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-% correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor}
-%\usepackage{graphicx}
-\bibliographystyle{IEEEtran}
-% argument is your BibTeX string definitions and bibliography database(s)
-%\bibliography{IEEEabrv,../bib/paper}
-%\bibliographystyle{elsarticle-num}
+\bibliographystyle{IEEEtran}
\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}
+\newcommand{\KG}[2][inline]{%
+ \todo[color=green!10,#1]{\sffamily\textbf{KG:} #2}\xspace}
+\newcommand{\AS}[2][inline]{%
+ \todo[color=orange!10,#1]{\sffamily\textbf{AS:} #2}\xspace}
\begin{document}
-%
-% paper title
-% Titles are generally capitalized except for words such as a, an, and, as,
-% at, but, by, for, in, nor, of, on, or, the, to and up, which are usually
-% not capitalized unless they are the first or last word of the title.
-% Linebreaks \\ can be used within to get better formatting as desired.
-% Do not put math or special symbols in the title.
-\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
+\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
-% author names and affiliations
-% use a multiple column layout for up to three different
-% affiliations
-\author{\IEEEauthorblockN{Michael Shell}
-\IEEEauthorblockA{School of Electrical and\\Computer Engineering\\
-Georgia Institute of Technology\\
-Atlanta, Georgia 30332--0250\\
-Email: http://www.michaelshell.org/contact.html}
+\author{\IEEEauthorblockN{Kahina Guidouche, Abderrahmane Sider }
+ \IEEEauthorblockA{Laboratoire LIMED\\
+ Faculté des sciences exactes\\
+ Université de Bejaia, 06000, Algeria\\
+Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
\and
-\IEEEauthorblockN{Homer Simpson}
-\IEEEauthorblockA{Twentieth Century Fox\\
-Springfield, USA\\
-Email: homer@thesimpsons.com}
-\and
-\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
-\IEEEauthorblockA{Starfleet Academy\\
-San Francisco, California 96678--2391\\
-Telephone: (800) 555--1212\\
-Fax: (888) 555--1212}}
-
-% conference papers do not typically use \thanks and this command
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-%\author{\IEEEauthorblockN{Michael Shell\IEEEauthorrefmark{1},
-%Homer Simpson\IEEEauthorrefmark{2},
-%James Kirk\IEEEauthorrefmark{3},
-%Montgomery Scott\IEEEauthorrefmark{3} and
-%Eldon Tyrell\IEEEauthorrefmark{4}}
-%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
-%Georgia Institute of Technology,
-%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
-%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
-%Email: homer@thesimpsons.com}
-%\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\
-%Telephone: (800) 555--1212, Fax: (888) 555--1212}
-%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}
-
-
-
-
-% use for special paper notices
-%\IEEEspecialpapernotice{(Invited Paper)}
+\IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
+\IEEEauthorblockA{FEMTO-ST Institute\\
+ University of Bourgogne Franche-Comte, France\\
+Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
-
-
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\maketitle
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-% in the abstract
\begin{abstract}
-\LZK{J'ai un peu modifié l'abstract. Sinon à revoir pour le degré max des polynômes après 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 dernière 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
+\LZK{Faut pas mettre des keywords?}
-
-
-% For peer review papers, you can put extra information on the cover
-% page as needed:
-% \ifCLASSOPTIONpeerreview
-% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
-% \fi
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-% For peerreview papers, this IEEEtran command inserts a page break and
-% creates the second title. It will be ignored for other modes.
\IEEEpeerreviewmaketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
-Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and expressing any outcome as a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree $n$ having $n$ coefficients in the complex plane $\mathbb{C}$ is:
-\begin{equation}
-p(x)=\sum_{i=0}^{n}{a_ix^i}.
-\end{equation}
-\LZK{Dans ce cas le polynôme a $n+1$ coefficients et non pas $n$!}
-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 zeros or roots of $p$. If zeros are $\{\alpha_{i}\}_{1\leq i\leq n}$, then $p(x)$ can be written as :
-\begin{equation}
- p(x)=a_n\prod_{i=1}^n(x-\alpha_i), a_0 a_n\neq 0.
-\end{equation}
-\LZK{C'est $a_n\neq 0$ (polynôme de degré $n$) et non pas $a_0 a_n\neq 0$, non?}
-\LZK{Est-ce $\alpha_i$ sont les $z_i$ définis dans la suite du papier?}
-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 ones, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial zeros. 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. The first method of this group is Durand-Kerner method:
+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}
-\label{DK}
- DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})}, i = 1, \ldots, n,
+p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0,
\end{equation}
-where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the iteration $k$. Another method discovered by Borsch-Supan~\cite{ Borch-Supan63} and also described by Ehrlich~\cite{Ehrlich67} and Aberth~\cite{Aberth73} uses a different iteration form as follows:
-%%\begin{center}
+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}
-\label{Eq:EA}
- EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, \ldots, n,
+ p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
\end{equation}
-%%\end{center}
-where $p'(z)$ is the polynomial derivative of $p$ evaluated in the point $z$.
-
-%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 DK, EA 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 (DK and EA) 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 $z^{k}_{i}$ 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 \textit{OpenMP} and for a distributed memory one with \textit{MPI}. They are able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only by using 8 personal computers and 2 communications per iteration.
-\LZK{je suppose que c'est pour la version mpi (only by using 8 personal computers and 2 communications per iteration). A t on utilisé le même nombre de procs pour les deux versions openmp et mpi} The authors showed an interesting speedup comparing to the sequential implementation that takes up-to 3,300 seconds to obtain same results.
-
-Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of 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 and 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.
-
-%Finding polynomial roots rapidly and accurately is the main objective of our work. We consider two architectures: shared-memory computers with OpenMP API and distributed-memory computers with MPI API. The first approach is based on threads from the same system process, with each thread attached to one GPU and after the various memory allocations, each thread launches its part of computations. To do this we must first load on the GPU required data and after the computations are carried, repatriate the result on the host. The second approach i.e distributed memory with MPI relies on the MPI library which is often used for parallel programming~\cite{Peter96} in cluster systems because it is a message-passing programming language. Each GPU is attached to one MPI process, and a loop is in charge of the distribution of tasks between the MPI processes. This solution can be used on one GPU, or executed on a distributed cluster of GPUs, employing the Message Passing Interface (MPI) to communicate between separate CUDA cards. This solution permits scaling of the problem size to larger classes than would be possible on a single device and demonstrates the performance which users might expect from future HPC architectures where accelerators are deployed.
-
-Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. We consider two architectures: shared memory and distributed memory computers. The first parallel algorithm is implemented on shared memory computers by using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory. The second parallel algorithm uses the MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. This latter approach is more used on distributed memory clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
-\LZK{Cette partie est réécrite. \\ Sinon qu'est ce qui a été fait pour l'accuracy dans ce papier (Finding polynomial roots rapidly and accurately is the main objective of our work.)?}
-\LZK{Les contributions ne sont pas définies !!}
-
-%This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section~\ref{sec3} we present EA algorithm on single GPU. In section~\ref{sec4} we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In sectioné\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}
-The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA.
-\LZK{A revoir toute cette organization}
+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}.
+
+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.
+\RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?}
+\LZK{Supprimons ces détails et mettons une référence s'il y en a une}
+
+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.
+
+In this paper we propose the parallelization of Ehrlich-Aberth method which has a good convergence and it is suitable to be implemented in parallel computers. We use two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA-MPI and CUDA-OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
+\LZK{J'ai ajouté une phrase pour justifier notre choix de la méthode Ehrlich-Aberth. A revérifier.}
+ \begin{itemize}
+
+\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a shared memory using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory.
+\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a distributed memory using MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications.
+ \end{itemize}
+This latter approach is more used on clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
+\LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.}
+
+The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA. In Section~\ref{sec3} we present the implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on multiple GPUs using the OpenMP and MPI approaches. In section~\ref{sec5} we present our experiments and discuss them. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.
+%\LZK{A revoir toute cette organization: je viens de la revoir}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Parallel programming models}
\label{sec2}
-\LZK{Toute cette section a été réécrite. Donc à relire et à améliorer si possible.}
-
-In this section we present three different parallel programming models: OpenMP, MPI and CUDA.
+Our objective consists in implementing a root-finding algorithm of polynomials on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We investigate two parallel paradigms: OpenMP and MPI. In this case, the GPU indices are defined according to the identifiers of the OpenMP threads or the ranks of the MPI processes. In this section we present the parallel programming models: OpenMP, MPI and CUDA.
\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 allocated 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 usage of OpenMP threads and managed data explicitly done with MPI can be considered, this approcache undermines the advantages of OpenMP.
-%\subsection{OpenMP}
-%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].
-OpenMP (Open Multi-processing) is an application programming interface for shared memory parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications.
+OpenMP (Open Multi-processing) is an application programming interface for parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications.
\subsection{MPI}
-%The MPI (Message Passing Interface) library allows to create computer programs that run on a distributed memory architecture. The various processes have their own environment of execution and execute their code in a asynchronous way, according to the MIMD model (Multiple Instruction streams, Multiple Data streams); they communicate and synchronize by exchanging messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a multi-thread programming environment like OpenMP or Pthreads.
+
MPI (Message Passing Interface) is a portable message passing style of the parallel programming designed especially for the distributed memory architectures~\cite{Peter96}. In most MPI implementations, a computation contains a fixed set of processes created at the initialization of the program in such way one process is created per processor. The processes synchronize their computations and communicate by sending/receiving messages to/from other processes. In this case, the data are explicitly exchanged by message passing while the data exchanges are implicit in a multithread programming model like OpenMP and Pthreads. However in the MPI programming model, the processes may either execute different programs referred to as multiple program multiple data (MPMD) or every process executes the same program (SPMD). The MPI approach is one of most used HPC programming model to solve large scale and complex applications.
\subsection{CUDA}
-%CUDA (is an acronym of the Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA10}.The unit of execution in CUDA is called a thread. Each thread executes a kernel by the streaming processors in parallel. In CUDA, a group of threads that are executed together is called a thread block, and the computational grid consists of a grid of thread blocks. Additionally, a thread block can use the shared memory on a single multiprocessor while the grid executes a single CUDA program logically in parallel. Thus in CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency and a proper usage of shared memory, since it can be shared only in a thread block scope. The effective bandwidth of each memory space depends on the memory access pattern. Since the global memory has lower bandwidth than the shared memory, the global memory accesses should be minimized.
-
-CUDA (Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA15} for GPUs. It provides a high level GPGPU-based programming model to program GPUs for general purpose computations and non-graphic applications. The GPU is viewed as an accelerator such that data-parallel operations of a CUDA program running on a CPU are off-loaded onto GPU and executed by this later. The data-parallel operations executed by GPUs are called kernels. The same kernel is executed in parallel by a large number of threads organized in grids of thread blocks, such that each GPU multiprocessor executes one or more thread blocks in SIMD fashion (Single Instruction, Multiple Data) and in turn each core of the multiprocessor executes one or more threads within a block. Threads within a block can cooperate by sharing data through a fast shared memory and coordinate their execution through synchronization points. In contrast, within a grid of thread blocks, there is no synchronization at all between blocks. The GPU only works on data filled in the global memory and the final results of the kernel executions must be transferred out of the GPU. In the GPU, the global memory has lower bandwidth than the shared memory associated to each multiprocessor. Thus in the CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency, a proper usage of the shared memory and the global memory accesses should be minimized.
-
+CUDA (Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA15} for GPUs. It provides a high level GPGPU-based programming model to program GPUs for general purpose computations and non-graphic applications. The GPU is viewed as an accelerator such that data-parallel operations of a CUDA program running on a CPU are off-loaded onto GPU and executed by this later. The data-parallel operations executed by GPUs are called kernels. The same kernel is executed in parallel by a large number of threads organized in grids of thread blocks, such that each GPU multiprocessor executes one or more thread blocks in SIMD fashion (Single Instruction, Multiple Data) and in turn each core of the multiprocessor executes one or more threads within a block. Threads within a block can cooperate by sharing data through a fast shared memory and coordinate their execution through synchronization points. In contrast, within a grid of thread blocks, there is no synchronization at all between blocks. The GPU only works on data filled in the global memory and the final results of the kernel executions must be transferred out of the GPU. In the GPU, the global memory has lower bandwidth than the shared memory associated to each multiprocessor. Thus in the CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency and a proper usage of the shared memory, and the global memory accesses should be minimized.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
-
-\section{The EA algorithm on a single GPU}
+\section{The Ehrlich-Aberth algorithm on a GPU}
\label{sec3}
-\subsection{The EA method}
-A cubically convergent iteration method to find zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. The
-Ehrlich-Aberth (EA is short) 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.%
+\subsection{The Ehrlich-Aberth method}
+The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
+\begin{equation}
+\label{Eq:EA1}
+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,\ldots,n
+\end{equation}
-\subsubsection{Polynomials Initialization}
-The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:
+This method contains 4 steps. The first step consists in the initializing the polynomial. The second step initializes the solution vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all distinct from each other. In step 3, the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is applied. In our case, the Ehrlich-Aberth is applied as in~(\ref{Eq:EA1}). Iterations of the Ehrlich-Aberth method will converge to the roots of the considered polynomial. In order to stop the iterative function, a stop condition is applied, this is the 4th step. This condition checks that all the root modules are lower than a fixed value $\epsilon$.
\begin{equation}
-\label{eq:SimplePolynome}
- p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
+\label{eq:Aberth-Conv-Cond}
+\forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
\end{equation}
+\subsection{Improving Ehrlich-Aberth method}
+With high degree polynomials, the Ehrlich-Aberth method suffers from floating point overflows due to the mantissa of floating points representations. This induces errors in the computation of $p(z)$ when $z$ is large.
+
-\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-distant 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}.
+In order to solve this problem, we propose to modify the iterative
+function by using the logarithm and the exponential of a complex and
+we propose a new version of the Ehrlich-Aberth method. This method
+allows us to exceed the computation of the polynomials of degree
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
\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}
+\label{Log_H2}
+z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
\end{equation}
-Where:
+
+where:
+
\begin{equation}
-u_{i}=2.|a_{i}|^{\frac{1}{i}};
-v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
+\label{Log_H1}
+Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
\end{equation}
-\subsubsection{Iterative Function}
-The operator used by the Aberth method corresponds to the
-equation~\ref{Eq:EA1}, it enables the convergence towards
-the polynomials zeros, 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:EA1}
-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}
+Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower values in absolute values~\cite{Karimall98}.
-\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}
+\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
+Our objective consists in implementing a root finding polynomial
+algorithm on multiple GPUs. To this end, it is primordial to know how
+to manage CUDA contexts of different GPUs. A direct method for
+controlling the various GPUs is to use as many threads or processes as
+GPU devices. We can choose the GPU index based on the identifier of
+OpenMP thread or the rank of the MPI process. Both approaches will be
+investigated. \LZK{Répétition! Le même texte est déjà écrit comme
+ intro dans la section II. Sinon ici on parle seulement de
+ l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
+ revoir après, quand les 2 parties suivantes seront plus stables}}
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
-%\caption{The Ehrlich-Aberth algorithm on single GPU}
-%\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 $ε$
-
-
-\subsection{EA parallel implementation on CUDA}
-Like any parallel code, a GPU parallel implementation first
-requires to determine the sequential tasks and the
-parallelizable parts of the sequential version of the
-program/algorithm. In our case, all the operations that are easy
-to execute in parallel must be made by the GPU to accelerate
-the execution of the application, like the step 3 and step 4. On the other hand, all the
-sequential operations and the operations that have data
-dependencies between threads or recursive computations must
-be executed by only one CUDA or CPU thread (step 1 and step 2). Initially, we specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block.
-
-The code is organzed by what is named kernels, portions o code that are run on GPU devices. For step 3, there are two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde one is named
-\textit{update} and is used to update the $Z^{K}$ vector. For step 4, a kernel
-tests the convergence of the method. In order to
-compute the function H, we have two possibilities: either to use
-the Jacobi mode, or the Gauss-Seidel mode of iterating which uses the
-most recent computed roots. It is well known that the Gauss-
-Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
-parallelize the code, we created kernels and many functions to
-be executed on the GPU for all the operations dealing with the
-computation on complex numbers and the evaluation of the
-polynomials. As said previously, we managed both functions
-of evaluation of a polynomial: the normal method, based on
-the method of Horner and the method based on the logarithm
-of the polynomial. All these methods were rather long to
-implement, as the development of corresponding kernels with
-CUDA is longer than on a CPU host. This comes in particular
-from the fact that it is very difficult to debug CUDA running
-threads like threads on a CPU host. In the following paragraph
-Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Aberth method.
-
-\begin{enumerate}
-\begin{algorithm}[htpb]
-\label{alg1-cuda}
-%\LinesNumbered
-\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-%\BlankLine
+Like any parallel code, a GPU parallel implementation first requires to determine the sequential code and the data-parallel operations of a algorithm. In fact, all the operations that are easy to execute in parallel must be made by the GPU to accelerate the execution, like the steps 3 and 4. On the other hand, all the sequential operations and the operations that have data dependencies between CUDA threads or recursive computations must be executed by only one CUDA thread or a CPU thread (the steps 1 and 2).\LZK{La méthode est déjà mal présentée, dans ce cas c'est encore plus difficile de comprendre que représentent ces différentes étapes!} Initially, we specify the organization of parallel threads by specifying the dimension of the grid \verb+Dimgrid+, the number of blocks per grid \verb+DimBlock+ and the number of threads per block.
+
+The code is organized as kernels which are parts of code that are run on GPU devices. For step 3, there are two kernels, the first is named \textit{save} is used to save vector $Z^{K-1}$ and the second one is
+named \textit{update} and is used to update the $Z^{K}$ vector. For
+step 4, a kernel tests the convergence of the method. In order to
+compute the function H, we have two possibilities: either to use the
+Jacobi mode, or the Gauss-Seidel mode of iterating which uses the most
+recent computed roots. It is well known that the Gauss-Seidel mode
+converges more quickly. So, we use Gauss-Seidel iterations. To
+parallelize the code, we create kernels and many functions to be
+executed on the GPU for all the operations dealing with the
+computation on complex numbers and the evaluation of the
+polynomials. As said previously, we manage both functions of
+evaluation: the normal method, based on the method of
+Horner and the method based on the logarithm of the polynomial. All
+these methods were rather long to implement, as the development of
+corresponding kernels with CUDA is longer than on a CPU host. This
+comes in particular from the fact that it is very difficult to debug
+CUDA running threads like threads on a CPU host. In the following
+paragraph Algorithm~\ref{alg1-cuda} shows the GPU parallel
+implementation of Ehrlich-Aberth method.
+\LZK{Vaut mieux expliquer l'implémentation en faisant référence à l'algo séquentiel que de parler des différentes steps.}
+
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\item Allocate and copy initial data to the GPU global memory\;
-\item k=0\;
-\While {$\Delta z_{max} > \epsilon$}{
-\item Let $\Delta z_{max}=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu)$\;
-\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+\begin{algorithm}[htpb]
+\LinesNumbered
+\SetAlgoNoLine
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
+\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold)}
+\KwOut{$Z$ (solution vector of roots)}
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Copy $P$, $P'$ and $Z$ from CPU to GPU\;
+\While{\emph{not convergence}}{
+ $Z^{prev}$ = KernelSave($Z,n$)\;
+ $Z$ = KernelUpdate($P,P',Z,n$)\;
+ $\Delta Z$ = KernelComputeError($Z,Z^{prev},n$)\;
+ $\Delta Z_{max}$ = CudaMaxFunction($\Delta Z,n$)\;
+ TestConvergence($\Delta Z_{max},\epsilon$)\;
}
-\item Copy results from GPU memory to CPU memory\;
+Copy $Z$ from GPU to CPU\;
+\label{alg1-cuda}
+\LZK{J'ai modifié l'algo. Sinon, est ce qu'on doit mettre en paramètre
+ $Z^{prev}$ ou $Z$ tout court (dans le cas où on exploite
+ l'asynchronisme des threads cuda!) pour le Kernel\_Update? }
+\RC{Le $Z_{prev}$ sert à calculer l'erreur donc j'ai remis Z. La ligne
+avec TestConvergence ca fait une ligne de plus.}
\end{algorithm}
-\end{enumerate}
-~\\
-
-\section{The EA algorithm on Multi-GPU}
+\section{The EA algorithm on Multiple GPUs}
\label{sec4}
-\subsection{MGPU : 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).
-
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{OpenMP-CUDA}
-%\caption{The OpenMP-CUDA architecture}
-%\label{fig:03}
-%\end{figure}
-%Each thread OpenMP compute the kernels on GPUs,than after each iteration they copy out the data from GPU memory to CPU shared memory. The kernels are re-runs is up to the roots converge sufficiently. Here are below the corresponding algorithm:
-
-$num\_gpus$ OpenMP threads are created using \verb=omp_set_num_threads();=function (step $3$, Algorithm \ref{alg2-cuda-openmp}), the shared memory is created using \verb=#pragma omp parallel shared()= OpenMP function (line $5$, Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates memory and copies initial data from CPU memory to GPU global memory, executes the kernels on GPU, but computes only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-openmp}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After each iteration, all OpenMP threads synchronize using \verb=#pragma omp barrier;= to gather all the correct values of $\Delta z$, thus allowing the computation the maximum stop condition on vector $\Delta z$ (line 12, Algorithm \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots sufficiently converge.
-\begin{enumerate}
+\subsection{an OpenMP-CUDA approach}
+Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
+OpenMP and CUDA programming model. All the data are shared with
+OpenMP amoung all the OpenMP threads. The shared data are the solution
+vector $Z$, the polynomial to solve $P$, and the error vector $\Delta
+z$. The number of OpenMP threads is equal to the number of GPUs, each
+OpenMP thread binds to one GPU, and it controls a part of the shared
+memory. More precisely each OpenMP thread will be responsible to
+update its owns part of the vector Z. This part is call $Z_{loc}$ in
+the following. Then all GPUs will have a grid of computation organized
+according to the device performance and the size of data on which it
+runs the computation kernels.
+
+To compute one iteration of the EA method each GPU performs the
+followings steps. First roots are shared with OpenMP and the
+computation of the local size for each GPU is performed (lines 5-7 in
+Algo\ref{alg2-cuda-openmp}). Each thread starts by copying all the
+previous roots inside its GPU (line 9). Then each GPU will copy the
+previous roots (line 10) and it will compute an iteration of the EA
+method on its own roots (line 11). For that all the other roots are
+used. The convergence is checked on the new roots (line 12). At the end
+of an iteration, the updated roots are copied from the GPU to the
+CPU (line 14) by direcly updating its own roots in the shared memory
+arrays containing all the roots.
+
+
+
\begin{algorithm}[htpb]
+\LinesNumbered
+\SetAlgoNoLine
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
+\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold), $ngpu$ (number of GPUs)}
+\KwOut{$Z$ (solution vector of roots)}
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Start of a parallel part with OpenMP ($Z$, $\Delta Z$, $\Delta
+Z_{max}$, $P$, $P'$ are shared variables)\;
+$id_{gpu}$ = cudaGetDevice()\;
+$n_{loc}$ = $n/ngpu$ (local size)\;
+%$idx$ = $id_{gpu}\times n_{loc}$ (local offset)\;
+Copy $P$, $P'$ from CPU to GPU\;
+\While{\emph{not convergence}}{
+ Copy $Z$ from CPU to GPU\;
+ $Z^{prev}$ = KernelSave($Z,n$)\;
+ $Z_{loc}$ = KernelUpdate($P,P',Z^{prev},n_{loc}$)\;
+ $\Delta Z_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc},n_{loc}$)\;
+ $\Delta Z_{max}[id_{gpu}]$ = CudaMaxFunction($\Delta Z_{loc},n_{loc}$)\;
+ Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
+ $max$ = MaxFunction($\Delta Z_{max},ngpu$)\;
+ TestConvergence($max,\epsilon$)\;
+}
\label{alg2-cuda-openmp}
-%\LinesNumbered
-\caption{CUDA-OpenMP Algorithm to find roots with the Ehrlich-Aberth method}
+\LZK{J'ai modifié l'algo. Le $P$ est mis shared. Qu'en est-il pour
+ $P'$?}\RC{Je l'ai rajouté. Bon sinon le n\_loc ne remplace pas
+ vraiment un offset et une taille mais bon... et là il y a 4 lignes
+ pour la convergence, c'est bcp ... Zloc, Zmax, max et
+ testconvergence. On pourrait faire mieux}
+\end{algorithm}
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z$ ( Vector of errors for stop condition), $num_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
-\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
-\BlankLine
-\item Initialization of P\;
-\item Initialization of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\verb=omp_set_num_threads(num_gpus);=
-\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
-\verb=cudaGetDevice(gpu_id);=
-\item Allocate and copy initial data from CPU memory to the GPU global memories\;
-\item index= $Size/num\_gpus$\;
-\item k=0\;
-\While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
-%\verb=#pragma omp barrier;=
-\item error= Max($\Delta z$)\;
-}
-
-\item Copy results from GPU memories to CPU memory\;
-\end{algorithm}
-\end{enumerate}
-~\\
+\subsection{a MPI-CUDA approach}
+Our parallel implementation of EA to find root of polynomials using a
+CUDA-MPI approach follows a similar computing approach to the one used
+in CUDA-OpenMP. Each process is responsible to compute its own part of
+roots using all the roots computed by other processors at the previous
+iteration. The difference between both approaches lies in the way
+processes communicate and exchange data. With MPI processors need to
+send and receive data explicitely. So in
+Algorithm~\ref{alg2-cuda-mpi}, after the initialization all the
+processors have the same $Z$ vector. Then they need to compute the
+parameters used by the $MPI\_AlltoAll$ routines (line 4). In practise,
+each processor needs to compute its offset and its local size. Then
+processors need to allocate memory on their GPU (line 5). At the
+beginning of each iteration, a processor starts by transfering the
+whole vector Z from the CPU to the GPU (line 7). Then only the local
+part of $Z^{prev}$ is saved (line 8). After that, a processor is able
+to compute its own roots (line 9). Next, the local error can be
+computed (ligne 10) and the global error (line 11). Then the local
+roots are transfered from the GPU memory to the CPU memory (line 12)
+before being exchanged between all processors (linge 13) in order to
+give to all processors the last version of the roots. If the
+convergence is not statisfied, an new iteration is executed.
-\subsection{Multi-GPU : an MPI-CUDA approach}
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
-%\caption{The MPI-CUDA architecture }
-%\label{fig:03}
-%\end{figure}
-Our parallel implementation of EA to find root of polynomials using a CUDA-MPI approach is a data parallel approach. It splits input data of the polynomial to solve among MPI processes. In Algorithm \ref{alg2-cuda-mpi}, input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $ZPrev$, and the value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the degree of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $\lceil n/p \rceil$ roots to find per MPI process, for each $Z$ and $ZPrec$. Consequently, each MPI process of rank $k$ will have its own solution vector $Z_{k}$ and $ZPrec$, the error related to the stop condition $\Delta z_{k}$, enabling each MPI process to compute $\lceil n/p \rceil$ roots.
-Since a GPU works only on data already allocated in its memory, all local input data, $Z_{k}$, $ZPrec$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterwards, the same EA algorithm (Algorithm \ref{alg1-cuda}) is run by all processes but on different polynomial subset of roots $ p(x)_{k}=\sum_{i=1}^{n} a_{i}x^{i}, k=1,...,p$. Each MPI process executes the loop \verb=(While(...)...do)= containing the CUDA kernels but each MPI process computes only its own portion of the roots according to the rule ``''owner computes``''. The local range of roots is indicated with the \textit{index} variable initialized at (line 5, Algorithm \ref{alg2-cuda-mpi}), and passed as an input variable to $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize (\verb=MPI_Allreduce= function) by a reduction on $\Delta z_{k}$ in order to compute the maximum error related to the stop condition. Finally, processes copy the values of new computed roots from GPU memories to CPU memories, then communicate their results to other processes with \verb=MPI_Alltoall= broadcast. If the stop condition is not verified ($error > \epsilon$) then processes stay withing the loop \verb= while(...)...do= until all the roots sufficiently converge.
-\begin{enumerate}
\begin{algorithm}[htpb]
\label{alg2-cuda-mpi}
-%\LinesNumbered
+\LinesNumbered
\caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
+ threshold), P (Polynomial to solve), P' (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition)}
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+\KwOut {$Z$ (Solution root's vector)}
\BlankLine
-\item Initialization of P\;
-\item Initialization of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\item Allocate and copy initial data from CPU memories to GPU global memories\;
-\item $index= Size/num_gpus$\;
-\item k=0\;
+Initialization of P\;
+Initialization of Pu\;
+Initialization of the solution vector $Z^{0}$\;
+Computation of the parameters for the $MPI\_AlltoAll$\;
+Allocate memory to GPU\;
\While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
-\item ComputeMaxError($\Delta z$,error)\;
-\item Copy results from GPU memories to CPU memories\;
-\item Send $Z[id]$ to all processes\;
-\item Receive $Z[j]$ from every other process j\;
+copy Z from CPU to GPU\;
+$Z^{Prev}_{loc}=kernel\_save(Z_{loc})$\;
+$Z_{loc}=kernel\_update(Z,P,P')$\;
+$\Delta z=kernel\_testConv(Z_{loc},Z^{prev}_{loc})$\;
+$error=MPI\_Reduce(\Delta z)$\;
+Copy $Z_{loc}$ from GPU to CPU\;
+$Z=MPI\_AlltoAll(Z_{loc})$\;
}
+\RC{A uniformiser avec les autres algos, mais les grandes lignes sont là}
\end{algorithm}
-\end{enumerate}
-~\\
+
\section{Experiments}
\label{sec5}
\begin{equation}
{\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
\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 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 $10^{-7}$.
-%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}.
+For our test, 4 cards GPU tesla Kepler K40 are used. In order to
+evaluate both the GPU and Multi-GPU approaches, we performed a set of
+experiments on a single GPU and multiple GPUs using OpenMP or MPI with
+the EA algorithm, for both sparse and full polynomials of different
+sizes. All experimental results obtained are perfomed with double
+precision float data and the convergence threshold of the EA method is
+set to $10^{-7}$. The initialization values of the vector solution of
+the methods are given by Guggenheimer method~\cite{Gugg86}.
+
-\subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
+\subsection{Evaluation of the 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}
+Here we report some experiments witt full and sparse polynomials of
+different degrees with multiple GPUs.
+\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
-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.
+In this experiments we report the execution time of the EA algorithm, on single GPU and multi-GPUs 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 to
+ solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
\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.
+Figure~\ref{fig:01} shows that the CUDA-OpenMP approach scales well
+with multiple GPUs. This version allows us to solve sparse 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 times of the EA method to solve full polynomials on multiple GPUs}
-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.
+These experiments show the execution times 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}
-\label{fig:03}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to
+ solve full polynomials on multiple GPUs with CUDA-OpenMP.}
+\label{fig:02}
\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.
+In Figure~\ref{fig:02}, we can observe that with full polynomials the EA version with
+CUDA-OpenMP scales also well. Using 4 GPUs allows us to achieve a
+quasi-linear speedup.
-\subsection{Evaluting 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.
+\subsection{Evaluation of the CUDA-MPI approach}
+In this part we perform some experiments to evaluate the CUDA-MPI
+approach to solve 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 times of the EA method to solve sparse polynomials on multiple GPUs}
\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}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to
+ solve sparse polynomials on multiple GPUs with CUDA-MPI.}
+\label{fig:03}
\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.
-%%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}
+Figure~\ref{fig:03} shows the execution times of te EA algorithm,
+for a single GPU, and multiple GPUs (2, 3, 4) with the CUDA-MPI approach.
+
+\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 multiple GPUs with CUDA-MPI.}
\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.
-
-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.
-
-
-
-
-%\begin{figure}[htbp]
-%\centering
- % \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}
-%\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}
-%\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}
-%\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}
-
-% An example of a floating figure using the graphicx package.
-% Note that \label must occur AFTER (or within) \caption.
-% For figures, \caption should occur after the \includegraphics.
-% Note that IEEEtran v1.7 and later has special internal code that
-% is designed to preserve the operation of \label within \caption
-% even when the captionsoff option is in effect. However, because
-% of issues like this, it may be the safest practice to put all your
-% \label just after \caption rather than within \caption{}.
-%
-% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
-% option should be used if it is desired that the figures are to be
-% displayed while in draft mode.
-%
-%\begin{figure}[!t]
-%\centering
-%\includegraphics[width=2.5in]{myfigure}
-% where an .eps filename suffix will be assumed under latex,
-% and a .pdf suffix will be assumed for pdflatex; or what has been declared
-% via \DeclareGraphicsExtensions.
-%\caption{Simulation results for the network.}
-%\label{fig_sim}
-%\end{figure}
-
-% Note that the IEEE typically puts floats only at the top, even when this
-% results in a large percentage of a column being occupied by floats.
-
-
-% An example of a double column floating figure using two subfigures.
-% (The subfig.sty package must be loaded for this to work.)
-% The subfigure \label commands are set within each subfloat command,
-% and the \label for the overall figure must come after \caption.
-% \hfil is used as a separator to get equal spacing.
-% Watch out that the combined width of all the subfigures on a
-% line do not exceed the text width or a line break will occur.
-%
-%\begin{figure*}[!t]
-%\centering
-%\subfloat[Case I]{\includegraphics[width=2.5in]{box}%
-%\label{fig_first_case}}
-%\hfil
-%\subfloat[Case II]{\includegraphics[width=2.5in]{box}%
-%\label{fig_second_case}}
-%\caption{Simulation results for the network.}
-%\label{fig_sim}
-%\end{figure*}
-%
-% Note that often IEEE papers with subfigures do not employ subfigure
-% captions (using the optional argument to \subfloat[]), but instead will
-% reference/describe all of them (a), (b), etc., within the main caption.
-% Be aware that for subfig.sty to generate the (a), (b), etc., subfigure
-% labels, the optional argument to \subfloat must be present. If a
-% subcaption is not desired, just leave its contents blank,
-% e.g., \subfloat[].
-
-
-% An example of a floating table. Note that, for IEEE style tables, the
-% \caption command should come BEFORE the table and, given that table
-% captions serve much like titles, are usually capitalized except for words
-% such as a, an, and, as, at, but, by, for, in, nor, of, on, or, the, to
-% and up, which are usually not capitalized unless they are the first or
-% last word of the caption. Table text will default to \footnotesize as
-% the IEEE normally uses this smaller font for tables.
-% The \label must come after \caption as always.
-%
-%\begin{table}[!t]
-%% increase table row spacing, adjust to taste
-%\renewcommand{\arraystretch}{1.3}
-% if using array.sty, it might be a good idea to tweak the value of
-% \extrarowheight as needed to properly center the text within the cells
-%\caption{An Example of a Table}
-%\label{table_example}
-%\centering
-%% Some packages, such as MDW tools, offer better commands for making tables
-%% than the plain LaTeX2e tabular which is used here.
-%\begin{tabular}{|c||c|}
-%\hline
-%One & Two\\
-%\hline
-%Three & Four\\
-%\hline
-%\end{tabular}
-%\end{table}
-
-
-% Note that the IEEE does not put floats in the very first column
-% - or typically anywhere on the first page for that matter. Also,
-% in-text middle ("here") positioning is typically not used, but it
-% is allowed and encouraged for Computer Society conferences (but
-% not Computer Society journals). Most IEEE journals/conferences use
-% top floats exclusively.
-% Note that, LaTeX2e, unlike IEEE journals/conferences, places
-% footnotes above bottom floats. This can be corrected via the
-% \fnbelowfloat command of the stfloats package.
+In Figure~\ref{fig:04}, we can also observe that the CUDA-MPI approach
+is also efficient to solve full polynimails on multiple GPUs.
+\subsection{Comparison of the CUDA-OpenMP and the CUDA-MPI approaches}
+In the previuos section we saw that both approches are very effecient
+to reduce the execution times the sparse and full polynomials. In
+this section we try to compare these two approaches.
-\section{Conclusion}
-\label{sec5}
-In this paper, we have presented a parallel implementation of Ehrlich-Aberth algorithm for solving full and sparse polynomials, on single GPU with CUDA and on multiple GPUs using two parallel paradigms : shared memory with OpenMP and distributed memory with MPI. These architectures were addressed by a CUDA-OpenMP approach and CUDA-MPI approach, respectively.
-The experiments show that, using parallel programming model like (OpenMP, MPI), we can efficiently manage multiple graphics cards to work together to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree 1,000,000, four times faster than on single GPU, that is a quasi-linear speedup.
-
+\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{Execution times to solvs sparse polynomials of three
+ distinct sizes on multiple GPUs using MPI and OpenMP with the
+ Ehrlich-Aberth method}
+\label{fig:05}
+\end{figure}
+In Figure~\ref{fig:05} there is one curve for CUDA-MPI and another one
+for CUDA-OpenMP. We can see that the results are quite similar between
+OpenMP and MPI for the polynomials size of 200K. For the size of 800K,
+the MPI version is a little bit 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]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full}
+\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 when it comes to full polynomials, both approaches are almost equivalent.
-%In future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model
+\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
-Our next objective is to extend the model presented here at clusters of nodes featuring multiple GPUs, with a three-level scheme: inter-node communication via MPI processes (distributed memory), management of multi-GPU node by OpenMP threads (shared memory).
+In this experiment we compare the execution time of the EA algorithm
+according to the number of GPUs to solve sparse and full
+polynomials on multiples GPUs 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{Execution times to solve 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 times with sparse
+polynomial are very low compared to full polynomials. With sparse
+polynomials the number of monomials is reduced, consequently the number
+of operations is reduced and the execution time decreases.
-%present a communication approach between multiple GPUs. The comparison between MPI and OpenMP as GPUs controllers shows that these
-%solutions can effectively manage multiple graphics cards to work together
-%to solve the same problem
+\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{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
+\label{fig:08}
+\end{figure}
- %than we have presented two communication approach between multiple GPUs.(CUDA-OpenMP) approach and (CUDA-MPI) approach, in the objective to manage multiple graphics cards to work together and solve the same problem. in the objective to manage multiple graphics cards to work together and solve the same problem.
+Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact follows the same pattern, a difference in execution time in favor of the sparse polynomials.
+\subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
+These experiments report the execution times of the EA method for
+sparse and full polynomials ranging from 1,000,000 to 5,000,000.
+\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 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 approaches are scalable
+and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both
+approaches give very similar results.
+%SIDER JE viens de virer \c ca For sparse polynomials here are a noticeable difference in favour of MPI when the degree is
+%above 4 millions. Between 1 and 3 millions, OpenMP is more effecient.
+%Under 1 million, OpenMPI and MPI are almost equivalent.
+%SIDER : il faut une explication sur les différences ici aussi.
+
-% conference papers do not normally have an appendix
+\section{Conclusion}
+\label{sec6}
+In this paper, we have presented a parallel implementation of
+Ehrlich-Aberth algorithm to solve full and sparse polynomials, on
+single GPU with CUDA and on multiple GPUs using two parallel
+paradigms: shared memory with OpenMP and distributed memory with
+MPI. These architectures were addressed by a CUDA-OpenMP approach and
+CUDA-MPI approach, respectively. Experiments show that, using
+parallel programming model like (OpenMP, MPI). We can efficiently
+manage multiple graphics cards to solve the same
+problem and accelerate the parallel execution with 4 GPUs and solve a
+polynomial of degree up to 5,000,000, four times faster than on single
+GPU.
+
+
+Our next objective is to extend the model presented here with clusters
+of GPU nodes, with a three-level scheme: inter-node communication via
+MPI processes (distributed memory), management of multi-GPU node by
+OpenMP threads (shared memory).
-% use section* for acknowledgment
\section*{Acknowledgment}
-
-The authors would like to thank...
-
-
-
+Computations have been performed on the supercomputer facilities of
+the Mésocentre de calcul de Franche-Comté. We also would like to thank
+Nvidia for hardware donation under CUDA Research Center 2014.
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