On partial synchronization of iterative methods

Abstract

Rapidly growing field of parallel computing systems promotes study of parallel algorithms. Monte Carlo method and asynchronous iterations are among the most valuable ones. Advantages of these algorithms are as follows. There is no need of global time in parallel system (no need for synchronization) and all computational resources are efficiently loaded (minimal idle time of processors). Method of partial synchronization of iterations for systems of equations proposed earlier by authors is developed and generalized on the case of nonlinear equations of the form x = F(x), where x is unknown column-vector of length n, F is an operator from Rn to Rn. Operators that do not satisfy the sufficient conditions of convergence of asynchronous iterations but those with convergence of simple iterations are considered. In that case one can find an example of the operator and point out the properties of parallel system for which convergence of asynchronous iterations is violated. Partial synchronisation is one of the efficient solutions of such problem. Algorithm that garantees convergence of asynchronous iterations and Monte Carlo method is proposed. Rate of convergence of the method is estimated. Obtained results can be useful while solving high dimensional problems on the multiprocessor computational systems. Refs 4.