Research On GMRES Method And Its Variation Algorithm

The thesis mainly studies Krylov subspace methods for solving large sparse linear systems,especially the GMRES methods which is a classical method for solving large nonsymmetric linear equations.With the increase of iteration steps,the amount of storage and computational work required also increases,so GMRES(m)algorithm the restarted version of GMRES algorithm is usually used in the practical application,the GMRES(m)algorithm limits the maximum dimension of the subspace to m and restarts after m steps,the restart parameter m is generally chosen small relative to the size of coefficient matrix.The drawback of restarting is that information is discarded at each restart and the orthogonality to previously generated subspaces is not preserved at each restart,so the algorithm loses its global optimality which will slow down the convergence and even stagnation.Therefore,there are many strategies to accelerate the GMRES(m)algorithm convergence,the augmented methods is one of them,it accelerates the convergence of the GMRES(m)algorithm by retaining some of the information that is typically discarded at the time of restart.There are two kind of augmented methods,one is to add approximate eigenvectors corresponding to the smallest eigenvalues in magnitude to the subspace when restarting,that can effectively deflate corresponding eigenvalues which slow down the convergence from the spectrum,thus accelerating the convergence of GMRES(m)algorithm.The other is to retain some of correction vectors form the old loop and add them to the new loop's subspace,this method does not require additional calculations and only requires minimal changes to the standard GMRES(m)algorithm,this strategy can prevent the phenomenon that the residual vectors at the end of each restart cycle alternate direction in a cyclic pattern,thus accelerating the convergence of GMRES(m)algorithm.In this thesis,the study based on the Simpler GMRES algorithm which is a efficient implementation of GMRES algorithm.In order to reduce the negative influence of restarting and accelerate the convergence of algorithm,we have analyzed the characteristics and advantages of two kinds of augmented methods by theory analysis and numerical experiment?Based on the two types of the augmented method,we propose a new Krylov subspace algorithm which adopts a double augment strategy to extend the approximate subspace of SGMRES algorithm.The theory analysis and the numerical experiments have proved the high efficiency and applicability of the new algorithm.