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-\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
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\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}}
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-% make the title area
\maketitle
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\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
\LZK{Faut pas mettre des keywords?}
-
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
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.}
-
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}
\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
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}
+
\begin{algorithm}[htpb]
\LinesNumbered
Copy $P$, $P'$ and $Z$ from CPU to GPU\;
\While{\emph{not convergence}}{
$Z^{prev}$ = KernelSave($Z,n$)\;
- $Z$ = KernelUpdate($P,P',Z^{prev},n$)\;
+ $Z$ = KernelUpdate($P,P',Z,n$)\;
$\Delta Z$ = KernelComputeError($Z,Z^{prev},n$)\;
$\Delta Z_{max}$ = CudaMaxFunction($\Delta Z,n$)\;
TestConvergence($\Delta Z_{max},\epsilon$)\;
}
Copy $Z$ from GPU to CPU\;
\label{alg1-cuda}
-\RC{Si l'algo vous convient, il faudrait le détailler précisément\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? }}
+\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}
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]
-%% \label{alg2-cuda-openmp}
-%% \LinesNumbered
-%% \SetAlgoNoLine
-%% \caption{CUDA-OpenMP 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 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}
+
\begin{algorithm}[htpb]
\LinesNumbered
\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 ($Z$, $\Delta Z$, $\Delta Z_{max}$, $P$ are shared variables)\;
+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)\;
TestConvergence($max,\epsilon$)\;
}
\label{alg2-cuda-openmp}
-\LZK{J'ai modifié l'algo. Le $P$ est mis shared. Qu'en est-il pour $P'$?}
+\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}
\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.
%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}
