A unified approach to parallel space decomposition methods

Abstract

We consider (relaxed) additive and multiplicative iterative space decomposition methods for the minimization of sufficiently smooth functionals without constraints. We develop a general framework which unites existing approaches from both parallel optimization and finite elements. Specifically this work unifies earlier research on the parallel variable distribution method in minimization, space decomposition methods for convex functionals, algebraic Schwarz methods for linear systems and splitting methods for linear least squares. We develop a general convergence theory within this framework, which provides several new results as well as including known convergence results.