Nonstationary two-stage multisplitting methods for symmetric positive definite matrices

AbstractNonstationary synchronous two-stage multisplitting methods for the solution of the symmetric positive definite linear system of equations are considered. The convergence properties of these methods are studied. Relaxed variants are also discussed. The main tool for the construction of the two-stage multisplitting and related theoretical investigation is the diagonally compensated reduction (cf. [1])

Comparison results for parallel multisplitting methods with applications to AOR methods

AbstractIn this paper we present some comparison theorems between two general parallel multisplittings. These comparison theorems can be applied to many classical splittings, such as Jacobi, Gauss–Seidel and AOR splittings. Some significant improvements and generalizations of the existing comparison results are obtained

Asynchronous iterations with flexible communication: contracting operators

AbstractThe concept of flexible communication permits one to model efficient asynchronous iterations on parallel computers. This concept is particularly useful in two practical situations. Firstly, when communications are requested while a processor has completed the current update only partly, and secondly, in the context of inner/outer iterations, when processors are also allowed to make use of intermediate results obtained during the inner iteration in other processors.In the general case of nonlinear or linear fixed point problems, we give a global convergence results for asynchronous iterations with flexible communication whereby the iteration operators satisfy certain contraction hypotheses. In this manner we extend to a contraction context previous results obtained for monotone operators with respect to a partial ordering

A general asynchronous block iterative model with related convergence conditions

AbstractThis paper formulates a general time-varying asynchronous block-iterative model. A convergence condition for asynchronous block-iterations based on this model is given, compared to existing conditions for similar and shown to be strictly weaker