Serial and Parallel Iterative Splitting Methods: Algorithms and Applications

The properties of iterative splitting methods with serial versions have been analyzed since recent years, see [1] and [3]. We extend the iterative splitting methods to a class of parallel versions, which allow to reduce the computational time and keep the benefit of the higher accuracy with each iterative step. Parallel splitting methods are nowadays important to solve large problems, which can be splitted in subproblems and computed independently with the different processors. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, see [2], [10] and [15]. Such a flexibilisation with multisplitting methods allow to decompose large iterative splitting methods and recover the benefit of their underlying waveform-relaxation (WR) methods. We discuss the convergence results of the parallel iterative splitting methods, while we could reformulate such an error to a summation of the individual WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods and present different numerical applications to validate the benefit of the parallel versions.