Effective preconditioning techniques for eigenvalue problems

In the Davidson method, any preconditioner can be exploited for the iterative computation of eigen-pairs. However, the convergence of the eigenproblem solver may be poor if the quality of the preconditioner for linear systems solvers is good. Theoretically, this counter-intuitive phenomenon with the Davidson method is reme-died by the Jacobi-Davidson approach, where the preconditioned system is restricted to appropriate subspaces of co-dimension one. However, it is not clear how the restricted system can be solved accurately and efficiently in case of a good preconditioner. The obvious approach introduces instabilities that hampers convergence. In this paper, we show how an incomplete decomposition based on the MRILU approach can be used in a stable way. We also show how this preconditioner can be efficiently improved when better approximations for the eigenvalue of interest become available. Our approach leads to a good initial guess for the wanted eigenpair and to high quality preconditioners for nearby eigenvalues. The additional costs for updating the preconditioner are negligible.