Models of parallel chaotic iteration methods

Abstract

We consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations Ax = b. In the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed. In the second model any processor can update the global approximation which resides in the central processor at any time. This model is a generalization of a sequential iterative scheme due to Ostrowski called the free steering group Jacobi iterative scheme and a chaotic relaxation point iterative scheme due to Chazan and Miranker. We show that when A is a monotone matrix and all the splittings are weak regular, both models lead to convergent schemes.