Preconditioning Techniques For Linear Equations And Their Applications On Two-Dimensional And Three-Temperature Problem And Implementation

The solution of linear equations is a wide problem arising in many research areas. It is very important in theories and applications. For large scale linear equations, it is of order over several hundreds of thousands, even over one million. Normally, these large scale linear systems can only be efficiently solved with preconditioning techniques. Currently, preconditioning techniques is much less than satisfactory. The aim to enhance the speed of solution for linear equations is shared by both theory researchers and engineers. As an important part in solution of linear equations, the importance of preconditioning techniques will increase in the future.In this thesis, preconditioning techniques for some kinds of special matrices and problems are studied, and a Matlab software package for developing solution techniques for linear equations are designed and implemented. The main results are as follows:1. Preconditioning techniques for M-matrices and nonnegative matrices associated with Jacobi iterative method and Gauss-Seidel iterative methods are studied. Firstly, one I + S-type preconditoning technique is modified. Then, optimization models are proposed for selecting optimal parameters for I + S-type preconditioners. In addition, with LU methods, two kinds of Gauss-type preconditioning techniques including relative algorithms and algorithms comparison are put forward. By theory analysis, the reasonability and reliability of these preconditioning techniques is proved. Their efficiency and performance is further verified by numerical experiments.2. Through analysis of "good" preconditioner associated with sparse approximate inverse techniques, I + H_c-type preconditioners are put forward. With different preconditioning straties and considerations, several optimization models searching for parameters in preconditioners are presented, respectively. The remarkable merits of these techniques is of low computation cost and high parallelism. Numerical experiments shows the performance of these preconditioners with Krylov subspace iterative methods for M-matrices.3. Preconditioers for a kind of block-tirangular matrices arising from some two-dimensional PDE problem are studied. Firstly, by observation of sparsity pattern of the inverses of some corresponding problem matrices, we present block-seven diagonal preconditioners. Then, by a popular implementation of sparse approximate inverse tech- niques with Frobenius normal minimization: Minimal Residual Algorithm (MR), one modified MR algorithm are put to construct block-seven diagonal matrices. Modified MR algorithm (MMR) has more lower computation cost that original MR algorithm. Moreover, MMR algorithm has the nearly same accelerating rate than MR algorithm. All these preconditioning effectiveness are tested with Krylov subspace iterative methods. Numerical experiments further show that MMR algorithm is also adapt to some non-symmetric cases.4. In this thesis, preconditioning techniques for two-dimensional and three-temperature problem are studied from two ways. (1) Reordering methods of block-partioned system with red-black ordering are studied. With the help of one good reordering method (i.e. hyperplane reordering), a new class of ordering method named alterative-hyperplane (A-hyperplane). Numerical experiments shows, compared with RB,hyperplane preconditioning reordering method, A-hyperplane method largely enhance the accerlating rate when it is applied with ILU(k) preconditioning mehtod. (2) For the global preconditioning of two-dimension and three-temperature problem, a class of Domain Decomposition Method (DDM) are applied. Numerical experiments shows Domain Decomposition Method make the iteration number increase slowly as enlarging the domain number and increasing parallelism. All these work prepare well for the parallel research for two-dimension and three-temperature problem.5. A Matlab software package (LINSOLKIT) is developed. In order to enhance the developing efficiency of solution methods for linear equations, especially preconditioning techniques, with power functions of Matlab and analysis of current popular non-Matlab language packages for linear equations solution, a software package named LINSOLKIT is designed and implemented with Matlab. The interface of this package composes of preconditioning interface,preconditioned iterative method interface,preconditioner system solver interface and reordering performing interface. Because LINSOLKIT is based on Matlab software, the maintenance of LINSOLKIT such as increasing preconditioning techniques or iterative method is rather convenient. Most of numerical experiment in this thesis are performed on the interface of LINSOLKIT. In addition, a group of Fortran programmes based on SparsKit package are developed for testing two-dimensional and three-temperature problem and two-dimensional PDE problem.