Exploiting Multilevel Preconditioning Techniques in Eigenvalue Computations

In the Davidson method, any preconditioner can be exploited for the iterative computation of eigenpairs. However, the convergence of the eigenproblem solver may be poor for a high quality preconditioner. Theoretically, this counter-intuitive phenomenon with the Davidson method is remedied by the Jacobi--Davidson approach, where the preconditioned system is restricted to appropriate subspaces of codimension one. However, it is not clear how the restricted system can be solved accurately and efficiently in the case of a good preconditioner. The obvious approach introduces instabilities that hamper convergence. In this paper, we show how incomplete decomposition based on multilevel approaches can be used in a stable way. We also show how these preconditioners can be efficiently improved when better approximations for the eigenvalue of interest become available. The additional costs for updating the preconditioners are negligible. Furthermore, our approach leads to a good initial guess for the wanted eigenpair and to high quality preconditioners for nearby eigenvalues. We illustrate our ideas for the MRILU preconditioner.