The Research Of Numerical Solution For Large Sparse Algebraic Systems

Large-scale sparse algebraic systems arise in a wide variety of applications through-out computational science and engineering fields, such as computational ?uid dynam-ics,constrained optimization,computational electromagnetics,mixed finite elementapproximations of PDEs,nonlinear optimization,transport theory and so on. Hence, ithas important theoretic and practical significance to study on the numerical methods forsolving this type of algebraic systems. Many large-scale sparse algebraic systems arisingfrom practical problems often have special structures. Therefore, the experts and schol-ars from home and abroad have been attracted on the research of numerical methods forsolving large sparse algebraic systems with special structure. This thesis represents deepand systematic research on the numerical methods for solving large-scale sparse algebraicsystems which have special structures, especially on the iterative methods and precondi-tioning techniques for linear saddle point problems, iterative methods for nonlinear saddleproblems and algebraic Riccati equation. This thesis is divided into four parts, includingsix chapters.Uzawa type iterative methods are studied for solving linear saddle point problems.First, a modified nonlinear Uzawa iterative method is proposed and convergence of themodified method is discussed, and also theoretical and numerical comparison are given.Numerical experiments are provided to show the efficiency of the modified method. Sec-ond, a generalized modified local Hermitian and Skew-Hermitian splitting method is pre-sented for the generalized saddle point problems with nonzero (2, 2) block and conver-gence of the algorithm is discussed. Last, the inexact Uzawa method, the GSSOR methodand the MLHSS method are applied to solve singular saddle point systems and the semi-convergence analysis of these three methods are provided, respectively.Preconditioning techniques are investigated for nonsymmetric saddle point prob-lems. Firstly, inexact block triangular preconditioners which contain a parameter arepresented, and eigenpair properties of the preconditioned matrix are analyzed. Further-more, eigenvalue perturbation analysis of the preconditioned matrix is shown. Secondly,on the basis of partial HS and PS splitting of coefficient matrix, PHSS preconditioner andPPSS preconditioner are presented and analyzed for nonsymmetric saddle point prob- lems, and also the spectral properties of the preconditioned matrix are studied in detail.Moreover, it can be concluded that the eigenvalue of the preconditioned matrix will gatherinto two clusters: one is near (0, 0) and the other is near (2, 0) when the iteration param-eter becomes small enough. In addition, the theoretical results and effectiveness of thetwo preconditioners are proved by a lot of numerical experiments. Finally, SIMPLE pre-conditioner is given. Using eigenvalue theory, the relation of two different expression ofspectrum of preconditioned matrix is investigated.Iterative methods are considered for solving nonlinear saddle point problems. Basedon the iterative methods for solving linear saddle point problems, several iterative methodsare proposed for solving nonlinear saddle point problems, and their convergence analysisare given, respectively. Numerical experiments are provided to reveal the performance ofthose proposed iterative methods.Iterative methods are investigated for the algebraic Riccati equation. In fact, alge-braic Riccati equation arising in transport theory can be rewritten as a vector equation.Firstly, based on the relaxation technique and Newton method, a relaxed Newton-likemethod containing a relaxation parameter is proposed for solving the vector equation,and some convergence results are given. Secondly, using quasi-Newton idea and combin-ing the existing Newton type methods, two modified Newton type methods are presentedfor solving vector equation and some convergence results are obtained. The results of ex-periments show that our methods can effectively improve the convergence of the existingalgorithms.