Parallelizable approximate solvers for recursions arising in preconditioning Original Research Article

For the recursions used in the modified incomplete LU preconditioner, namely, the construction of incomplete decomposition, the forward elimination, and the back substitution, parallelizable approximate solvers are introduced. The analysis shows that the solutions of the recursions depend only weakly on their initial conditions, which may be interpreted to indicate that the inexact solution is close, in some sense, to the exact one. The method is based on a domain decomposition approach, suitable for parallel implementations on message-passing architectures. It requires a small number of communication steps per preconditioned iteration, independent of the number of subdomains and the size of the problem. The overlapping subdomains are either squares (suitable for mesh-connected arrays of processors) or defined by the data-flow rule of the recursions (suitable for line-connected arrays of SIMD or vector processors). Numerical examples show that, in both cases, the overhead in the number of preconditioned iterations required for convergence is small in comparison with the speedup gained.