$v=e^{-a}.u$ represents the general change of variables such that $a=\frac{b^{t}(x,y,z)}{2\eta}$.
Consequently, the numerical resolution of the diffusion problem (the self-adjoint
operator~(\ref{ch13:eq:04})) is done by optimization algorithms, in contrast to that
-of the convection-diffusion problem (non self-adjoint operator~(\ref{ch13:eq:03}))
+of the convection-diffusion problem (nonself-adjoint operator~(\ref{ch13:eq:03}))
which is done by relaxation algorithms. In the case of our studied algorithm, the convergence\index{convergence}
is ensured by M-matrix property; then, the performance is linked to the magnitude of
the spectral radius of the iteration matrix, which is independent of the condition
\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}
relaxed components to be used in the computational process may be guided by any criterion,
and in particular, a natural criterion is to pickup the most recently available
values of the components computed by the processors. Furthermore, the asynchronous
-iterations are implemented by means of nonblocking MPI communication subroutines\index{MPI subroutines!nonblocking}
+iterations are implemented by means of nonblocking MPI communication subroutines\index{MPI!nonblocking}
(asynchronous communications).
The important property ensuring the convergence of the parallel projected Richardson
\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
and \verb+cublasGetVectorAsync()+ in the asynchronous algorithm. Moreover, we
use the communication routines of the MPI library to carry out the data exchanges
between the neighboring nodes. We use the following communication routines: \verb+MPI_Isend()+
-and \verb+MPI_Irecv()+ to perform nonblocking\index{MPI subroutines!nonblocking}
+and \verb+MPI_Irecv()+ to perform nonblocking\index{MPI!nonblocking}
sends and receives, respectively. For the synchronous algorithm, we use the MPI
routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node in
blocking status until all data exchanges with neighboring nodes (sends and receives)
are completed. In contrast, for the asynchronous algorithms, we use the MPI routine
\verb+MPI_Test()+ which tests the completion of a data exchange (send or receives)
-without putting the MPI process in blocking status\index{MPI subroutines!blocking}.
+without putting the MPI process in blocking status\index{MPI!blocking}.
The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{ch13:alg:02})
computes, at each iteration, the new elements of the iterate vector $U$. Its general code
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:
$$
conv \leftarrow true;
\end{array}
$$
-where the function $AllReduce()$ uses the MPI global reduction subroutine\index{MPI subroutines!global}
+where the function $AllReduce()$ uses the MPI global reduction subroutine\index{MPI!global}
\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
\begin{table}
\centering
+\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous} & \multicolumn{3}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-7}
$800^{3}$ & $3,950.87$ & $899,088$ & $56.22$ & $3,636.57$ & $834,900$ & $51.91$ & $7.95$ \\ \hline
\end{tabular}
+\end{scriptsize}
\vspace{0.5cm}
\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.}
\label{ch13:tab:02}
%%--------------------------%%
\section{Red-black ordering technique}
\label{ch13:sec:06}
-As is wellknown, the Jacobi method\index{iterative method!Jacobi} is characterized
+As is well known, the Jacobi method\index{iterative method!Jacobi} is characterized
by a slow convergence\index{convergence} rate compared to some iterative methods\index{iterative method}
(for example, Gauss-Seidel method\index{iterative method!Gauss-Seidel}). So, in this
section, we present some solutions to reduce the execution time and the number of
we apply the point red-black ordering\index{iterative method!red-black ordering}
accordingly to the $y$-coordinate, as is shown in Figure~\ref{ch13:fig:06.02}. In
this case, the vector elements having even $y$-coordinate are computed in parallel
-using the values of those having odd $y$-coordinate and then viceversa. Moreover,
+using the values of those having odd $y$-coordinate and then vice-versa. Moreover,
in the GPU implementation of the parallel projected Richardson method (Section~\ref{ch13:sec:04}),
we have shown that a subproblem of size $(NX\times ny\times nz)$ is decomposed into
$nz$ grids of size $(NX\times ny)$. Then, each kernel is executed in parallel by
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.
