-
-%% bare_conf.tex
-%% V1.4b
-%% 2015/08/26
-%% by Michael Shell
-%% See:
-%% http://www.michaelshell.org/
-%% for current contact information.
-%%
-%% This is a skeleton file demonstrating the use of IEEEtran.cls
-%% (requires IEEEtran.cls version 1.8b or later) with an IEEE
-%% conference paper.
-%%
-%% Support sites:
-%% http://www.michaelshell.org/tex/ieeetran/
-%% http://www.ctan.org/pkg/ieeetran
-%% and
-%% http://www.ieee.org/
-
-%%*************************************************************************
-%% Legal Notice:
-%% This code is offered as-is without any warranty either expressed or
-%% implied; without even the implied warranty of MERCHANTABILITY or
-%% FITNESS FOR A PARTICULAR PURPOSE!
-%% User assumes all risk.
-%% In no event shall the IEEE or any contributor to this code be liable for
-%% any damages or losses, including, but not limited to, incidental,
-%% consequential, or any other damages, resulting from the use or misuse
-%% of any information contained here.
-%%
-%% All comments are the opinions of their respective authors and are not
-%% necessarily endorsed by the IEEE.
-%%
-%% This work is distributed under the LaTeX Project Public License (LPPL)
-%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
-%% distributed and modified. A copy of the LPPL, version 1.3, is included
-%% in the base LaTeX documentation of all distributions of LaTeX released
-%% 2003/12/01 or later.
-%% Retain all contribution notices and credits.
-%% ** Modified files should be clearly indicated as such, including **
-%% ** renaming them and changing author support contact information. **
-%%*************************************************************************
-
-
-% *** Authors should verify (and, if needed, correct) their LaTeX system ***
-% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
-% *** with production work. The IEEE's font choices and paper sizes can ***
-% *** trigger bugs that do not appear when using other class files. *** ***
-% The testflow support page is at:
-% http://www.michaelshell.org/tex/testflow/
-
-
-
\documentclass[conference]{IEEEtran}
-% Some Computer Society conferences also require the compsoc mode option,
-% but others use the standard conference format.
-%
-% If IEEEtran.cls has not been installed into the LaTeX system files,
-% manually specify the path to it like:
-% \documentclass[conference]{../sty/IEEEtran}
-
-
-
-
-
-% Some very useful LaTeX packages include:
-% (uncomment the ones you want to load)
-
-
-% *** MISC UTILITY PACKAGES ***
-%
-%\usepackage{ifpdf}
-% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
-% compilation based on whether the output is pdf or dvi.
-% usage:
-% \ifpdf
-% % pdf code
-% \else
-% % dvi code
-% \fi
-% The latest version of ifpdf.sty can be obtained from:
-% http://www.ctan.org/pkg/ifpdf
-% Also, note that IEEEtran.cls V1.7 and later provides a builtin
-% \ifCLASSINFOpdf conditional that works the same way.
-% When switching from latex to pdflatex and vice-versa, the compiler may
-% have to be run twice to clear warning/error messages.
-
-
-
-
-
-
-% *** CITATION PACKAGES ***
-%
-%\usepackage{cite}
-% cite.sty was written by Donald Arseneau
-% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
-% \cite{} output to follow that of the IEEE. Loading the cite package will
-% result in citation numbers being automatically sorted and properly
-% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
-% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
-% \cite will automatically add leading space, if needed. Use cite.sty's
-% noadjust option (cite.sty V3.8 and later) if you want to turn this off
-% such as if a citation ever needs to be enclosed in parenthesis.
-% cite.sty is already installed on most LaTeX systems. Be sure and use
-% version 5.0 (2009-03-20) and later if using hyperref.sty.
-% The latest version can be obtained at:
-% http://www.ctan.org/pkg/cite
-% The documentation is contained in the cite.sty file itself.
-
-
-
-
-
-
-% *** GRAPHICS RELATED PACKAGES ***
-%
-\ifCLASSINFOpdf
- \usepackage[pdftex]{graphicx}
-
- % declare the path(s) where your graphic files are
- % \graphicspath{{../pdf/}{../jpeg/}}
- % and their extensions so you won't have to specify these with
- % every instance of \includegraphics
- % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
-\else
- % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
- % will default to the driver specified in the system graphics.cfg if no
- % driver is specified.
- % \usepackage[dvips]{graphicx}
- % declare the path(s) where your graphic files are
- % \graphicspath{{../eps/}}
- % and their extensions so you won't have to specify these with
- % every instance of \includegraphics
- % \DeclareGraphicsExtensions{.eps}
-\fi
-% graphicx was written by David Carlisle and Sebastian Rahtz. It is
-% required if you want graphics, photos, etc. graphicx.sty is already
-% installed on most LaTeX systems. The latest version and documentation
-% can be obtained at:
-% http://www.ctan.org/pkg/graphicx
-% Another good source of documentation is "Using Imported Graphics in
-% LaTeX2e" by Keith Reckdahl which can be found at:
-% http://www.ctan.org/pkg/epslatex
-%
-% latex, and pdflatex in dvi mode, support graphics in encapsulated
-% postscript (.eps) format. pdflatex in pdf mode supports graphics
-% in .pdf, .jpeg, .png and .mps (metapost) formats. Users should ensure
-% that all non-photo figures use a vector format (.eps, .pdf, .mps) and
-% not a bitmapped formats (.jpeg, .png). The IEEE frowns on bitmapped formats
-% which can result in "jaggedy"/blurry rendering of lines and letters as
-% well as large increases in file sizes.
-%
-% You can find documentation about the pdfTeX application at:
-% http://www.tug.org/applications/pdftex
-
-
-
-
-
-% *** MATH PACKAGES ***
-%
-\usepackage{amsmath}
-% A popular package from the American Mathematical Society that provides
-% many useful and powerful commands for dealing with mathematics.
-%
-% Note that the amsmath package sets \interdisplaylinepenalty to 10000
-% thus preventing page breaks from occurring within multiline equations. Use:
-%\interdisplaylinepenalty=2500
-% after loading amsmath to restore such page breaks as IEEEtran.cls normally
-% does. amsmath.sty is already installed on most LaTeX systems. The latest
-% version and documentation can be obtained at:
-% http://www.ctan.org/pkg/amsmath
-
-
-
-
-
-% *** SPECIALIZED LIST PACKAGES ***
-%
-
-% algorithmic.sty was written by Peter Williams and Rogerio Brito.
-% This package provides an algorithmic environment fo describing algorithms.
-% You can use the algorithmic environment in-text or within a figure
-% environment to provide for a floating algorithm. Do NOT use the algorithm
-% floating environment provided by algorithm.sty (by the same authors) or
-% algorithm2e.sty (by Christophe Fiorio) as the IEEE does not use dedicated
-% algorithm float types and packages that provide these will not provide
-% correct IEEE style captions. The latest version and documentation of
-% algorithmic.sty can be obtained at:
-% http://www.ctan.org/pkg/algorithms
-% Also of interest may be the (relatively newer and more customizable)
-% algorithmicx.sty package by Szasz Janos:
-% http://www.ctan.org/pkg/algorithmicx
\usepackage[ruled,vlined]{algorithm2e}
-
-
-
-% *** ALIGNMENT PACKAGES ***
-%
-%\usepackage{array}
-% Frank Mittelbach's and David Carlisle's array.sty patches and improves
-% the standard LaTeX2e array and tabular environments to provide better
-% appearance and additional user controls. As the default LaTeX2e table
-% generation code is lacking to the point of almost being broken with
-% respect to the quality of the end results, all users are strongly
-% advised to use an enhanced (at the very least that provided by array.sty)
-% set of table tools. array.sty is already installed on most systems. The
-% latest version and documentation can be obtained at:
-% http://www.ctan.org/pkg/array
-
-
-% IEEEtran contains the IEEEeqnarray family of commands that can be used to
-% generate multiline equations as well as matrices, tables, etc., of high
-% quality.
-
-
-
-
-% *** SUBFIGURE PACKAGES ***
-%\ifCLASSOPTIONcompsoc
-% \usepackage[caption=false,font=normalsize,labelfont=sf,textfont=sf]{subfig}
-%\else
-% \usepackage[caption=false,font=footnotesize]{subfig}
-%\fi
-% subfig.sty, written by Steven Douglas Cochran, is the modern replacement
-% for subfigure.sty, the latter of which is no longer maintained and is
-% incompatible with some LaTeX packages including fixltx2e. However,
-% subfig.sty requires and automatically loads Axel Sommerfeldt's caption.sty
-% which will override IEEEtran.cls' handling of captions and this will result
-% in non-IEEE style figure/table captions. To prevent this problem, be sure
-% and invoke subfig.sty's "caption=false" package option (available since
-% subfig.sty version 1.3, 2005/06/28) as this is will preserve IEEEtran.cls
-% handling of captions.
-% Note that the Computer Society format requires a larger sans serif font
-% than the serif footnote size font used in traditional IEEE formatting
-% and thus the need to invoke different subfig.sty package options depending
-% on whether compsoc mode has been enabled.
-%
-% The latest version and documentation of subfig.sty can be obtained at:
-% http://www.ctan.org/pkg/subfig
-
-
-
-
-% *** FLOAT PACKAGES ***
-%
-%\usepackage{fixltx2e}
-% fixltx2e, the successor to the earlier fix2col.sty, was written by
-% Frank Mittelbach and David Carlisle. This package corrects a few problems
-% in the LaTeX2e kernel, the most notable of which is that in current
-% LaTeX2e releases, the ordering of single and double column floats is not
-% guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
-% single column figure to be placed prior to an earlier double column
-% figure.
-% Be aware that LaTeX2e kernels dated 2015 and later have fixltx2e.sty's
-% corrections already built into the system in which case a warning will
-% be issued if an attempt is made to load fixltx2e.sty as it is no longer
-% needed.
-% The latest version and documentation can be found at:
-% http://www.ctan.org/pkg/fixltx2e
-
-
-%\usepackage{stfloats}
-% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
-% the ability to do double column floats at the bottom of the page as well
-% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
-% LaTeX2e). It also provides a command:
-%\fnbelowfloat
-% to enable the placement of footnotes below bottom floats (the standard
-% LaTeX2e kernel puts them above bottom floats). This is an invasive package
-% which rewrites many portions of the LaTeX2e float routines. It may not work
-% with other packages that modify the LaTeX2e float routines. The latest
-% version and documentation can be obtained at:
-% http://www.ctan.org/pkg/stfloats
-% Do not use the stfloats baselinefloat ability as the IEEE does not allow
-% \baselineskip to stretch. Authors submitting work to the IEEE should note
-% that the IEEE rarely uses double column equations and that authors should try
-% to avoid such use. Do not be tempted to use the cuted.sty or midfloat.sty
-% packages (also by Sigitas Tolusis) as the IEEE does not format its papers in
-% such ways.
-% Do not attempt to use stfloats with fixltx2e as they are incompatible.
-% Instead, use Morten Hogholm'a dblfloatfix which combines the features
-% of both fixltx2e and stfloats:
-%
-% \usepackage{dblfloatfix}
-% The latest version can be found at:
-% http://www.ctan.org/pkg/dblfloatfix
-
-
-
-
-% *** PDF, URL AND HYPERLINK PACKAGES ***
-%
-%\usepackage{url}
-% url.sty was written by Donald Arseneau. It provides better support for
-% handling and breaking URLs. url.sty is already installed on most LaTeX
-% systems. The latest version and documentation can be obtained at:
-% http://www.ctan.org/pkg/url
-% Basically, \url{my_url_here}.
-
-
-
-
-% *** Do not adjust lengths that control margins, column widths, etc. ***
-% *** Do not use packages that alter fonts (such as pslatex). ***
-% There should be no need to do such things with IEEEtran.cls V1.6 and later.
-% (Unless specifically asked to do so by the journal or conference you plan
-% to submit to, of course. )
-
-
-% 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}
-
-
-
\usepackage{amsfonts}
\usepackage[utf8]{inputenc}
\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{Kahina Guidouche, Abderrahmane Sider }
\IEEEauthorblockA{Laboratoire LIMED\\
Faculté des sciences exactes\\
University of Bourgogne Franche-Comte, France\\
Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
-% conference papers do not typically use \thanks and this command
-% is locked out in conference mode. If really needed, such as for
-% the acknowledgment of grants, issue a \IEEEoverridecommandlockouts
-% after \documentclass
-
-% for over three affiliations, or if they all won't fit within the width
-% of the page, use this alternative format:
-%
-%\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)}
-
-
-
-% make the title area
\maketitle
-% As a general rule, do not put math, special symbols or citations
-% in the abstract
\begin{abstract}
Finding roots of polynomials is a very important part of solving
real-life problems but it is not so easy for polynomials of high
\end{abstract}
% no keywords
-\LZK{Faut pas mettre des keywords?}
+\LZK{Faut pas mettre des keywords?\KG{Oui d'après ça: "no keywords" qui se trouve dans leur fichier source!!, mais c'est Bizzard!!!}}
-
-
-% 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
-%
-% 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 issue of finding the roots of polynomials of very high degrees arises in many complex problems in various fields, such as algebra, biology, finance, physics or climatology [1]. In algebra for example, finding eigenvalues or eigenvectors of any real/complex matrix amounts to that of finding the roots of the so-called characteristic polynomial.
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}
\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 }
-%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 références pour les deux méthodes?, c'est fait}
-
-%The first method of this group is Durand-Kerner method:
-%\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,
-%\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}
-%\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,
-%\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
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
+iteration. Couturier and 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.
+roots of sparse polynomials of degree 10,000. The authors showed an interesting
+speedup that is 20 times as fast as 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}
-
+\KG{Je viens de supprimer les détails, la référence existe déja, a reverifier}
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.
-%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 !!}
-
-%In this paper we propose the parallelization of Ehrlich-Aberth method using 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{Pourquoi la méthode Ehrlich-Aberth et pas autres? the Ehrlich-Aberth have very good convergence and it is suitable to be implemented in parallel computers.}
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 An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.\RC{j'ai envie de virer ca, car c'est pas la nouveauté dans ce papier}
- %\item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.\RC{idem}
+
\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.}
-
-%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.}
+\LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.
+\AS{La résolution du problème pour des polynomes pleins de degré 6M est une contribution aussi à mon avis}}
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}
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 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 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec3}
\subsection{The Ehrlich-Aberth 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.%
-
-
-%\subsubsection{Polynomials Initialization}
-%The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients %$a_{i}$:
-
-%\begin{equation}
-%\label{eq:SimplePolynome}
-% p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
-%\end{equation}
-
-
-%\subsubsection{Vector $Z^{(0)}$ Initialization}
-%\label{sec:vec_initialization}
-%As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , %n.$
-%The initial guess is very important since the number of steps needed by the iterative method to %reach
-%a given approximation strongly depends on it.
-%In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
-%equi-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}.
-
-%\begin{equation}
-%\label{eq:radiusR}
-%%\begin{align}
-%\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
-%v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
-%%\end{align}
-%\end{equation}
-%Where:
-%\begin{equation}
-%u_{i}=2.|a_{i}|^{\frac{1}{i}};
-%v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
-%\end{equation}
-
-%\subsubsection{Iterative Function}
-%The operator used by the Aberth method 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:EA-1}
-%EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
-%{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, %i=1,. . . .,n
-%\end{equation}
-
-%\subsubsection{Convergence Condition}
-%The convergence condition determines the termination of the algorithm. It consists in stopping the %iterative function when the roots are sufficiently stable. We consider that the method converges %sufficiently when:
-
-%\begin{equation}
-%\label{eq:AAberth-Conv-Cond}
-%\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
-%\end{equation}
-
-
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
-%\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~\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{Aberth73} using the following iteration
\begin{equation}
{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}
-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 $\epsilon$.
+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:Aberth-Conv-Cond}
\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.
-%Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as:
-
-%\begin{equation}
-%\label{R.EL}
-%R = exp(log(DBL\_MAX)/(2*n) );
-%\end{equation}
-
-
-% where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
-
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
\end{equation}
-%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
+
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}.
-%This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
-%propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
-%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. \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}}
-
-
-
-
-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.}
-
-%\begin{algorithm}[htpb]
-%\label{alg1-cuda}
-%\LinesNumbered
-%\SetAlgoNoLine
-%\caption{CUDA Algorithm to find polynomial roots with the Ehrlich-Aberth method}
-%\KwIn{$Z^{0}$ (Initial vector of roots), $\epsilon$ (Error tolerance threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z_{max}$ (Maximum value of stop condition)}
-%\KwOut{$Z$ (Solution vector of roots)}
-
-%\BlankLine
-
-%Initialization of P\;
-%Initialization of Pu\;
-%Initialization of the solution vector $Z^{0}$\;
-%Allocate and copy initial data to the GPU global memory\;
-%\While {$\Delta z_{max} > \epsilon$}{
-% $ ZPres=kernel\_save(Z)$\;
-% $ Z=kernel\_update(Z,P,Pu)$\;
-% $\Delta z_{max}=kernel\_testConv(Z,ZPrec)$\;
-
-%}
-%Copy results from GPU memory to CPU memory\;
-%\end{algorithm}
+
+
+
+
+The code is organized as kernels which are parts of code that are run
+on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
+implementation of the Ehrlich-Aberth on a GPU. This algorithms starts
+by initializinf the polynomial and its derivative (line 1). The
+initialization of the roots is performed (line 2). Data are transfered
+from the CPU to the GPU (after allocation of the required memory on
+the GPU) (line 3). Then at each iteration, if the error is greater
+than a threshold, the following operations are performed. The previous
+roots are saved using a kernel (line 5). Then the new root with the
+new iterations are computed using the EA method with a Gauss Seidel
+iteration modes in order to use the lastest roots updated (line
+6). This improves the convergence. This kernel is, in pratice, very
+long since it performs all the operations with complex numbers with
+the normal mode of the EA method but also with the
+logarithm-exponential one. Then the error is computed with a final
+kernel (line 7). Finally when the EA method has converged, the roots
+are transferred from the GPU to the CPU.
\begin{algorithm}[htpb]
+\label{alg1-cuda}
\LinesNumbered
\SetAlgoNoLine
\caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
-\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold)}
+\KwIn{ $\epsilon$ (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 the CPU memory to the GPU memory\;
-\While{\emph{not convergence}}{
- $Z_{prev}$ = Kernel\_Save($Z$)\;
- $Z$ = Kernel\_Update($P,P',Z$)\;
- Kernel\_Test\_Convergence($Z,Z_{prev},\epsilon$)\;
+Copy $P$, $P'$ and $Z$ from CPU to GPU\;
+\While{$error > \epsilon$}{
+ $Z^{prev}$ = KernelSave($Z$)\;
+ $Z$ = KernelUpdate($P,P',Z$)\;
+ $error$ = KernelComputeError($Z,Z^{prev}$)\;
}
-Copy $Z$ from the GPU memory to the CPU memory\;
-\label{alg1-cuda}
+Copy $Z$ from GPU to CPU\;
\end{algorithm}
+The development of this code is a rather long task, 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
+section the GPU parallel implementation of Ehrlich-Aberth method with
+OpenMP and MPI is presented.
-
-
-
-
-\RC{Si l'algo vous convient, il faudrait le détailler précisément}
-\section{The EA algorithm on Multiple GPUs}
+\section{The EA algorithm on multiple GPUs}
\label{sec4}
\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.
+OpenMP and CUDA programming model. This algorithm is presented in
+Algorithm~\ref{alg2-cuda-openmp}. All the data are shared with OpenMP
+amoung all the OpenMP threads. The shared data are the solution vector
+$Z$, the polynomial to solve $P$, its derivative $P'$, and the error
+vector $error$. 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
+called $Z_{loc}$ 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.
-
-%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:
-
-%% \RC{Surement à virer ou réécrire pour etre compris sans algo}
-%% $num\_gpus$ 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{algorithm}[h]
+computation of the local size for each GPU is performed (line 4). Each
+thread starts by copying all the previous roots inside its GPU (line
+5). At each iteration, the following operations are performed. First
+the vector $Z$ is transferred from the CPU to the GPU (line 7). Each
+GPU copies the previous roots (line 8) and it computes an iteration of
+the EA method on its own roots (line 9). For that all the other roots
+are used. The local error is computed on the new roots (line 10) and
+the max of the local errors is computed on all OpenMP threads (lien 11). At
+the end of an iteration, the updated roots are copied from the GPU to
+the CPU (line 12) by direcly updating its own roots in the shared
+memory arrays containing all the roots.
+
+
+
+\begin{algorithm}[htpb]
\label{alg2-cuda-openmp}
\LinesNumbered
\SetAlgoNoLine
-\caption{CUDA-OpenMP Algorithm to find roots with the Ehrlich-Aberth method}
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
+\KwIn{ $\epsilon$ (tolerance threshold)}
+\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 ($Z$, $error$, $P$, $P'$ are shared variables)\;
+Determine the local part of the OpenMP thread\;
+Copy $P$, $P'$ from CPU to GPU\;
+\While{$error > \epsilon$}{
+ Copy $Z$ from CPU to GPU\;
+ $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
+ $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
+ $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
+ $error = max(error_{loc})$\;
+ Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
+}
+\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
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Start of a parallel part with OpenMP (Z, $\Delta z$, P are shared variables)\;
-gpu\_id=cudaGetDevice()\;
-Allocate memory on GPU\;
-Compute local size and offet according to gpu\_id\;
-\While {$error > \epsilon$}{
- copy Z from CPU to GPU\;
-$ ZPrec_{loc}=kernel\_save(Z_{loc})$\;
-$ Z_{loc}=kernel\_update(Z,P,Pu)$\;
-$\Delta z[gpu\_id] = kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
-$ error= Max(\Delta z)$\;
- copy $Z_{loc}$ from GPU to Z in CPU
-}
-\end{algorithm}
+\subsection{a MPI-CUDA approach}
+Our parallel implementation of EA to find roots of polynomials using a
+CUDA-MPI approach follows a similar 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. Processors need to allocate memory on their GPU and need to copy
+their data on the 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). Only the local part of $Z^{prev}$ is saved (line
+8). After that, a processor is able to compute an updated version of
+its own roots (line 9) with the EA method. The local error is computed
+(ligne 10) and the global error using $MPI\_Reduce$ (line 11). Then
+the local roots are transfered from the GPU memory to the CPU memory
+(line 12) before being exchanged between all processors (lige 13) in
+order to give to all processors the last version of the roots (with
+the MPI\_AlltoAll routine). If the convergence is not statisfied, an
+new iteration is executed.
-\subsection{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{algorithm}[htpb]
\label{alg2-cuda-mpi}
-%\LinesNumbered
+\LinesNumbered
+\SetAlgoNoLine
\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)}
+\KwIn{ $\epsilon$ (tolerance threshold)}
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+\KwOut {$Z$ (solution vector of roots)}
\BlankLine
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Distribution of Z\;
-Allocate memory to GPU\;
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Determine the local part of the MPI process\;
+Computation of the parameters for the $MPI\_AlltoAll$\;
+Copy $P$, $P'$ from CPU to GPU\;
\While {$error > \epsilon$}{
-copy Z from CPU to GPU\;
-$ZPrec_{loc}=kernel\_save(Z_{loc})$\;
-$Z_{loc}=kernel\_update(Z,P,Pu)$\;
-$\Delta z=kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
-$error=MPI\_Reduce(\Delta z)$\;
-Copy $Z_{loc}$ from GPU to CPU\;
-$Z=MPI\_AlltoAll(Z_{loc})$\;
+ Copy $Z$ from CPU to GPU\;
+ $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
+ $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
+ $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
+ $error=MPI\_Reduce(error_{loc})$\;
+ Copy $Z_{loc}$ from GPU to CPU\;
+ $Z=MPI\_AlltoAll(Z_{loc})$\;
}
\end{algorithm}
%SIDER : il faut une explication sur les différences ici aussi.
-%for sparse and full polynomials
-% An example of a floating figure using the graphicx package.
-% Note that \label must occur AFTER (or within) \caption.
-% 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.
-
-
-
\section{Conclusion}
\label{sec6}
GPU.
-%In future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model
-
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).
-%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
-
-
- %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.
-
-
-
-
-% conference papers do not normally have an appendix
-
-% use section* for acknowledgment
\section*{Acknowledgment}
Computations have been performed on the supercomputer facilities of
-
-
-
-% trigger a \newpage just before the given reference
-% number - used to balance the columns on the last page
-% adjust value as needed - may need to be readjusted if
-% the document is modified later
-%\IEEEtriggeratref{8}
-% The "triggered" command can be changed if desired:
-%\IEEEtriggercmd{\enlargethispage{-5in}}
-
-% references section
-
-% can use a bibliography generated by BibTeX as a .bbl file
-% BibTeX documentation can be easily obtained at:
-% http://mirror.ctan.org/biblio/bibtex/contrib/doc/
-% The IEEEtran BibTeX style support page is at:
-% http://www.michaelshell.org/tex/ieeetran/bibtex/
-%\bibliographystyle{IEEEtran}
-% argument is your BibTeX string definitions and bibliography database(s)
-%\bibliography{IEEEabrv,../bib/paper}
-%\bibliographystyle{./IEEEtran}
\bibliography{mybibfile}
-%
-% <OR> manually copy in the resultant .bbl file
-% set second argument of \begin to the number of references
-% (used to reserve space for the reference number labels box)
-%\begin{thebibliography}{1}
-
-%\bibitem{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
- % 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
-
-%\bibitem{IEEEhowto:NVIDIA12}
- %NVIDIA Corporation, \textit{Whitepaper NVIDA’s Next Generation CUDATM Compute
-%Architecture: KeplerTM }, 1st ed., 2012.
-
-%\end{thebibliography}
-
-
-
-
-% that's all folks
\end{document}
