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-\section{The EA algorithm on Multi-GPU}
+\section{The EA algorithm on Multiple GPUs}
\label{sec4}
-\subsection{MGPU : an OpenMP-CUDA approach}
+\subsection{M-GPU : an OpenMP-CUDA approach}
Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
-Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$, and the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU, and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
+Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$, and the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU, and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory. Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
%\begin{figure}[htbp]
%\centering
For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
%SIDER : Une meilleure présentation de l'architecture est à faire ici.
-In order to evaluate both the MGPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
-All experimental results obtained are made in double precision data, the convergence threshold of the methods is set to $10^{-7}$.
+In order to evaluate both the M-GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
+All experimental results obtained are made in double precision data whereas the convergence threshold of the EA method is set to $10^{-7}$.
%Since we were more interested in the comparison of the
%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
%CPUs versus on GPUs.
\subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
-We report here the results of the set of experiments with M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
-\subsubsection{Execution times in seconds of the EA method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+We report here the results of the set of experiments with the M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
+\subsubsection{Execution time of the EA method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
\label{fig:01}
\end{figure}
-This figure~\ref{fig:01} shows that the (CUDA-OpenMP) Multi-GPU approach reduces the execution time by a factor up to 100 w.r.t the single GPU apparaoch and a by a factor of 1000 for polynomials exceeding degree 1,000,000. It shows the advantage to use the OpenMP parallel paradigm to gather the capabilities of several GPUs and solve polynomials of very high degrees.
+This figure~\ref{fig:01} shows that the (CUDA-OpenMP) M-GPU approach reduces the execution time by a factor up to 100 w.r.t the single GPU approach and a by a factor of 1000 for polynomials exceeding degree 1,000,000. It shows the advantage to use the OpenMP parallel paradigm to gather the capabilities of several GPUs and solve polynomials of very high degrees.
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+\subsubsection{Execution time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU approach}
The experiments shows the execution time of the EA algorithm, on a single GPU and on multiple GPUs using the CUDA OpenMP approach for full polynomials of degrees ranging from 100,000 to 1,400,000.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using shared memory paradigm with OpenMP}
+\caption{Execution time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU appraoch}
\label{fig:03}
\end{figure}
Results with full polynomials show very important savings in execution time. For a polynomial of degree 1,4 million, the CUDA-OpenMP approach with 4 GPUs solves it 4 times as fast as single GPU, thus achieving a quasi-linear speedup.
\subsection{Evaluating the Multi-GPU (CUDA-MPI) approach}
-In this part we perform a set of experiments to compare Multi-GPU (CUDA MPI) approach with single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.
+In this part we perform a set of experiments to compare the Multi-GPU (CUDA MPI) approach with a single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
+\subsubsection{Execution time of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
+\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}
\label{fig:02}
\end{figure}
~\\
-This figure shows 4 curves of execution time of EA algorithm, a curve with single GPU, 3 curves with multiple GPUs (2, 3, 4). We can clearly see that the curve with single GPU is above the other curves, which shows consumption in execution time compared to the Multi-GPU. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 100 for polynomials of degree more than 1,000,000 whereas a single GPU is of the scale 1000.
+Figure~\ref{fig:02} shows execution time of EA algorithm, for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. We can clearly see that the curve for a single GPU is above the other curves, which shows overtime in execution time compared to the Multi-GPU approach. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 10 for polynomials of degree more than 1,000,000. For example, at degree 1000000, the execution time with a single GPU amounted to 10 thousand seconds, while with 4 GPUs, it is lowered to about just one thousand seconds which makes it for a tenfold speedup.
%%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
\\cette figure montre 4 courbes de temps d'exécution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constaté clairement que la courbe à un seul GPU est au-dessus des autres courbes, vue sa consomation en temps d'exècution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exècution jusqu'à l'echelle 100 pour le polynômes qui dépasse 1,000,000 tandis que Single GPU est de l'echelle 1000.
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using distributed memory paradigm with MPI}
+\subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using distributed memory paradigm with MPI}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using the Multi-GPU}
\label{fig:04}
\end{figure}
-%SIDER : Corriger le point de la courbe 3-GPUs qui correpsond à un degré de 600000
-Figure \ref{fig:04} shows the execution time of the algorithm on single GPU and on multipe GPUs with (2, 3, 4) GPUs for full polynomials. With the CUDA-MPI approach, we notice that the three curves are distinct from each other, more we use GPUs more the execution time decreases. On the other hand the curve with a single GPU is well above the other curves.
+
+ Figure \ref{fig:04} shows execution time for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. With the CUDA-MPI approach, we notice that the three curves are distinct from each other, more we use GPUs more the execution time decreases. On the other hand the curve for a single GPU is well above the other curves.
This is due to the use of MPI parallel paradigm that divides the problem computations and assigns portions to each GPU. But unlike the single GPU which carries all the computations on a single GPU, data communications are introduced, consequently engendering more execution time. But experiments show that execution time is still highly reduced.
-\subsection{Comparative between (CUDA-OpenMP) approach and (CUDA-MPI) approach}
-In this part we present some experiment comparing the two Multi-GPU approach (OpenMP versus MPI) for solving sparse polynomial, full polynomials than we compare the execution time of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with MPI and with OpenMP.
+\subsection{Comparing the CUDA-OpenMP approach and the CUDA-MPI approach}
+
+In the previuos section we saw that both approches are very effective in reducing execution time for sparse as well as full polynomials. At this stage, the interesting question is which approach is better. In the fellowing, we present appropriate experiments comparing the two Multi-GPU approaches to answer the question.
-\subsubsection{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse polynomials on GPUs}
-In this experiment we chose three polynomials of different size like (200K, 800K, 1,4M). We compare their execution time according to the number of the GPUs.
+\subsubsection{Solving sparse polynomials}
+In this experiment three sparse polynomials of size 200K, 800K and 1,4M are investigated.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
-\caption{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse polynomials on GPUs.}
+\caption{Execution time for solving sparse polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
\label{fig:05}
\end{figure}
-in figure ~\ref{fig:05} we have two curves: MPI curve and OpenMP curve for each polynomials size. We can see that the results are similar between OpenMP curves and MPI curves for the polynomials size (200K, 1,4M), but there is a slight different between MPI curve and OpenMP curve for the polynomial of size 800K. ...
+In Figure~\ref{fig:05} there two curves for each polynomial size : one for the MPI-CUDA and another for the OpenMP. We can see that the results are similar between OpenMP and MPI for the polynomials size of 200K. For the size of 800K, the MPI version is a little slower than the OpenMP approach but for for the 1,4M size, there is a slight advantage for the MPI version.
-\subsubsection{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
+\subsubsection{Solving full polynomials}
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full}
-\caption{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs.}
+\caption{Execution time for solving full polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
\label{fig:06}
\end{figure}
-in figure ~\ref{fig:06}, we can see that the two paradigm MPI and OpenMP give the same result for solving full polynomials with EA algorithm.
-% size (200k,800K, 1,4M) are very similar for solving full polynomials with the EA algorithm.
+In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
-\subsubsection{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
-in this experiment we compare the execution time of EA algorithm according to the number of the GPU for solving sparse and full polynomials on Multi-GPU using MPI. We chose three sparse and full polynomials of different size like (200K, 800K, 1,4M).
+\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
+In this experiment we compare the execution time of the EA algorithm according to the number of GPUs for solving sparse and full polynomials on Multi-GPU using MPI. We chose three sparse and full polynomials of size 200K, 800K and 1,4M.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{MPI}
-\caption{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI.}
+\caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using MPI}
\label{fig:07}
\end{figure}
-in figure ~\ref{fig:07} we can see that (CUDA-MPI) can solve sparse and full polynomials of high degrees, the execution time with sparse polynomial are very low comparing to full polynomials. with sparse polynomials the number of monomial are reduce, consequently the number of operation are reduce than the execution time decrease.
+in figure ~\ref{fig:07} we can see that CUDA-MPI can solve sparse and full polynomials of high degrees, the execution time with sparse polynomial are very low comparing to full polynomials. with sparse polynomials the number of monomial are reduce, consequently the number of operation are reduce than the execution time decrease.
-\subsubsection{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
+\subsubsection{Solving sparse and full polynomials of the same size with CUDA-OpenMP}
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{OMP}
-\caption{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP.}
+\caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
\label{fig:08}
\end{figure}
-in figure ~\ref{fig:08}
-\subsection{Scalability of the EA method on Multi-GPU to solve very high polynomials degrees}
- This experiment we report the execution time according to the degrees polynomials ranging from 1,000,000 to 5,000,000 for both approaches (cuda-OpenMP) and (CUDA-MPI) with sparse and full polynomials.
+Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact fellows the same pattern, a difference in execution time in favor of the sparse polynomials.
+%SIDER : il faut une explication ici. je ne vois pas de prime abords, qu'est-ce qui engendre cette différence, car quelques soient les coefficients nulls ou non nulls, c'est toutes les racines qui sont calculées qu'elles soient similaires ou non (degrés de multiplicité).
+\subsection{Scalability of the EA method on Multi-GPU to solve very high degree polynomials}
+These experiments report the execution time according to the degrees of polynomials ranging from 1,000,000 to 5,000,000 for both approaches with sparse and full polynomials.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{big}
- \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degrees on 4 GPUs.}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degree on 4 GPUs for sizes ranging from 1M to 5M}
\label{fig:09}
\end{figure}
-in figure ~\ref{fig:09} we can see that both (cuda-OpenMP) and (CUDA-MPI) approaches are scalable can solve very high polynomials degrees. with full polynomial the both approaches give very interesting ans similar results for polynomials of 5,000,000 degrees we not reach 30,000 s
+In figure ~\ref{fig:09} we can see that both approaches are scalable and can solve very high degree polynomials. With full polynomial both approaches give interestingly very similar results. For the sparse case however, there are a noticeable difference in favour of MPI when the degree is above 4M. Between 1M and 3M, the OMP approach is more effective and under 1M degree, OMP and MPI approaches are almost equivalent.
+
+%SIDER : il faut une explication sur les différences ici aussi.
+
%for sparse and full polynomials
% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
