Parallel Iterative Solvers with Localized ILU Preconditioning

Publisher Summary This chapter has illustrated implementation of “Localized” ILU(O) preconditioning method to various type of iterative solvers. In this method, ILU(O) factorization is carried out for each processor by zeroing out the matrix components whose column numbers are outside the processor domain. This method provides data locality on each processor and good parallelization effect. Developed system performance has been also evaluated on simulated parallel processors by workstation cluster with PVM. Most of preconditioned iterative processes are combination of matrix–vector product calculation, vector–vector inner product calculation, SAXPY operations and vector scaling, and preconditioning operation. The first 3 operations can be relatively easily parallelized. Generally speaking, preconditioning (back/forward substitution, BFS) operation requires almost 50% of the whole calculation if ILU(O) is implemented as preconditioner. Therefore, it is very important to get high degree of parallelization in the BFS operation. Localized ILU(O) preconditioner has been implemented to 3 types of iterative solvers. Localized ILU(O) method provides data locality on each processor and good parallelization effect because there are no inter-processor communications during ILU(O) operation. A convection-diffusion problem for simple geometry has been applied to evaluate the method. Various types of domain partition effect have been also applied. Results show that Localized ILU(O) is most effective when the cell number per domain (processor) is sufficiently large and hence, is very effective in solving large-scale sparse matrix on parallel computers. In the chapter, only simple geometry with simple boundary conditions has been evaluated.