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%\usepackage{graphicx}
-
-
+\bibliographystyle{IEEEtran}
+% argument is your BibTeX string definitions and bibliography database(s)
+%\bibliography{IEEEabrv,../bib/paper}
+\bibliographystyle{elsarticle-num}
\begin{document}
%
% paper title
\section{Parallel Programmings Model}
-\subsection{OpenMP}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications
-Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity [22]. OpenMP is
+\subsection{OpenMP}
+Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity~\cite{openmp13}. OpenMP is
a portable approach for parallel programming on shared memory systems based on compiler directives, that can be included in order
to parallelize a loop. In this way, a set of loops can be distributed along the different threads that will access to different data allo-
cated in local shared memory. One of the advantages of OpenMP is its global view of application memory address space that allows relatively fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of
performance in large scale applications. Although, in OpenMP a usage of threads ids and managing data explicitly as done in an MPI
code can be considered, it defeats the advantages of OpenMP.
-\subsection{OpenMP} %L'article en Français Programmation multiGPU – OpenMP versus MPI
-OpenMP is a shared memory programming API based on threads from
-the same system process. Designed for multiprocessor shared memory UMA or
-NUMA [10], it relies on the execution model SPMD ( Single Program, Multiple Data Stream )
-where the thread "master" and threads "slaves" asynchronously execute their codes
-communicate / synchronize via shared memory [7]. It also helps to build
-the loop parallelism and is very suitable for an incremental code parallelization
-Sequential natively. Threads share some or all of the available memory and can
-have private memory areas [6].
-
-\subsection{MPI} %L'article en Français Programmation multiGPU – OpenMP versus MPI
- The library MPI allows to use a distributed memory architecture. The various processes have their own environment of execution and execute their codes in a asynchronous way, according to the model MIMD (Multiple Instruction streams, Multiple Dated streams); they communicate and synchronize by exchanges of messages [17]. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a programming multi-thread (OpenMP/Pthreads).
+%\subsection{OpenMP} %L'article en Français Programmation multiGPU – OpenMP versus MPI
+%OpenMP is a shared memory programming API based on threads from
+%the same system process. Designed for multiprocessor shared memory UMA or
+%NUMA [10], it relies on the execution model SPMD ( Single Program, Multiple Data Stream )
+%where the thread "master" and threads "slaves" asynchronously execute their codes
+%communicate / synchronize via shared memory [7]. It also helps to build
+%the loop parallelism and is very suitable for an incremental code parallelization
+%Sequential natively. Threads share some or all of the available memory and can
+%have private memory areas [6].
+
+\subsection{MPI}
+ The library MPI allows to use a distributed memory architecture. The various processes have their own environment of execution and execute their codes in a asynchronous way, according to the model MIMD (Multiple Instruction streams, Multiple Dated streams); they communicate and synchronize by exchanges of messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a programming multi-thread (OpenMP/Pthreads).
\subsection{CUDA}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications
- CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA [28]. The
+ CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{NVIDIA12}. The
unit of execution in CUDA is called a thread. Each thread executes the kernel by the streaming processors in parallel. In CUDA,
a group of threads that are executed together is called thread blocks, and the computational grid consists of a grid of thread
blocks. Additionally, a thread block can use the shared memory on a single multiprocessor as while as the grid executes a single
\section{The EA algorithm on single GPU}
\subsection{the EA method}
+
+A cubically convergent iteration method to find zeros of
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The
+Ehrlich-Aberth method contains 4 main steps, presented in what
+follows.
+
+%The Aberth method is a purely algebraic derivation.
+%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
+
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}
+
+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}
+
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}
+
+%Differentiating the rational function $R_{i}(z)$ and applying the
+%Newton method, we have:
+
+%\begin{equation}
+%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
+%\end{equation}
+%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z
+%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.%
+
+
+\subsubsection{Polynomials Initialization}
+The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:
+
+\begin{equation}
+\label{eq:SimplePolynome}
+ p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
+\end{equation}
+
+
+\subsubsection{Vector $Z^{(0)}$ Initialization}
+\label{sec:vec_initialization}
+As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , n.$
+The initial guess is very important since the number of steps needed by the iterative method to reach
+a given approximation strongly depends on it.
+In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
+equi-spaced points on a circle of center 0 and radius r, where r is
+an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
+performed this choice by selecting complex numbers along different
+circles which relies on the result of~\cite{Ostrowski41}.
+
+\begin{equation}
+\label{eq:radiusR}
+%%\begin{align}
+\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
+v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
+%%\end{align}
+\end{equation}
+Where:
+\begin{equation}
+u_{i}=2.|a_{i}|^{\frac{1}{i}};
+v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
+\end{equation}
+
+\subsubsection{Iterative Function}
+The operator used by the Aberth method is corresponding to the
+following equation~\ref{Eq:EA} which will enable the convergence towards
+polynomial solutions, provided all the roots are distinct.
+
+%Here we give a second form of the iterative function used by the Ehrlich-Aberth method:
+
+\begin{equation}
+\label{Eq:EA}
+EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
+\end{equation}
+
+\subsubsection{Convergence Condition}
+The convergence condition determines the termination of the algorithm. It consists in stopping the iterative function when the roots are sufficiently stable. We consider that the method converges sufficiently when:
+
+\begin{equation}
+\label{eq:Aberth-Conv-Cond}
+\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
+\end{equation}
+
+
%\begin{figure}[htbp]
%\centering
% \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
%\label{fig:03}
%\end{figure}
-the Ehrlich-Aberth method is an iterative method, contain 4 steps, start from the initial approximations of all the
-roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
-are lower than a fixed value $ε$
+%the Ehrlich-Aberth method is an iterative method, contain 4 steps, start from the initial approximations of all the
+%roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
+%are lower than a fixed value $ε$
+
+
\subsection{EA parallel implementation on CUDA}
Like any parallel code, a GPU parallel implementation first
requires to determine the sequential tasks and the
\begin{enumerate}
\begin{algorithm}[htpb]
-\label{alg2-cuda}
+\label{alg1-cuda}
%\LinesNumbered
\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-\BlankLine
+%\BlankLine
\item Initialization of the of P\;
\item Initialization of the of Pu\;
%CPUs versus on GPUs.
The initialization values of the vector solution
of the methods are given in %Section~\ref{sec:vec_initialization}.
+
+\subsection{Test with Multi-GPU (CUDA OpenMP) approach}
+
+In this part we performed a set of experiments on Multi-GPU (CUDA OpenMP) approach for full and sparse polynomials of different degrees, compare it with Single GPU (CUDA).
+ \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+
+ 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_openmp}
+ \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}
\label{fig:01}
\end{figure}
+This figure~\ref{fig:01} shows that (CUDA OpenMP) Multi-GPU approach reduce the execution time up to the scale 100 whereas single GPU is of scale 1000 for polynomial who exceed 1,000,000. It shows the advantage to use OpenMP parallel paradigm to connect the performances of several GPUs and solve a polynomial of high degrees.
+
+\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+
+This experiments shows the execution time of the EA algorithm, on single GPU (CUDA) and Multi-GPU (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]{Sparse_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
-\label{fig:02}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+\label{fig:03}
\end{figure}
+The second test with full polynomial shows a very important saving of time, for a polynomial of degrees 1,4M (CUDA OpenMP) approach with 4 GPUs compute and solve it 4 times as fast as single GPU. We notice that curves are positioned one below the other one, more the number of used GPUs increases more the execution time decreases.
+
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Full_openmp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
-\label{fig:03}
+ \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}
+\label{fig:02}
\end{figure}
+
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpivsomp}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse plynomials on GPUs}
\label{fig:05}
\end{figure}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpivsomp}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full}
\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
\label{fig:06}
\end{figure}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{MPI_mpivsomp}
+ \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
\label{fig:07}
\end{figure}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{OMP_mpivsomp}
+ \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
\label{fig:08}
\end{figure}
\section{Conclusion}
-The conclusion goes here.
+The conclusion goes here~\cite{IEEEexample:bibtexdesign}.
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
+%\bibliographystyle{./IEEEtran}
+\bibliography{mybibfile}
+
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
-\begin{thebibliography}{1}
+%\begin{thebibliography}{1}
-\bibitem{IEEEhowto:kopka}
-H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
- 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
+%\bibitem{IEEEhowto:kopka}
+%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
+ % 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
+
+%\bibitem{IEEEhowto:NVIDIA12}
+ %NVIDIA Corporation, \textit{Whitepaper NVIDA’s Next Generation CUDATM Compute
+%Architecture: KeplerTM }, 1st ed., 2012.
-\end{thebibliography}
+%\end{thebibliography}
