A class of asynchronous multisplitting two-stage iterations for large sparse block systems of weakly nonlinear equations

For the block system of weakly nonlinear equations Ax=G(x), where A∈Rn×n is a large sparse block matrix and G:Rn→Rn is a block nonlinear mapping having certain smoothness properties, we present a class of asynchronous parallel multisplitting block two-stage iteration methods in this paper. These methods are actually the block variants and generalizations of the asynchronous multisplitting two-stage iteration methods studied by Bai and Huang (Journal of Computational and Applied Mathematics 93(1) (1998) 13–33), and they can achieve high parallel efficiency of the multiprocessor system, especially, when there is load imbalance. Under quite general conditions that A∈Rn×n is a block H-matrix of different types and G:Rn→Rn is a block P-bounded mapping, we establish convergence theories of these asynchronous multisplitting block two-stage iteration methods. Numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment.