Convergence Estimates for the Generalized Davidson Method for Symmetric Eigenvalue Problems I: The Preconditioning Aspect

The generalized Davidson (GD) method can be viewed as a generalization of the preconditioned steepest descent (PSD) method for solving symmetric eigenvalue problems. There are two aspects of this generalization. The most obvious one is that in the GD method the new approximation is sought in a larger subspace, namely the one that spans all the previous approximate eigenvectors, in addition to the current one and the preconditioned residual thereof. Another aspect relates to the preconditioning. Most of the available results for the PSD method are associated with the same view on preconditioning as in the case of linear systems. Consequently, they fail to detect the superlinear convergence for certain "ideal" preconditioners, such as the one corresponding to the "exact" version of the Jacobi--Davidson method---one of the most familiar instances of the GD method. Focusing on the preconditioning aspect, this paper advocates an alternative approach to measuring the quality of preconditioning for eigenvalue problems and presents corresponding non-asymptotic convergence estimates for the GD method in general and Jacobi--Davidson method in particular that correctly detect known cases of the superlinear convergence.

[1]  E. D'yakonov.  Optimization in Solving Elliptic Problems , 1995 .

[2]  P. Smit,et al.  THE EFFECTS OF INEXACT SOLVERS IN ALGORITHMS FOR SYMMETRIC EIGENVALUE PROBLEMS , 1999 .

[3]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .

[4]  Andrew V. Knyazev,et al.  Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method , 2001, SIAM J. Sci. Comput..

[5]  A. Knyazev.  Convergence rate estimates for iterative methods for a mesh symmetrie eigenvalue problem , 1987 .

[6]  Andrew V. Knyazev,et al.  A subspace preconditioning algorithm for eigenvector/eigenvalue computation , 1995, Adv. Comput. Math..

[7]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM J. Matrix Anal. Appl..

[8]  E. D'yakonov.  Iteration methods in eigenvalue problems , 1983 .

[9]  H. V. D. Vorst,et al.  Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems , 1995 .

[10]  S. Oliveira,et al.  On the Convergence Rate of a Preconditioned Subspace Eigensolver , 1999, Computing.

[11]  E. D'yakonov,et al.  Minimization of the computational labor in determining the first eigenvalues of differential operators , 1980 .

[12]  Andrew Knyazev,et al.  Preconditioned Eigensolvers - an Oxymoron? , 1998 .

[13]  Bernard Philippe,et al.  The Davidson Method , 1994, SIAM J. Sci. Comput..

[14]  Jasper van den Eshof,et al.  The convergence of Jacobi-Davidson iterations for Hermitian eigenproblems , 2002, Numer. Linear Algebra Appl..

[15]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

[16]  R. Varga,et al.  Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods , 1961 .

[17]  A. Knyazev,et al.  Efficient solution of symmetric eigenvalue problems using multigridpreconditioners in the locally optimal block conjugate gradient method , 2001 .

[18]  Evgueni E. Ovtchinnikov,et al.  Convergence Estimates for the Generalized Davidson Method for Symmetric Eigenvalue Problems II: The Subspace Acceleration , 2003, SIAM J. Numer. Anal..

[19]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[20]  Gerard L. G. Sleijpen,et al.  Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils , 1998, SIAM J. Sci. Comput..

[21]  B. Parlett.  The Rayleigh Quotient Iteration and Some Generalizations for Nonnormal Matrices , 1974 .

[22]  Yvan Notay,et al.  Convergence Analysis of Inexact Rayleigh Quotient Iteration , 2002, SIAM J. Matrix Anal. Appl..

[23]  Klaus Neymeyr,et al.  A geometric theory for preconditioned inverse iteration applied to a subspace , 2002, Math. Comput..

[24]  Yvan Notay,et al.  Combination of Jacobi–Davidson and conjugate gradients for the partial symmetric eigenproblem , 2002, Numer. Linear Algebra Appl..

[25]  H. A. V. D. Vorsty.  University Utrecht a Generalized Jacobi-davidson Iteration Method for Linear Eigenvalue Problems a Generalized Jacobi-davidson Iteration Method for Linear Eigenvalue Problems , 1994 .

[26]  H. V. D. Vorst,et al.  EFFICIENT EXPANSION OF SUBSPACES IN THE JACOBI-DAVIDSON METHOD FOR STANDARD AND GENERALIZED EIGENPROBLEMS , 1998 .

[27]  A. Knyazev.  A Preconditioned Conjugate Gradient Method for Eigenvalue Problems and its Implementation in a Subspace , 1991 .