Asynchronous Richardson iterations

TL;DR

This paper analyzes asynchronous Richardson iterative methods for solving linear systems, establishing convergence conditions for different parameter choices and demonstrating practical convergence through multithreaded implementation.

Contribution

First analysis of asynchronous second order Richardson methods, identifying parameter ranges for convergence and comparing with synchronous optimal parameters.

Findings

Optimal parameters for synchronous methods also work asynchronously for first order.

Asynchronous second order methods may converge with near-optimal parameters.

Practical implementation confirms convergence behavior in multithreaded settings.

Abstract

We consider asynchronous versions of the first and second order Richardson methods for solving linear systems of equations. These methods depend on parameters whose values are chosen a priori. We explore the parameter values that can be proven to give convergence of the asynchronous methods. This is the first such analysis for asynchronous second order methods. We find that for the first order method, the optimal parameter value for the synchronous case also gives an asynchronously convergent method. For the second order method, the parameter ranges for which we can prove asynchronous convergence do not contain the optimal parameter values for the synchronous iteration. In practice, however, the asynchronous second order iterations may still converge using the optimal parameter values, or parameter values close to the optimal ones, despite this result. We explore this behavior with a multithreaded parallel implementation of the asynchronous methods.