Preconditioned Gradient-Type Iterative Methods in
a Subspace for Partial Generalized Symmetric Eigenvalue Problems

Andrew V. Knyazev
Department of Mathematics
University of Colorado at Denver
P.O. Box 173364
Campus Box 170
Denver, CO 80217-3364

Alexander Skorokhodov
Institute of Numerical Mathematics
Russian Academy of Sciences
Moscow, Russia


We show that a modification of preconditioned gradient-type iterative methods for partial generalized eigenvalue problems makes it possible to implement them in a subspace. We propose such methods and estimate their convergence rate. We also describe iterative methods for finding a group of eigenvalues, propose preconditioners, suggest a practical way of computing the initial guess, and consider a model example. These methods are most effective for finding minimal eigenvalues of simple discretizations of elliptic operators with piece-wise constant coefficients in domains composed of rectangles or parallelepipeds. The iterative process is carried out on the interfaces between the subdomains. Its rate of convergence does not decrease when the mesh gets finer, and each iteration has a quite modest cost. This process is effective and parallelizable.

Contributed March 4, 1996.