Parallel preconditioned solvers for large sparse Hermitian eigenproblems

Parallel preconditioned solvers are presented to compute a few extreme eigenvalues and -vectors of large sparse Hermitian matrices based on the Jacobi-Davidson (JD) method by G.L.G. Sleijpen and H.A. van der Vorst. For preconditioning, an adaptive approach is applied using the QMR (Quasi-Minimal Residual) iteration. Special QMR versions have been developed for the real symmetric and the complex Hermitian case. To parallelise the solvers, matrix and vector partitioning is investigated with a data distribution and a communication scheme exploiting the sparsity of the matrix. Synchronization overhead is reduced by grouping inner products and norm computations within:the QMR and the JD iteration. The efficiency of these strategies is demonstrated on the massively parallel systems NEC Cenju-3 and Cray T3E.