GMRES convergence bounds for eigenvalue problems
For researchers and practitioners solving eigenvalue problems, this work offers more accurate convergence analysis and practical preconditioning strategies, though it is an incremental improvement over existing bounds.
The paper provides new, sharper convergence bounds for GMRES when solving linear systems arising in eigenvalue problems, explaining the initial rapid residual decrease and leading to improved preconditioners. Numerical results show the bounds are much sharper than conventional ones, and preconditioned subspace iteration with tuned or polynomial preconditioners is recommended.
The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds give rise to adapted preconditioners applied to the eigenvalue problems, e.g. tuned and polynomial preconditioners. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.