X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper2.git/blobdiff_plain/369ec1155d40a2375303ba29bfb493136cf7b944..eabad72090e474064cddadf1b21b1f4fc29ae590:/paper.tex?ds=inline

diff --git a/paper.tex b/paper.tex
index 9c918b8..9bad8af 100644
--- a/paper.tex
+++ b/paper.tex
@@ -326,6 +326,8 @@
 \usepackage[textsize=footnotesize]{todonotes}
 \newcommand{\LZK}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\newcommand{\RC}[2][inline]{%
+  \todo[color=blue!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
 
 
 
@@ -399,9 +401,17 @@ Fax: (888) 555--1212}}
 % As a general rule, do not put math, special symbols or citations
 % in the abstract
 \begin{abstract}
-\LZK{J'ai un peu modifiÃ© l'abstract. Sinon Ã  revoir pour le degrÃ© max des polynÃ´mes aprÃ¨s les tests de raph.}
-Finding roots of polynomials is a very important part of solving real-life problems but it is not so easy for polynomials of high degrees. In this paper, we present two different parallel algorithms of the Ehrlich-Aberth method to find roots of sparse and fully defined polynomials of high degrees. Both algorithms are based on CUDA technology to be implemented on multi-GPU computing platforms but each using different parallel paradigms: OpenMP or MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU devices to find roots of polynomials of degree up-to 1.4 billion. To our knowledge, this is the first paper to present this technology mix to solve such a highly demanding problem in parallel programming. 
-\LZK{Je n'ai pas bien saisi la derniÃ¨re phrase.}
+Finding roots of polynomials is a very important part of solving
+real-life problems but it is not so easy for polynomials of high
+degrees. In this paper, we present two different parallel algorithms
+of the Ehrlich-Aberth method to find roots of sparse and fully defined
+polynomials of high degrees. Both algorithms are based on CUDA
+technology to be implemented on multi-GPU computing platforms but each
+using different parallel paradigms: OpenMP or MPI. The experiments
+show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
+GPU to find roots of polynomials of degree up-to 1.4
+million. Moreover, other experiments show it is possible to find roots
+of polynomials of degree up to 5 millions.
 \end{abstract}
 
 % no keywords
@@ -433,17 +443,17 @@ Finding roots of polynomials is a very important part of solving real-life probl
 
 Finding roots of polynomials of very high degrees arises in many complex problems in various domains such as algebra, biology or physics. A polynomial $p(x)$ in $\mathbb{C}$ in one variable $x$ is an algebraic expression in $x$ of the form:
 \begin{equation}
-p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_0\neq 0. 
+p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0. 
 \end{equation}
 where $\{a_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $a_n\neq0$ then $n$ is called the degree of the polynomial. The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called roots of $p(x)$. Let $\{z_i\}_{1\leq i\leq n}$ be the roots of polynomial $p(x)$, then $p(x)$ can be written as :
 \begin{equation}
  p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
 \end{equation}
-\LZK{Pourquoi $a_0a_n\neq 0$ ?: $a_0$ pour la premiere equation et $a_n$ pour la deuxieme equation }
+%\LZK{Pourquoi $a_0a_n\neq 0$ ?: $a_0$ pour la premiere equation et $a_n$ pour la deuxieme equation }
 
 %The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
 Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous methods, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial roots. These methods start from the initial approximations of all $n$ polynomial roots and give a sequence of approximations that converge to the roots of the polynomial. Two examples of well-known simultaneous methods for root-finding problem of polynomials are  Durand-Kerner method~\cite{Durand60,Kerner66} and Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
-\LZK{Pouvez-vous donner des rÃ©fÃ©rences pour les deux mÃ©thodes?, c'est fait}
+%\LZK{Pouvez-vous donner des rÃ©fÃ©rences pour les deux mÃ©thodes?, c'est fait}
 
 %The first method of this group is Durand-Kerner method:
 %\begin{equation}
@@ -462,7 +472,31 @@ Most of the numerical methods that deal with the polynomial root-finding problem
 %Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
 %the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of %convergence.
 
-The main problem of the simultaneous methods is that the necessary time needed for the convergence increases with the increasing of the polynomial's degree. Many authors have treated the problem of implementing  simultaneous methods in parallel. Freeman~\cite{Freeman89} implemented and compared Durand-Kerner method, Ehrlich-Aberth method and another method of the fourth order of convergence proposed by Farmer and Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83} often diverges, but the first two methods (Durand-Kerner and Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman and Bane~\cite{Freemanall90} considered asynchronous algorithms in which each processor continues to update its approximations even though the latest values of other approximations $z^{k}_{i}$ have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with OpenMP and for a distributed memory one with MPI. They are able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with OpenMP and 135 seconds with MPI only by using 8 personal computers and 2 communications per iteration. The authors showed an interesting speedup comparing to the sequential implementation which takes up-to 3,300 seconds to obtain same results.
+The main problem of the simultaneous methods is that the necessary
+time needed for the convergence increases with the increasing of the
+polynomial's degree. Many authors have treated the problem of
+implementing  simultaneous methods in
+parallel. Freeman~\cite{Freeman89} implemented and compared
+Durand-Kerner method, Ehrlich-Aberth method and another method of the
+fourth order of convergence proposed by Farmer and
+Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials
+of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83}
+often diverges, but the first two methods (Durand-Kerner and
+Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman
+and Bane~\cite{Freemanall90} considered asynchronous algorithms in
+which each processor continues to update its approximations even
+though the latest values of other approximations $z^{k}_{i}$ have not
+been received from the other processors, in contrast with synchronous
+algorithms where it would wait those values before making a new
+iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods
+of parallelization for a shared memory architecture with OpenMP and
+for a distributed memory one with MPI. They are able to compute the
+roots of sparse polynomials of degree 10,000 in 116 seconds with
+OpenMP and 135 seconds with MPI only by using 8 personal computers and
+2 communications per iteration. \RC{si on donne des temps faut donner
+  le proc, comme c'est vieux Ã  mon avis faut supprimer ca, votre avis?} The authors showed an interesting
+speedup comparing to the sequential implementation which takes up-to
+3,300 seconds to obtain same results. 
 \LZK{``only by using 8 personal computers and 2 communications per iteration''. Pour MPI? et Pour OpenMP: Rep: c'est MPI seulement}
 
 Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
@@ -620,7 +654,7 @@ CUDA (Compute Unified Device Architecture) is a parallel computing architecture
 
 %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~\cite{,}, witch will make it possible to converge to the roots solution, provided that all the root are different.
 
-The Ehrlich-Aberth method is a simultaneous method~\cite{} using the following iteration
+The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
 \begin{equation}
 \label{Eq:EA1}
 EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
@@ -666,7 +700,7 @@ This method allows, indeed, to exceed the computation of the polynomials of degr
 
 \begin{equation}
 \label{Log_H2}
-EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
+z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
 p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
 \end{equation}
 
@@ -735,8 +769,8 @@ Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Abert
 
 %\BlankLine
 
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
+\item Initialization of P\;
+\item Initialization of Pu\;
 \item Initialization of the solution vector $Z^{0}$\;
 \item Allocate and copy initial data to the GPU global memory\;
 \item k=0\;
@@ -870,14 +904,14 @@ We study two categories of polynomials: sparse polynomials and full polynomials.
 \end{equation}
 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 prÃ©sentation de l'architecture est Ã  faire ici.
-
+For our test, a cluster of computing with 72 nodes, 1116 cores, 4 cards GPU tesla Kepler K40 are used,
 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.
 The initialization values of the vector solution
-of the methods are given in %Section~\ref{sec:vec_initialization}.
+of the methods are given by Guggenheimer method~\cite{Gugg86} %Section~\ref{sec:vec_initialization}.
 
 \subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
 
@@ -920,9 +954,9 @@ In this part we perform a set of experiments to compare the Multi-GPU (CUDA MPI)
 \label{fig:02}
 \end{figure}
 ~\\
-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.
+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 1,000,000, 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'exÃ©cution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constatÃ© clairement que la courbe Ã  un seul GPU est au-dessus des autres courbes, vue sa consomation en temps d'exÃ¨cution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exÃ¨cution jusqu'Ã  l'echelle 100 pour le polynÃ´mes qui dÃ©passe 1,000,000 tandis que Single GPU est de l'echelle 1000.
+\\cette figure montre 4 courbes de temps d'exÃ©cution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constatÃ© clairement que la courbe Ã  un seul GPU est au-dessus des autres courbes, vue sa consommation en temps d'exÃ¨cution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exÃ¨cution jusqu'Ã  l'echelle 100 pour le polynÃ´mes qui dÃ©passe 1,000,000 tandis que Single GPU est de l'echelle 1000.
 
 \subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}
 
@@ -952,7 +986,7 @@ In this experiment three sparse polynomials of size 200K, 800K and 1,4M are inve
 \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} 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.
+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 the 1,4 millions size, there is a slight advantage for the MPI version.
 
 \subsubsection{Solving full polynomials}
 \begin{figure}[htbp]
