+
+\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}
+
+ 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. That means that in this version the cost of communications is relatively constant regardless the number of computing nodes in the cluster.\\
+
+ Finally, as far as we are concerned, the parallel solving of a linear system can be easy to optimize when the associated matrix is regular. This is unfortunately not the case for many real-world applications. When the matrix has an irregular structure, the amount of communication between processors is not the same. Another important parameter is the size of the matrix bandwidth which has a huge influence on the amount of communication. In this work, we have generated different kinds of matrices in order to analyze several difficulties. With a bandwidth as large as possible, involving communications between all processors, which is the most difficult situation, we proposed to use two heuristics. Unfortunately, there is no fast method that optimizes the communication in any situation. For systems of non linear equations, there are different algorithms but most of them consist in linearizing the system of equations. In this case, a linear system needs to be solved. The big interest is that the matrix is the same at each step of the non linear system solving, so the partitioning method which is a time consuming step is performed only once.
+
+
+
+Another very important issue, which might be ignored by too many people, is that the communications have a greater influence on a cluster of GPUs than on a cluster of CPUs. There are two reasons for that. The first one comes from the fact that with a cluster of GPUs, the CPU/GPU data transfers slow down communications between two GPUs that are not on the same machines. The second one is due to the fact that with GPUs the ratio of the computation time over the communication time decreases since the computation time is reduced. So the impact of the communications between GPUs might be a very important issue that can limit the scalability of a parallel algorithm.
