\end{array}
\label{ch12:eq:13}
\end{equation}
-GMRES uses the Arnoldi process~\cite{ch12:ref5}\index{iterative method!Arnoldi process} to construct an
+GMRES uses the Arnoldi iterations~\cite{ch12:ref5}\index{iterative method!Arnoldi iterations} to construct an
orthonormal basis $V_k$ for the Krylov subspace $\mathcal{K}_k$ and an upper Hessenberg matrix\index{Hessenberg matrix}
$\bar{H}_k$ of order $(k+1)\times k$:
\begin{equation}
Algorithm~\ref{ch12:alg:02} shows the key points of the GMRES method with restarts.
It solves the left-preconditioned\index{sparse linear system!preconditioned} sparse linear
system~(\ref{ch12:eq:11}), such that $M$ is the preconditioning matrix. At each iteration
-$k$, GMRES uses the Arnoldi process\index{iterative method!Arnoldi process} (defined from
+$k$, GMRES uses the Arnoldi iterations\index{iterative method!Arnoldi iterations} (defined from
line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper
Hessenberg matrix\index{Hessenberg matrix} $\bar{H}_m$ of size $(m+1)\times m$. Then, it
solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$
elements necessary to compute this multiplication. First, each computing node determines, in its
local subvector, the vector elements needed by other nodes. Then, the neighboring nodes exchange
between them these shared vector elements. The data exchanges are implemented by using the MPI
-point-to-point communication routines: blocking\index{MPI subroutines!blocking} sends with \verb+MPI_Send()+
-and nonblocking\index{MPI subroutines!nonblocking} receives with \verb+MPI_Irecv()+. Figure~\ref{ch12:fig:02}
+point-to-point communication routines: blocking\index{MPI!blocking} sends with \verb+MPI_Send()+
+and nonblocking\index{MPI!nonblocking} receives with \verb+MPI_Irecv()+. Figure~\ref{ch12:fig:02}
shows an example of data exchanges between \textit{Node 1} and its neighbors \textit{Node 0}, \textit{Node 2},
and \textit{Node 3}. In this example, the iterate matrix $A$ split between these four computing
nodes is that presented in Figure~\ref{ch12:fig:01}.
and vice versa before and after the synchronization operation between CPUs. We have used the CUBLAS\index{CUBLAS}
communication subroutines to perform the data transfers between a CPU core and its GPU: \verb+cublasGetVector()+
and \verb+cublasSetVector()+. Finally, in addition to the data exchanges, GPU nodes perform reduction operations
-to compute in parallel the dot products and Euclidean norms. This is implemented by using the MPI global communication\index{MPI subroutines!global}
+to compute in parallel the dot products and Euclidean norms. This is implemented by using the MPI global communication\index{MPI!global}
\verb+MPI_Allreduce()+.
All tests are made on double-precision floating point operations. The parameters of both linear
solvers are initialized as follows: the residual tolerance threshold $\varepsilon=10^{-12}$, the
maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$, and the
-initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi process\index{iterative method!Arnoldi process}
+initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi iterations\index{iterative method!Arnoldi iterations}
used in the GMRES method to $16$ iterations ($m=16$). For the sake of simplicity, we have chosen
the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows us to easily
compute the required inverse matrix $M^{-1}$, and it provides a relatively good preconditioning for
the present chapter, the bandwidth of a sparse matrix is defined as the number of matrix columns separating
the first and the last nonzero value on a matrix row.
+
\begin{table}
\centering
+\begin{small}
\begin{tabular}{|c|c|c|c|c|}
\hline
{\bf Matrix Type} & {\bf Matrix Name} & {\bf \# Rows} & {\bf \# Nonzeros} & {\bf Bandwidth} \\ \hline \hline
& torso3 & $259,156$ & $4,429,042$ & $216,854$ \\ \hline
\end{tabular}
+\end{small}
\caption{Main characteristics of sparse matrices chosen from the University of Florida collection.}
\label{ch12:tab:01}
\end{table}
+
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\begin{table}[!h]
\begin{center}
+\begin{small}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\#~Iter.}$ & $\mathbf{Prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline
torso3 & $4.242s$ & $2.030s$ & $2.09$ & $175$ & $2.69e$-$10$ & $1.78e$-$14$ \\ \hline
\end{tabular}
+\end{small}
\caption{Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.}
\label{ch12:tab:03}
\end{center}
CG method is characterized by a better convergence\index{convergence} rate and a shorter execution
time of an iteration than those of the GMRES method. Moreover, an iteration of the parallel GMRES
method requires more data exchanges between computing nodes compared to the parallel CG method.
-
+\clearpage
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\begin{table}[!h]
\begin{center}
+\begin{small}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\#~Iter.}$ & $\mathbf{Prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline
torso3 & $31.463s$ & $3.681s$ & $8.55$ & $175$ & $2.69e$-$10$ & $2.66e$-$14$ \\ \hline
\end{tabular}
+\end{small}
\caption{Performances of the parallel GMRES method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs.
on a cluster of 12 GPUs.}
\label{ch12:tab:06}
