Massively Parallel Preconditioners for Krylov Subspace Methods

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The goal of this research is to investigate highly parallel preconditioning techniques for solving general large sparse linear systems of equations. Of particular interest are systems that arise from the discretization of partial differential equations on unstructured grids. The focuses on efficient preconditioners which scale well when number of processors is very large. In this situation, the standard incomplete factorization preconditioners are not satisfactory. The project investigates a number of non-standard alternatives which emphasize approximations based on ``high level of accuracy.'' These are similar to ILU factorizations with high level of fill-in. Many of the parallel preconditioners, suffer from the disadvantage that entail a larger number of iterations than standard preconditioners to achieve convergence. This often outweighs potential gains arising from the higher degree of parallelism. However, recent experience suggests that increasing the of the preconditioner is a good strategy for achieving faster execution by reducing the number of iterations. The use of multi-coloring techniques is also investigated.