The Study Of Preconditioners For Saddle Point Problems

Matrix computation plays an important role in scientific and engineering computation, because many problems require numerical results by matrix computation. Saddle point problems or generalized saddle point problems are often solved in many fields, such as fluid dynamics, optimization, economics, finance, circuit network, electromagnetics, mixed finite element approximation of elliptic partial differential equations, and so on.The coefficient matrix is often large and sparse in solving the saddle point problems. Hence, the iterative methods are more prefer. However, the direct method is an integral method, such as in the optimization problem. Moreover, the direct methods are often used in solving some sub-problems, such as preconditioned equations. As for saddle point problems, the Uzawa method is the most classical approach in the iterative methods, but it has the disadvantage that every step of the iterative algorithm involves solving a linear equations. Krylov subspace methods are also a good way, such as MINRES, GMRES, QMR, GCG, CR, and so on. However, if we cannot select the proper preconditioners, the convergence speed is also very slow. Usually in the practical problems, the saddle point matrix has a certain structure. For example, the block matrix A is often a block diagonal and each also has a special block diagonal structure. So we should choose the saddle point matrix for the different structures of the different iterative algorithm, making sure that the solution process has certain advantages in computation and storage.In this paper, we study the preconditioning methods for solving saddle point problems. There are a lot of pre-treatment methods for solving saddle point problems now. By combining with the existing method, some improvements and generalizations have been attained. Firstly, we consider solving a discrete PDE constrained optimizatior. problems. The coefficient matrix of the discrete saddle point problems is a sparse matrix, and its structure is a block symmetric. According to the characteristics of the structure of the coefficient matrix, we present an optimal constraint preconditioners for solving this type of saddle point problems, which can effectively reduce the amount of computation. Then by using of the idea of HSS iteration method, we extended the single-parameter dimension splitting method of preconditioner to a two-parameter form, and obtain a two-parameter splitting preconditioner.