Research On Accelerating A Class Of Krylov Subspace Methods

Within some special cases,the residual descent curves of Krylov subspace methods may appear some unstable phenomena,such as stagnation,irregular oscillation,which delay the convergence rate of Krylov subspace methods seriously.This thesis aims at improving above-mentioned unstable issues and constructing Krylov subspace methods of high performance for solving large sparse linear systems with multiple right-hand sides.Consequently,we present two kinds of numerical methods for symmetric positive definite linear system solvers.That are the adaptive restart procedures for the breakdown-free block conjugate gradient(AR-BFBCG)method as well as the projected variant of deflated block conjugate gradient(PD-BCG)method.Detailed contents are as follows.Proposed the AR-BFBCG method.Based on Powell's restart proposed by Powell in 1977 as well as its improved version done by Dai et al.in 2004,which both designed new restart techniques to accelerate the convergence rate of the conjugate gradient(CG)method,we expand such kind of restart techniques to the BCG method to solve the linear systems with multiple right-hand sides.Meanwhile,we make use of the main idea of the breakdown-free block conjugate gradient(BFBCG)method to avoid the breakdown issue cased by rank deficiency.Consequently,we present the AR-BFBCG method for solving the large sparse symmetric positive definite linear system with multiple right-hand sides,which can effectively handle the breakdown problem cased by rank deficiency,and potentially accelerate the convergence rate of BFBCG by escaping the unstable phenomena mentioned above and reducing the abandoned information at restart as demonstrated by the numerical experiments.Moreover,the advantages of AR-BFBCG are more obvious especially when solving the ill-conditioned system or system with rank deficiency righthand sides.Devised the PD-BCG method.In 2011,Chen derived a deflated version of the block conjugate gradient(D-BCG)algorithm to accelerate the convergence rate of BCG by deflating the extreme eigenvalues(such as the smallest ones)to reduce the condition number of the preconditioned matrix.However,under the framework of finite arithmetic,the orthogonality between the block residual vectors and the deflation subspace is gradually lost along with the process of the underlying algorithm implementation,which usually causes the algorithm to be unstable or possibly have delayed convergence by long-term stagnation and even ruins the convergence completely when such loss of orthogonality is too serious.In order to maintain such orthogonality to keep within a certain level,full reorthogonalization could be employed as a remedy,but the required expense is quite costly.Therefore,we present a new projected variant of the deflated block conjugate gradient(PD-BCG)method to mitigate the loss of this orthogonality,which is helpful to deal with the delay of convergence and thus further achieve the theoretically faster convergence rate of D-BCG.Meanwhile,the proposed PD-BCG method is almost mathematically equal to D-BCG and shown to scarcely have any extra computational cost,and have the same theoretical properties as D-BCG in exact arithmetic.Finally,the effectiveness of PD-BCG is illustrated by the numerical experiments,which shows that the proposed method can be used to control the level of such orthogonality and handle convergence stagnation or even the Non-Convergence problem.