Parallel Jacobi-Davidson for Solving Generalized Eigenvalue Problems

We study the Jacobi-Davidson method for the solution of large generalised eigenproblems as they arise in MagnetoHydroDynamics. We have combined Jacobi-Davidson (using standard Ritz values) with a shift and invert technique. We apply a complete LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelisable algorithm. Moreover, we describe a variant of Jacobi-Davidson in which harmonic Ritz values are used. In this variant the same parallel LU decomposition is used, but this time as a preconditioner to solve the ‘correction‘ equation. The size of the relatively small projected eigenproblems which have to be solved in the Jacobi-Davidson method is controlled by several parameters. The influence of these parameters on both the parallel performance and convergence behaviour will be studied. Numerical results of Jacobi-Davidson obtained with standard and harmonic Ritz values will be shown. Executions have been performed on a Cray T3E.