+\textcolor{red}{\bf Hereafter, we show the influence of the communications on a GPU cluster compared to a CPU cluster. In Tables~\ref{tab:10},~\ref{tab:11} and~\ref{tab:12}, we compute the ratios between the computation time over the communication time of three versions of the parallel GMRES algorithm for solving sparse linear systems associated to matrices of Table~\ref{tab:06}. These tables show that the hypergraph partitioning and the compressed format of the vectors increase the ratios either on the GPU cluster or on the CPU cluster. This means that the two optimization techniques allow the minimization of the total communication volume between the computing nodes. However, we can notice that the ratios obtained on the GPU cluster are lower than those obtained on the CPU cluster. Indeed, GPUs compute faster than CPUs and communications are more time-consuming while the computation time remains unchanged.}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c||c|c|c||c|c|c|}
+\hline
+\multirow{2}{*}{Matrix} & \multicolumn{3}{c||}{GPU version} & \multicolumn{3}{c|}{CPU version} \\ \cline{2-7}
+ & $Time_{comput}$ & $Time_{comm}$ & $Ratio$ & $Time_{comput}$ & $Time_{comm}$ & $Ratio$ \\ \hline \hline
+2cubes\_sphere & 37.067 s & 1434.512 s & {\bf 0.026} & 312.061 s & 3453.931 s & {\bf 0.090}\\
+ecology2 & 4.116 s & 501.327 s & {\bf 0.008} & 60.776 s & 1216.607 s & {\bf 0.050}\\
+finan512 & 7.170 s & 386.742 s & {\bf 0.019} & 72.464 s & 932.538 s & {\bf 0.078}\\
+G3\_circuit & 4.797 s & 537.343 s & {\bf 0.009} & 66.011 s & 1407.378 s & {\bf 0.047}\\
+shallow\_water2 & 3.620 s & 411.208 s & {\bf 0.009} & 51.294 s & 973.446 s & {\bf 0.053}\\
+thermal2 & 6.902 s & 511.618 s & {\bf 0.013} & 77.255 s & 1281.979 s & {\bf 0.060}\\ \hline \hline
+cage13 & 12.837 s & 625.175 s & {\bf 0.021} & 139.178 s & 1518.349 s & {\bf 0.092}\\
+crashbasis & 48.532 s & 3195.183 s & {\bf 0.015} & 623.686 s & 7741.777 s & {\bf 0.081}\\
+FEM\_3D\_thermal2 & 37.211 s & 1584.650 s & {\bf 0.023} & 370.297 s & 3810.255 s & {\bf 0.097}\\
+language & 22.912 s & 2242.897 s & {\bf 0.010} & 286.682 s & 5348.733 s & {\bf 0.054}\\
+poli\_large & 13.618 s & 1722.304 s & {\bf 0.008} & 190.302 s & 4059.642 s & {\bf 0.047}\\
+torso3 & 74.194 s & 4454.936 s & {\bf 0.017} & 190.302 s & 10800.787 s & {\bf 0.083}\\ \hline
+\end{tabular}
+\caption{Ratios of the computation time over the communication time obtained from the parallel GMRES algorithm using row-by-row partitioning on 12 GPUs and 24 CPUs.}
+\label{tab:10}
+\end{center}
+\end{table}
+
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c||c|c|c||c|c|c|}
+\hline
+\multirow{2}{*}{Matrix} & \multicolumn{3}{c||}{GPU version} & \multicolumn{3}{c|}{CPU version} \\ \cline{2-7}
+ & $Time_{comput}$ & $Time_{comm}$ & $Ratio$ & $Time_{comput}$ & $Time_{comm}$ & $Ratio$ \\ \hline \hline
+2cubes\_sphere & 27.386 s & 154.861 s & {\bf 0.177} & 342.255 s & 42.100 s & {\bf 8.130}\\
+ecology2 & 3.822 s & 53.131 s & {\bf 0.072} & 69.956 s & 15.019 s & {\bf 4.658}\\
+finan512 & 6.366 s & 41.155 s & {\bf 0.155} & 79.592 s & 8.604 s & {\bf 9.251}\\
+G3\_circuit & 4.543 s & 63.132 s & {\bf 0.072} & 76.540 s & 27.371 s & {\bf 2.796}\\
+shallow\_water2 & 3.282 s & 43.080 s & {\bf 0.076} & 58.348 s & 8.088 s & {\bf 7.214}\\
+thermal2 & 5.986 s & 57.100 s & {\bf 0.105} & 87.682 s & 28.544 s & {\bf 3.072}\\ \hline \hline
+cage13 & 10.227 s & 70.388 s & {\bf 0.145} & 152.718 s & 30.785 s & {\bf 4.961}\\
+crashbasis & 41.527 s & 369.071 s & {\bf 0.113} & 701.040 s & 158.916 s & {\bf 4.411}\\
+FEM\_3D\_thermal2 & 28.691 s & 167.140 s & {\bf 0.172} & 403.510 s & 50.935 s & {\bf 7.922}\\
+language & 22.408 s & 242.589 s & {\bf 0.092} & 333.119 s & 64.409 s & {\bf 5.172}\\
+poli\_large & 13.710 s & 179.208 s & {\bf 0.077} & 215.934 s & 30.903 s & {\bf 6.987}\\
+torso3 & 58.455 s & 480.315 s & {\bf 0.122} & 993.609 s & 152.173 s & {\bf 6.529}\\ \hline
+\end{tabular}
+\caption{Ratios of the computation time over the communication time obtained from the parallel GMRES algorithm using row-by-row partitioning and compressed format for vectors on 12 GPUs and 24 CPUs.}
+\label{tab:11}
+\end{center}
+\end{table}
+
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c||c|c|c||c|c|c|}
+\hline
+\multirow{2}{*}{Matrix} & \multicolumn{3}{c||}{GPU version} & \multicolumn{3}{c|}{CPU version} \\ \cline{2-7}
+ & $Time_{comput}$ & $Time_{comm}$ & $Ratio$ & $Time_{comput}$ & $Time_{comm}$ & $Ratio$ \\ \hline \hline
+2cubes\_sphere & 28.440 s & 7.768 s & {\bf 3.661} & 327.109 s & 63.788 s & {\bf 5.128}\\
+ecology2 & 3.652 s & 0.757 s & {\bf 4.823} & 63.632 s & 13.520 s & {\bf 4.707}\\
+finan512 & 7.579 s & 4.569 s & {\bf 1.659} & 74.120 s & 22.505 s & {\bf 3.294}\\
+G3\_circuit & 4.876 s & 8.745 s & {\bf 0.558} & 72.280 s & 28.395 s & {\bf 2.546}\\
+shallow\_water2 & 3.146 s & 0.606 s & {\bf 5.191} & 52.903 s & 11.177 s & {\bf 4.733}\\
+thermal2 & 6.473 s & 4.325 s & {\bf 1.497} & 81.171 s & 20.907 s & {\bf 3.882}\\ \hline \hline
+cage13 & 11.676 s & 7.723 s & {\bf 1.512} & 145.755 s & 46.547 s & {\bf 3.131}\\
+crashbasis & 42.799 s & 29.399 s & {\bf 1.456} & 650.386 s & 203.918 s & {\bf 3.189}\\
+FEM\_3D\_thermal2 & 29.875 s & 8.915 s & {\bf 3.351} & 382.887 s & 93.252 s & {\bf 4.106}\\
+language & 20.991 s & 11.197 s & {\bf 1.875} & 310.679 s & 82.480 s & {\bf 3.767}\\
+poli\_large & 13.817 s & 102.760 s & {\bf 0.134} & 197.508 s & 151.672 s & {\bf 1.302}\\
+torso3 & 57.469 s & 16.828 s & {\bf 3.415} & 926.588 s & 242.721 s & {\bf 3.817}\\ \hline
+\end{tabular}
+\caption{Ratios of the computation time over the communication time obtained from the parallel GMRES algorithm using hypergraph partitioning and compressed format for vectors on 12 GPUs and 24 CPUs.}
+\label{tab:12}
+\end{center}
+\end{table}
+
+\begin{figure}
+\centering
+ \includegraphics[width=120mm,keepaspectratio]{weak}
+\caption{Weak scaling of the parallel GMRES algorithm on a GPU cluster.}
+\label{fig:09}
+\end{figure}
+
+\textcolor{red}{\bf Figure~\ref{fig:09} presents the weak scaling of four versions of the parallel GMRES algorithm on a GPU cluster. We fixed the size of a sub-matrix to 5 million of rows per GPU computing node. We used matrices having five bands generated from the symmetric matrix thermal2. This figure shows that the parallel GMRES algorithm in its naive version or using either the compression format for vectors or the hypergraph partitioning is not scalable on a GPU cluster due to the large amount of communications between GPUs. In contrast, we can see that the algorithm using both optimization techniques is fairly scalable. This means in this version the cost of communications is relatively constant regardless the number of computing nodes in the cluster.}
