Scientific Computing Seminar

Inner-outer iterative methods are potentially powerful (EIGIFP, RQI, JDCG) as they exploit inner methods with three term recurrence, if one can fine tune the inner stopping criterion. Similar to work of Simoncini and Elden for RQI, Notay developed a way to cheaply monitor the eigenvector residual during PCG in the correction phase of JD, and stop when it starts to converge to the RQI iterate. Further inner iterations would be wasteful. The method, JDCG, is equivalent to Newton-Grassmann with impressive convergence results.

First, we improve on JDCG by extending Notay's results to the symmetric QMR of Freund and Nachtigal. The method, JDQMR, has two more vector updates per iteration than JDCG, but it works with indefinite matrices (interior eigenvalues) and more importantly with indefinite symmetric preconditioners. In addition, it provides a smooth, monotonic convergence of the residual which facilitates better testing for early stopping. Our experiments show JDQMR to converge faster than JDCG, even more so without preconditioning.

JDQMR/JDCG, however, demonstrate a highly targeted convergence to a specific eigenvalue with small benefits to other eigenvalues because QMR/PCG has to rebuilt the space for each of them. When requiring many eigenvalues, the subspace built by the outer method should be accumulating this information.

In our [ETNA 98] paper, we employed JD without inner method, restarting with several Ritz vectors (thick restarting) and more importantly with k Ritz vectors from the last two successive iterations. We proved that because of closeness to k Conjugate Gradient recurrences, the method behaves similarly to the unrestarted method for the targeted eigenpairs. Moreover, it accumulates significant information for nearby eigenvalues. We called this method JD+k. A non-accelerated special case of our method was proposed in 2001 by Knyazev as LOBPCG.