This paper introduces the multiplicative variant of the recently proposed asynchronous additive coarse-space correction method. Definition of an asynchronous extension of multiplicative correction is not straightforward, however, our analysis allows for usual asynchronous programming approaches. General asynchronous iterative models are explicitly devised both for shared or replicated coarse problems and for centralized or distributed ones. Convergence conditions are derived and shown to be satisfied for M-matrices, as also done for the additive case. Implementation aspects are discussed, which reveal the need for non-blocking synchronization for building the successive right-hand-side vectors of the coarse problem. Optionally, a parameter allows for applying each coarse solution a maximum number of times, which has an impact on the algorithm efficiency. Numerical results on a high-speed homogeneous cluster confirm the practical efficiency of the asynchronous two-level method over its synchronous counterpart, even when it is not the case for the underlying one-level methods.

翻译：本文提出了近期提出的异步加性粗空间校正方法的乘性变体。乘性校正的异步扩展定义并非直接明了，然而我们的分析允许采用常规的异步编程方法。针对共享或复制的粗问题，以及集中式或分布式场景，本文明确构建了一般性的异步迭代模型。推导了收敛条件，并证明该条件与加性情形一样，对M矩阵成立。讨论了实现层面，揭示了构建粗问题连续右端向量需要非阻塞同步。可选地，引入参数允许对每个粗解施加最大应用次数，这会影响算法效率。在高速同构集群上的数值结果表明，即使底层单层方法并非如此，异步双层方法相对于其同步对应方法仍具有实际效率优势。