show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
GPU to find roots of polynomials of degree up-to 1.4
million. Moreover, other experiments show it is possible to find roots
-of polynomials of degree up to 5 millions.
+of polynomials of degree up-to 5 millions.
\end{abstract}
% no keywords
+\LZK{Faut pas mettre des keywords?}
Finding roots of polynomials of very high degrees arises in many complex problems in various domains such as algebra, biology or physics. A polynomial $p(x)$ in $\mathbb{C}$ in one variable $x$ is an algebraic expression in $x$ of the form:
\begin{equation}
-p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0.
+p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0,
\end{equation}
where $\{a_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $a_n\neq0$ then $n$ is called the degree of the polynomial. The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called roots of $p(x)$. Let $\{z_i\}_{1\leq i\leq n}$ be the roots of polynomial $p(x)$, then $p(x)$ can be written as :
\begin{equation}
for a distributed memory one with MPI. They are able to compute the
roots of sparse polynomials of degree 10,000 in 116 seconds with
OpenMP and 135 seconds with MPI only by using 8 personal computers and
-2 communications per iteration. \RC{si on donne des temps faut donner
- le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?} The authors showed an interesting
+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.
-\LZK{``only by using 8 personal computers and 2 communications per iteration''. Pour MPI? et Pour OpenMP: Rep: c'est MPI seulement}
+\RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?}
+\LZK{Supprimons ces détails et mettons une référence s'il y en a une}
Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
%\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 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 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. 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.
+\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}
-\LZK{Pas d'autres contributions possibles?: j'ai rajouté 2}
+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 Multi-GPU using the OpenMP and MPI approaches. In section\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
\begin{equation}
\label{Eq:EA1}
-EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
-{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
+z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,\ldots,n
\end{equation}
-This methods contains 4 steps. The first step consists of the initial
-approximations of all the roots of the polynomial. The second step
-initializes the solution vector $Z$ using the Guggenheimer
-method~\cite{Gugg86} to ensure the distinction of the initial vector
-roots. In step 3, the iterative function based on the Newton's
-method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
-applied. With this step the computation of roots will converge,
-provided that all roots are different.
+This method contains 4 steps. The first step consists of the initial approximations of all the roots of the polynomial.\LZK{Pas compris??}
+The second step initializes the solution vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure the distinction of the initial vector roots.\LZK{Quelle est la différence entre la 1st step et la 2nd step? Que veut dire " to ensure the distinction of the initial vector roots"?}
+In step 3, the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is applied. With this step the computation of roots will converge, provided that all roots are different.\LZK{On ne peut pas expliquer un peu plus comment? Donner des formules comment elle se base sur la méthode de Newton et de l'opérateur de Weiestrass?}
+\LZK{Elle est où la 4th step??}
+\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!}
In order to stop the iterative function, a stop condition is
applied. This condition checks that all the root modules are lower
-than a fixed value $\xi$.
+than a fixed value $\epsilon$.
\begin{equation}
\label{eq:Aberth-Conv-Cond}
-\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
+\forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
\end{equation}
+
+\LZK{On ne dit pas plutôt "the relative errors" à la place de "root modules"? Raph nous confirmera quelle critère d'arrêt a utilisé.}
+
\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.
+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:
function by using the logarithm and the exponential of a complex and
we propose a new version of the Ehrlich-Aberth method. This method
allows us to exceed the computation of the polynomials of degree
-100,000 and to reach a degree up to more than 1,000,000. This new
-version of the Ehrlich-Aberth method with exponential and logarithm is
-defined as follows:
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
\begin{equation}
\label{Log_H2}
-z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
-p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
+z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
\end{equation}
where:
-\begin{eqnarray}
+\begin{equation}
\label{Log_H1}
-Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
-\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right) \nonumber \\
-i=1,...,n
-\end{eqnarray}
+Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
+\end{equation}
%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 absolute values~\cite{Karimall98}.
+Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette dernière phrase?}
%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{Ehrlich-Aberth parallel implementation on CUDA}
+\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
%We introduced three paradigms of parallel programming.
Our objective consists in implementing a root finding polynomial
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.
+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!}
-Like any parallel code, a GPU parallel implementation first requires
-to determine the sequential tasks and the parallelizable parts of the
-sequential version of the program/algorithm. In our case, all the
-operations that are easy to execute in parallel must be made by the
-GPU to accelerate the execution of the application, like the step 3
-and step 4. On the other hand, all the sequential operations and the
-operations that have data dependencies between threads or recursive
-computations must be executed by only one CUDA or CPU thread (step 1
-and step 2). Initially, we specify the organization of parallel
-threads, by specifying the dimension of the grid Dimgrid, the number
-of blocks per grid DimBlock and the number of threads per block.
+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 kernels which are part of code that are run on
-GPU devices. For step 3, there are two kernels, the first named
-\textit{save} is used to save vector $Z^{K-1}$ and the second one is
+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
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}
\section{The EA algorithm on Multiple GPUs}
\label{sec4}
-\subsection{M-GPU : an OpenMP-CUDA approach}
+\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
%% roots sufficiently converge.
-\begin{algorithm}[htpb]
+\begin{algorithm}[h]
\label{alg2-cuda-openmp}
\LinesNumbered
\SetAlgoNoLine
Initialization of Pu\;
Initialization of the solution vector $Z^{0}$\;
omp\_set\_num\_threads(num\_gpus)\;
-\#pragma omp parallel shared(Z,$\Delta$ z,P)\;
+\#pragma omp parallel shared(Z,$\Delta z$,P)\;
\Indp
{
gpu\_id=cudaGetDevice()\;
copy $Z_{loc}$ from GPU to Z in CPU
}
\Indm}
+\RC{Est ce qu'on fait apparaitre le pragma? J'hésite...}
\end{algorithm}
-\subsection{Multi-GPU : an MPI-CUDA approach}
+\subsection{an MPI-CUDA approach}
%\begin{figure}[htbp]
%\centering
% \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
Since a GPU works only on data already allocated in its memory, all local input data, $Z_{k}$, $ZPrec$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterwards, the same EA algorithm (Algorithm \ref{alg1-cuda}) is run by all processes but on different polynomial subset of roots $ p(x)_{k}=\sum_{i=1}^{n} a_{i}x^{i}, k=1,...,p$. Each MPI process executes the loop \verb=(While(...)...do)= containing the CUDA kernels but each MPI process computes only its own portion of the roots according to the rule ``''owner computes``''. The local range of roots is indicated with the \textit{index} variable initialized at (line 5, Algorithm \ref{alg2-cuda-mpi}), and passed as an input variable to $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize (\verb=MPI_Allreduce= function) by a reduction on $\Delta z_{k}$ in order to compute the maximum error related to the stop condition. Finally, processes copy the values of new computed roots from GPU memories to CPU memories, then communicate their results to other processes with \verb=MPI_Alltoall= broadcast. If the stop condition is not verified ($error > \epsilon$) then processes stay withing the loop \verb= while(...)...do= until all the roots sufficiently converge.
-%% \begin{enumerate}
-%% \begin{algorithm}[htpb]
-%% \label{alg2-cuda-mpi}
-%% %\LinesNumbered
-%% \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)}
-
-%% \KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-
-%% \BlankLine
-%% \item Initialization of P\;
-%% \item Initialization of Pu\;
-%% \item Initialization of the solution vector $Z^{0}$\;
-%% \item Allocate and copy initial data from CPU memories to GPU global memories\;
-%% \item $index= Size/num_gpus$\;
-%% \item k=0\;
-%% \While {$error > \epsilon$}{
-%% \item Let $\Delta z=0$\;
-%% \item $kernel\_save(ZPrec,Z)$\;
-%% \item k=k+1\;
-%% \item $kernel\_update(Z,P,Pu,index)$\;
-%% \item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
-%% \item ComputeMaxError($\Delta z$,error)\;
-%% \item Copy results from GPU memories to CPU memories\;
-%% \item Send $Z[id]$ to all processes\;
-%% \item Receive $Z[j]$ from every other process j\;
-%% }
-%% \end{algorithm}
-%% \end{enumerate}
-%% ~\\
-
-%% \RC{ENCORE ENCORE PIRE}
+\begin{algorithm}[htpb]
+\label{alg2-cuda-mpi}
+%\LinesNumbered
+\caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
+
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+ threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
+
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+
+\BlankLine
+Initialization of P\;
+Initialization of Pu\;
+Initialization of the solution vector $Z^{0}$\;
+Distribution of Z\;
+Allocate memory to GPU\;
+\While {$error > \epsilon$}{
+copy Z from CPU to GPU\;
+$ZPrec_{loc}=kernel\_save(Z_{loc})$\;
+$Z_{loc}=kernel\_update(Z,P,Pu)$\;
+$\Delta z=kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
+$error=MPI\_Reduce(\Delta z)$\;
+$Copy Z_{loc} from GPU to CPU$\;
+$Z=MPI\_AlltoAll(Z_{loc})$\;
+}
+\end{algorithm}
+
\section{Experiments}
\label{sec5}
