Parallelizable Approximate Solvers for Recursions Arising in Preconditioning

For the recursions used in the Modiied Incomplete LU preconditioner (incomplete decomposition , forward elimination and back substitution), a parallelizable approximate solver is presented. It is motivated by an analysis showing that the solutions of these recursions depend only weakly on their initial conditions, which indicates that the solution of a truncated recursion is a fair approximation to that of the original one. The method is based on a domain decomposition approach, suitable for parallel implementations with message passing architectures. It requires a xed number of communication steps per preconditioned iteration, independently of the number of subdomains or the size of the problem. The overlapping subdomains are either cubes (suitable for mesh-connected arrays of processors) or constructed by the data-ow rule of the recursions (suitable for line-connected arrays with possibly SIMD or vector processors). Numerical examples show that, in both cases, the overhead in the number of iterations required for convergence of the preconditioned iteration is small relatively to the speed-up gained.