GMRES在特征值聚集时的超线性收敛及其在最小二乘问题中的应用
Superlinear Convergence of GMRES for clustered eigenvalues and its application to least squares problems
AI总结：
本文分析GMRES在系数矩阵特征值聚集时的超线性收敛，利用范德蒙德矩阵推导残差上界，并应用于解释最小二乘问题中内迭代预条件GMRES的收敛。
摘要：
本文旨在理解当系数矩阵具有聚集特征值时GMRES方法的超线性收敛行为。为了理解这一现象，我们利用由系数矩阵的特征值定义的范德蒙德矩阵来分析收敛性。尽管仅凭特征值无法解释收敛性，但它们结合右端项向量和系数矩阵的特征向量，可以为残差提供上界。我们证明，当系数矩阵可对角化时，如果其特征值聚集，且将右端项向量分解为特征向量分量所得矩阵的范数不是很大，则收敛曲线的上界表现出超线性收敛。我们将该分析应用于解释最小二乘问题中内迭代预条件GMRES的收敛性。
英文摘要：
The objective of this paper is to understand the superlinear convergence behavior of the GMRES method when the coefficient matrix has clustered eigenvalues. In order to understand the phenomenon, we analyze the convergence using the Vandermonde matrix which is defined using the eigenvalues of the coefficient matrix. Although eigenvalues alone cannot explain the convergence, they may provide an upper bound of the residual, together with the right hand side vector and the eigenvectors of the coefficient matrix. We show that when the coefficient matrix is diagonalizable, if the eigenvalues of the coefficient matrix are clustered, the upper bound of the convergence curve shows superlinear convergence, when the norm of the matrix obtained by decomposing the right hand side vector into the eigenvector components is not so large. We apply the analysis to explain the convergence of inner-iteration preconditioned GMRES for least squares problems.
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