A geometric theory for preconditioned inverse iteration. I : Extrema of the Rayleigh quotient
By Klaus Neymeyr. 2001
78 citations recorded by the source corpus.
Abstract The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix-free” method) derives from the well-known inverse iteration procedure in such a way that the associated system of linear equations is solved approximately by using a (multigrid) preconditioner. A new convergence analysis for preconditioned inverse iteration is presented. The preconditioner is assumed to satisfy some bound for the spectral radius of the error propagation matrix resulting in a simple geometric setup. In this first part the case of poorest convergence depending on the choice of the preconditioner is analyzed. In the second part the dependence on all initial vectors having a fixed Rayleigh quotient is considered. The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigenvalue/vector computations can be done with comparable efficiency as known from multigrid methods for boundary value problems.
Subfields: Atomic and Molecular Physics, and Optics, Computational Mechanics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/ea6c33de9f9d932ff947b51dddca39c48d9dddd4
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In Atomic and Molecular Physics, and Optics
In Computational Mechanics
In Computational Theory and Mathematics
Citations in the corpus
- Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format (2013, methodology): “(6) This method is know for a long time and was, among others, extensively investigated by Knyazev and Neymeyr, see [6, 12, 13, 8].”