Controlling Inner Iterations in the Jacobi-Davidson Method

The Jacobi–Davidson method is an eigenvalue solver which uses the iterative (and in general inaccurate) solution of inner linear systems to progress, in an outer iteration, towards a particular solution of the eigenproblem. In this paper we prove a relation between the residual norm of the inner linear system and the residual norm of the eigenvalue problem. We show that the latter may be estimated inexpensively during the inner iterations. On this basis, we propose a stopping strategy for the inner iterations to maximally exploit the strengths of the method. These results extend previous results obtained for the special case of Hermitian eigenproblems with the conjugate gradient or the symmetric QMR method as inner solver. The present analysis applies to both standard and generalized eigenproblems, does not require symmetry, and is compatible with most iterative methods for the inner systems. It is also easily extended to other type of inner-outer eigenvalue solvers, as inexact inverse iteration or inexact Rayleigh quotient iteration. The effectiveness of our approach is illustrated by a few numerical experiments, including the comparison of a standard Jacobi–Davidson code with the same code enhanced by our stopping strategy.