Study On Global Krylov Subspace Method And Its Application

Global Krylov subspace methods for solving large-scale sparse systems of linear equations with multiple right-hand sides are investigated in this thesis. Since solution of such linear systems with multiple right-hand sides are required in many practical fields, such as solution of partial differential equations, fluid dynamics, circuit simulation, electromagnetic computation, linear control theory, it becomes significantly paramount to establish efficient and stable numerical methods to solve linear equations with multiple right-hand sides.Firstly, Krylov subspace methods based on Arnoldi process, including the full orthogonal method(FOM) and the generalized minimal residual method(GMRES), are illustrated thoroughly and systematically with numerical experiments showing that GMRES is superior to FOM. And then, following the deriving way of these two methods, we analyze Krylov subspace methods based on global Arnoldi process, which are the global full orthogonal method(GL-FOM) and the global generalized minimal residual method(GL-GMRES). Consequently, the representations for iterates and the corresponding residual matrix norm are given with Givens rotation, which keep similar structures with the counterparts of FOM and GMRES. It is shown by numerical experiments that GL-GMRES outperforms GL-FOM.Based on the global Arnoldi process and weighted technique, we propose a restated weighted global generalized minimum residual method to solve systems of linear equations with multiple right-hand sides. Two theorems and two propositions are given in order to: 1. Ensure the D-inner product and its associated D-norm are well-defined, and represent the D-inner product with vectorized technique and Kronecker product; 2. Make iterates and the corresponding residual matrix norm obtained by the proposed restated weighted global generalized minimum residual method be further computed; 3. Ensure the iterates and the corresponding residual matrix norm obtained by the proposed restated weighted global generalized minimum residual method have agreement in structure with those counterparts of the global generalized minimum residual method and the generalized minimum residual method; 4. Assure a set of D-orthogonal basis is established for the block Krylov subspace by the weighted global Arnoldi process; 5. Intend to illustrate the scaling-invariant property of the presented restated weighted global generalized minimum residual method. In the end, numerical experiments in terms of convergence curves, iteration counts, CPU consuming time verify the efficiency of the proposed method.