  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.
