We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new...We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new starting vector,such that the occurrence of stagnation be mitigated or the convergence be accelerated.The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector,which is discussed in detail.Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.展开更多
The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the e...The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m + 1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modi- fied m-step Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m + 1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m+ 1)-step restarted ones.展开更多
The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz ...The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m+1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small 'dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m+1)-step iterative one.展开更多
文摘We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new starting vector,such that the occurrence of stagnation be mitigated or the convergence be accelerated.The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector,which is discussed in detail.Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.
基金the China State Key Project for Basic Researchesthe National Natural Science Foundation of ChinaThe Research Fund for th
文摘The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m + 1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modi- fied m-step Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m + 1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m+ 1)-step restarted ones.
文摘The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m+1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small 'dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m+1)-step iterative one.
