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.

我们考虑一阶和二阶Richardson方法的异步版本，用于求解方程的线性系统。这些方法取决于优先选择其值的参数。我们探索了可以证明可以使异步方法收敛的参数值。这是对异步二阶方法的首次此类分析。我们发现对于一阶方法，同步情况下的最佳参数值也给出了异步收敛的方法。对于二阶方法，可以证明异步收敛的参数范围不包含用于同步迭代的最佳参数值。但是实际上，异步二阶迭代仍可以使用最佳参数值或接近最佳参数值的参数值收敛，尽管有这个结果。我们通过异步方法的多线程并行实现来探索这种行为。