffusion equation.The theoretical analysis about the new preconditioner is given.Furthermore,based on the single-step HSS(SHSS)iterative method and Krylov-Plus-Inverted-Krylov(KPIK)subspace iterative method,we propose an SHSS-KPIK itera-tive method for solving the low-rank Sylvester equation arising from the discretization of two-dimensional time-periodic space fractional diffusion problem.The theoretical properties of the SHSS-KPIK method are analyzed in detail.The validity of all the theoretical results and new methods is verified by the experimental examples.