\label{ch13:eq:15}
\end{equation}
-The previous asynchronous scheme\index{asynchronous} of the projected Richardson
+The previous asynchronous scheme\index{asynchronous iterations} of the projected Richardson
method models computations that are carried out in parallel without order or
synchronization (according to the behavior of the parallel iterative method) and
describes a subdomain method without overlapping. It is a general model that takes
into account all possible situations of parallel computations and nonblocking message
-passing. So, the synchronous iterative scheme\index{synchronous} is defined by
+passing. So, the synchronous iterative scheme\index{synchronous iterations} is defined by
\begin{equation}
\forall j\in\{1,\ldots,\alpha\} \mbox{,~} \forall p\in\mathbb{N} \mbox{,~} \rho_j(p)=p.
\label{ch13:eq:16}
\end{itemize}
As mentioned previously, we develop the \emph{synchronous} and \emph{asynchronous}
algorithms of the projected Richardson method. Obviously, in this scope, the
-synchronous\index{synchronous} or asynchronous\index{asynchronous} communications
+synchronous\index{synchronous iterations} or asynchronous\index{asynchronous iterations} communications
refer to the communications between the CPU cores (MPI processes) on the GPU cluster,
in order to exchange the vector elements associated to subdomain boundaries. For
the memory copies between a CPU core and its GPU, we use the synchronous communication
other hand, an iteration is the update of at least all vector components with
$F_i$.
-In the synchronous\index{synchronous} algorithm, the global convergence is detected
+In the synchronous\index{synchronous iterations} algorithm, the global convergence is detected
when the maximal value of the absolute error, $error$, is sufficiently small and/or
the maximum number of relaxations, $MaxRelax$, is reached, as follows:
$$
\verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the local
absolute errors, $error$, of all computing nodes, and $p$ (in Algorithm~\ref{ch13:alg:02})
is used as a counter of the local relaxations carried out by a computing node. In
-the asynchronous\index{asynchronous} algorithms, the global convergence is detected
+the asynchronous\index{asynchronous iterations} algorithms, the global convergence is detected
when all computing nodes locally converge. For this, we use a token ring architecture
around which a boolean token travels, in one direction, from a computing node to another.
Starting from node $0$, the boolean token is set to $true$ by node $i$ if the local
between the time of the computation over that of the communication is reduced when
the computations are performed on GPUs. Indeed, GPUs compute faster than CPUs and
communications are more time-consuming. In this context, asynchronous algorithms
-are more scalable than synchronous ones. So, with large scale GPU clusters, synchronous\index{synchronous}
+are more scalable than synchronous ones. So, with large scale GPU clusters, synchronous\index{synchronous iterations}
algorithms might be more penalized by communications, as can be deduced from Figure~\ref{ch13:fig:07}.
-That is why we think that asynchronous\index{asynchronous} iterative algorithms
+That is why we think that asynchronous\index{asynchronous iterations} iterative algorithms
are all the more interesting in this case.
