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% paper title
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-\title{A parallel implementation of Ehrlich-Aberth algorithm for root finding of polynomials
-on Multi-GPU with OpenMP/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
% As a general rule, do not put math, special symbols or citations
% in the abstract
\begin{abstract}
-The abstract goes here.
+Finding roots of polynomials is a well-known important but not so very easy problem to solve especially for high degrees.
+In this paper, we present two different parallel approaches to achieve this gaol for sparse and fully defined polynomials of up to 1.4 billion degree. Our two approaches are based on the well known parallel paradigms of OpenMP and MPI but combined with the novel CUDA GPU technology. Our results show a quasi-linear speedup using up to 4 GPU devices to solve four times faster a polynomial root finding problem. To our knowledge, this is the first paper to present this technology mix to solve such a highly demanding problem in parallel programming.
+
\end{abstract}
% no keywords
Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and by expressing any outcome as a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
%%\begin{center}
\begin{equation}
- {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
+ {\Large p(x)=\sum_{i=0}^{n-1}{a_{i}x^{i}}}.
\end{equation}
%%\end{center}
%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 is increased with the increasing of the degree of the polynomial. Many authors have treated the problem of implementing simultaneous methods in parallel. Freeman [10] implemented and compared DK, EA and another method of the fourth order proposed by Farmer
-and Loizou [9], on a 8-processor linear chain, for polynomials of degree up to 8.
-The third method often diverges, but the first two methods have speed-up equal to 5.5. Later, Freeman and Bane [11] considered asynchronous algorithms, in which each processor continues to update its approximations even though the latest values of other $z^{k}_{i}$ have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before
-making a new iteration. Couturier and al. [12] proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for distributed memory one with \textit{MPI}. They were able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only by using 8 personal computers and 2 communications per iteration. Comparing to the sequential implementation where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.
+The main problem of the simultaneous methods is that the necessary time needed for the convergence is increased with the increasing of the degree of the polynomial. Many authors have treated the problem of implementing simultaneous methods in parallel. Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed by Farmer
+and Loizou~\cite{Loizou83}, on a 8-processor linear chain, for polynomials of degree up to 8.
+The third method often diverges, but the first two methods have speed-up equal to 5.5. Later, Freeman and Bane~\cite{Freemanall90} considered asynchronous algorithms, in which each processor continues to update its approximations even though the latest values of other $z^{k}_{i}$ have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before
+making a new iteration. Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for distributed memory one with \textit{MPI}. They were able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only by using 8 personal computers and 2 communications per iteration. Comparing to the sequential implementation where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.
-Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA) [13], a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche and al [14] proposed an implementation of the Durand-Kerner method on GPU. Their main result 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.
+Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche and al~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on GPU. Their main result 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 parallel programming paradigms (OpenMP, MPI) on GPUs. We consider two architectures: shared memory with OpenMP API and distributed memory MPI API. The first approach is based on threads from the same system process, with each thread attached to one GPU and after the various memory allocations, each thread launches its part of computations. To do this we must first load on the GPU required data and after the computations are carried, repatriate the result on the host. The second approach i.e distributed memory with MPI relies on the MPI library which is often used for parallel programming [11] in
+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 parallel programming paradigms (OpenMP, MPI) on GPUs. We consider two architectures: shared memory with OpenMP API and distributed memory MPI API. The first approach is based on threads from the same system process, with each thread attached to one GPU and after the various memory allocations, each thread launches its part of computations. To do this we must first load on the GPU required data and after the computations are carried, repatriate the result on the host. The second approach i.e distributed memory with MPI relies on the MPI library which is often used for parallel programming~\cite{Peter96} in
cluster systems because it is a message-passing programming language. Each GPU is attached to one MPI process, and a loop is in charge of the distribution of tasks between the MPI processes. This solution can be used on one GPU, or executed on a distributed cluster of GPUs, employing the Message Passing Interface (MPI) to communicate between separate CUDA cards. This solution permits scaling of the problem size to larger classes than would be possible on a single device and demonstrates the performance which users might expect from future
HPC architectures where accelerators are deployed.
-This paper is organized as follows, in section 2 we recall the Ehrlich-Aberth method. In section 3 we present EA algorithm on single GPU. In section 4 we propose the EA algorithm implementation on MGPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In section 5 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.
+This paper is organized as follows, in section 2 we recall the Ehrlich-Aberth method. In section 3 we present EA algorithm on single GPU. In section 4 we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In section 5 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.
\section{Parallel Programmings Model}
\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 allo-
-cated 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, in OpenMP a usage of threads ids and managing data explicitly as done in an MPI
-code can be considered, it defeats the advantages of OpenMP.
+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} %L'article en Français Programmation multiGPU – OpenMP versus MPI
+%\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 )
%have private memory areas [6].
\subsection{MPI}
- The library MPI allows to use a distributed memory architecture. The various processes have their own environment of execution and execute their codes in a asynchronous way, according to the model MIMD (Multiple Instruction streams, Multiple Dated streams); they communicate and synchronize by exchanges of messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a programming multi-thread (OpenMP/Pthreads).
+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.
-\subsection{CUDA}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications
- CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{NVIDIA12}. The
-unit of execution in CUDA is called a thread. Each thread executes the kernel by the streaming processors in parallel. In CUDA,
-a group of threads that are executed together is called thread blocks, and the computational grid consists of a grid of thread
-blocks. Additionally, a thread block can use the shared memory on a single multiprocessor as while as the grid executes a single
+\subsection{CUDA}
+CUDA (an acronym for 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.
-We introduced three paradigms of parallel programming. Our objective consist to implement an algorithm of root finding polynomial on multiple GPUs. It primordial to know how manage CUDA context of different GPUs. A direct method for controlling the various GPU is to use as many threads or processes that GPU. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be created.
+We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
-\section{The EA algorithm on single GPU}
-\subsection{the EA method}
+\section{The EA algorithm on a single GPU}
+\subsection{The EA method}
A cubically convergent iteration method to find zeros of
polynomials was proposed by O. Aberth~\cite{Aberth73}. The
-Ehrlich-Aberth method contains 4 main steps, presented in what
+Ehrlich-Aberth (EA is short) method contains 4 main steps, presented in what
follows.
%The Aberth method is a purely algebraic derivation.
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-spaced points on a circle of center 0 and radius r, where r is
+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}.
\end{equation}
\subsubsection{Iterative Function}
-The operator used by the Aberth method is corresponding to the
-following equation~\ref{Eq:EA} which will enable the convergence towards
-polynomial solutions, provided all the roots are distinct.
+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}
+\label{Eq:EA1}
EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
\end{equation}
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 specifies the organization of threads in parallel, need to specify the dimension of the grid Dimgrid: the number of block per grid and block by DimBlock: the number of threads per block required to process a certain task.
-
-we create the kernel, for step 3 we have two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the kernel
-\textit{update} is used to update the $Z^{K}$ vector. In step 4 a kernel is
-created to test the convergence of the method. In order to
-compute function H, we have two possibilities: either to use
-the Jacobi method, or the Gauss-Seidel method which uses the
+be executed by only one CUDA or CPU thread (step 1 and step 2). Initially, we specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block.
+
+The code is organzed by what is named kernels, portions o code that are run on GPU devices. For step 3, there are two kernels, the
+first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde one is named
+\textit{update} and is used to update the $Z^{K}$ vector. For step 4, a kernel
+tests the convergence of the method. In order to
+compute the function H, we have two possibilities: either to use
+the Jacobi mode, or the Gauss-Seidel mode of iterating which uses the
most recent computed roots. It is well known that the Gauss-
Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
parallelize the code, we created kernels and many functions to
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 1 shows the GPU parallel implementation of Ehrlich-Aberth method.
-
-Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
+Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Aberth method.
\begin{enumerate}
\begin{algorithm}[htpb]
\section{The EA algorithm on Multi-GPU}
-\subsection{MGPU (OpenMP-CUDA) approach}
-Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works
-as follows.
-Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$. vector of error of stop condition $\Delta z$. Let(T\_omp) number of OpenMP threads is equal to the number of GPUs, each threads OpenMP checks one GPU, and control a part of the shared memory, that is a part of the vector Z like: $(n/num\_gpu)$ roots, n: the polynomial's degrees, $num\_gpu$ the number of GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Than every GPU will have a grid of computation organized with its performances and the size of data of which it checks and compute kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
+\subsection{MGPU : an OpenMP-CUDA approach}
+Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
+Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$, and the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU, and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
%\begin{figure}[htbp]
%\centering
%\end{figure}
%Each thread OpenMP compute the kernels on GPUs,than after each iteration they copy out the data from GPU memory to CPU shared memory. The kernels are re-runs is up to the roots converge sufficiently. Here are below the corresponding algorithm:
-$num\_gpus$ thread OpenMP are created using \verb=omp_set_num_threads();=function (line,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}), than each OpenMP threads allocate and copy initial data from CPU memory to the GPU global memories, execute the kernels on GPU, and compute 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, OpenMP threads synchronize using \verb=#pragma omp barrier;= to recuperate all values of vector $\Delta z$, to compute the maximum stop condition in vector $\Delta z$(line 12, Algorithm \ref{alg2-cuda-openmp}).Finally,they copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots converge sufficiently.
+$num\_gpus$ OpenMP threads are created using \verb=omp_set_num_threads();=function (step $3$, Algorithm \ref{alg2-cuda-openmp}), the shared memory is created using \verb=#pragma omp parallel shared()= OpenMP function (line $5$, Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates memory and copies initial data from CPU memory to GPU global memory, executes the kernels on GPU, but computes only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-openmp}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After each iteration, all OpenMP threads synchronize using \verb=#pragma omp barrier;= to gather all the correct values of $\Delta z$, thus allowing the computation the maximum stop condition on vector $\Delta z$ (line 12, Algorithm \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots sufficiently converge.
\begin{enumerate}
\begin{algorithm}[htpb]
\label{alg2-cuda-openmp}
\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 degrees), $\Delta z$ ( Vector of errors of stop condition), $num_gpus$ (number of OpenMP threads/ number of GPUs), $Size$ (number of roots)}
+ 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 {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
\BlankLine
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
+\item Initialization of P\;
+\item Initialization of Pu\;
\item Initialization of the solution vector $Z^{0}$\;
\verb=omp_set_num_threads(num_gpus);=
\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
-\subsection{Multi-GPU (MPI-CUDA) approach}
+\subsection{Multi-GPU : an MPI-CUDA approach}
%\begin{figure}[htbp]
%\centering
% \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
%\caption{The MPI-CUDA architecture }
%\label{fig:03}
%\end{figure}
-Our parallel implementation of the Ehrlich-Aberth method to find root polynomial using (CUDA-MPI) approach, splits input data of the polynomial to solve between MPI processes. From Algorithm 3, the 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 size of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $⌈n/p⌉$ roots to find per MPI process, for each element mentioned above. Consequently, each MPI process $k$ will have its own solution vector $Z_{k}$,polynomial to be solved $p_{k}$, the error of stop condition $\Delta z_{k}$, Than each MPI processes compute only $⌈n/p⌉$ roots.
+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 $⌈n/p⌉$ 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 $⌈n/p⌉$ roots.
-Since a GPU works only on data of its memory, all local input data, $Z_{k}, p_{k}$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterward, the same EA algorithm (Algorithm 1) is run by all processes but on different sub-polynomial root $ p(x)_{k}=\sum_{i=k(\frac{n}{p})}^{k+1(\frac{n}{p})} a_{i}x^{i}, k=1,...,p$. Each processes MPI execute the loop \verb=(While(...)...do)= contain the kernels. Than each process MPI compute only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-mpi}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize using \verb=MPI_Allreduce= function, in order to compute the maximum error stops condition $\Delta z_{k}$ computed by each process MPI line (line, Algorithm\ref{alg2-cuda-mpi}), and copy the values of new roots computed from GPU memories to CPU memories, than communicate her results to the neighboring processes,using \verb=MPI_Alltoallv=. If maximum stop condition $error > \epsilon$ the processes stay to execute the loop \verb= while(...)...do= until all the roots converge sufficiently.
+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]
\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
\BlankLine
-\item Initialization of the P\;
-\item Initialization of the Pu\;
+\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 the GPU global memories\;
+\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 $kernel\_save(ZPrec,Z)$\;
\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
+\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 neighboring processes\;
-\item Receive $Z[j]$ from neighboring process j\;
-
-
+\item Send $Z[id]$ to all processes\;
+\item Receive $Z[j]$ from every other process j\;
}
\end{algorithm}
\end{enumerate}
~\\
-\section{experiments}
+\section{Experiments}
We study two categories of polynomials: sparse polynomials and full polynomials.\\
{\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
{\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
\end{equation}
For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
+%SIDER : Une meilleure présentation de l'architecture est à faire ici.
-We performed a set of experiments on single GPU and Multi-GPU using (OpenMP/MPI) to find roots polynomials with EA algorithm, for both sparse and full polynomials of different sizes. We took into account the execution times and the polynomial size performed by sum or each experiment.
-All experimental results obtained from the simulations are made in
-double precision data, the convergence threshold of the methods is set
-to $10^{-7}$.
+In order to evaluate both the MGPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
+All experimental results obtained are made in double precision data, the convergence threshold of the methods is set to $10^{-7}$.
%Since we were more interested in the comparison of the
%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
%CPUs versus on GPUs.
The initialization values of the vector solution
of the methods are given in %Section~\ref{sec:vec_initialization}.
-\subsection{Test with Multi-GPU (CUDA OpenMP) approach}
+\subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
-In this part we performed a set of experiments on Multi-GPU (CUDA OpenMP) approach for full and sparse polynomials of different degrees, compare it with Single GPU (CUDA).
- \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+We report here the results of the set of experiments with M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
+\subsubsection{Execution times in seconds of the EA method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
- In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000
+In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
\begin{figure}[htbp]
\centering
\label{fig:01}
\end{figure}
-This figure~\ref{fig:01} shows that (CUDA OpenMP) Multi-GPU approach reduce the execution time up to the scale 100 whereas single GPU is of scale 1000 for polynomial who exceed 1,000,000. It shows the advantage to use OpenMP parallel paradigm to connect the performances of several GPUs and solve a polynomial of high degrees.
+This figure~\ref{fig:01} shows that the (CUDA-OpenMP) Multi-GPU approach reduces the execution time by a factor up to 100 w.r.t the single GPU apparaoch and a by a factor of 1000 for polynomials exceeding degree 1,000,000. It shows the advantage to use the OpenMP parallel paradigm to gather the capabilities of several GPUs and solve polynomials of very high degrees.
\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
-This experiments shows the execution time of the EA algorithm, on single GPU (CUDA) and Multi-GPU (CUDA OpenMP) approach for full polynomials of degrees ranging from 100,000 to 1,400,000
+The experiments shows the execution time of the EA algorithm, on a single GPU and on multiple GPUs using the CUDA OpenMP approach for full polynomials of degrees ranging from 100,000 to 1,400,000.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using shared memory paradigm with OpenMP}
\label{fig:03}
\end{figure}
-The second test with full polynomial shows a very important saving of time, for a polynomial of degrees 1,4M (CUDA OpenMP) approach with 4 GPUs compute and solve it 4 times as fast as single GPU. We notice that curves are positioned one below the other one, more the number of used GPUs increases more the execution time decreases.
+Results with full polynomials show very important savings in execution time. For a polynomial of degree 1,4 million, the CUDA-OpenMP approach with 4 GPUs solves it 4 times as fast as single GPU, thus achieving a quasi-linear speedup.
-\subsection{Test with Multi-GPU (CUDA MPI) approach}
-In this part we perform a set of experiment to compare Multi-GPU (CUDA MPI) approach with single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.
+\subsection{Evaluting the Multi-GPU (CUDA-MPI) approach}
+In this part we perform a set of experiments to compare Multi-GPU (CUDA MPI) approach with single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.
\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
\label{fig:02}
\end{figure}
~\\
-This figure shows 4 curves of execution time of EA algorithm, a curve with single GPU, 3 curves with Multi-GPUs (2, 3, 4) GPUs. We see clearly that the curve with single GPU is above the other curves, which shows consumption in execution time compared to the Multi-GPU. We can see the approach Multi-GPU (CUDA MPI) reduces the execution time up to the scale 100 for polynomial of degrees more than 1,000,000 whereas single GPU is of the scale 1000.
+This figure shows 4 curves of execution time of EA algorithm, a curve with single GPU, 3 curves with multiple GPUs (2, 3, 4). We can clearly see that the curve with single GPU is above the other curves, which shows consumption in execution time compared to the Multi-GPU. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 100 for polynomials of degree more than 1,000,000 whereas a single GPU is of the scale 1000.
+%%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
\\
\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using distributed memory paradigm with MPI}
\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using distributed memory paradigm with MPI}
\label{fig:04}
\end{figure}
+%SIDER : Corriger le point de la courbe 3-GPUs qui correpsond à un degré de 600000
-\begin{figure}[htbp]
-\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
-\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse plynomials on GPUs}
-\label{fig:05}
-\end{figure}
+Figure \ref{fig:04} shows the execution time of the algorithm on single GPU and on multipe GPUs with (2, 3, 4) GPUs for full polynomials. With the CUDA-MPI approach, we notice that the three curves are distinct from each other, more we use GPUs more the execution time decreases. On the other hand the curve with a single GPU is well above the other curves.
+
+This is due to the use of MPI parallel paradigm that divides the problem computations and assigns portions to each GPU. But unlike the single GPU which carries all the computations on a single GPU, data communications are introduced, consequently engendering more execution time. But experiments show that execution time is still highly reduced.
-\begin{figure}[htbp]
-\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Full}
-\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
-\label{fig:06}
-\end{figure}
-\begin{figure}[htbp]
-\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
-\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
-\label{fig:07}
-\end{figure}
-\begin{figure}[htbp]
-\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
-\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
-\label{fig:08}
-\end{figure}
+
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
+%\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse plynomials on GPUs}
+%\label{fig:05}
+%\end{figure}
+
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{Full}
+%\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
+%\label{fig:06}
+%\end{figure}
+
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
+%\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
+%\label{fig:07}
+%\end{figure}
+
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
+%\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
+%\label{fig:08}
+%\end{figure}
% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
\section{Conclusion}
-The conclusion goes here~\cite{IEEEexample:bibtexdesign}.
+In this paper, we have presented a parallel implementation of Ehrlich-Aberth algorithm for solving full and sparse polynomials, on single GPU with CUDA and on multiple GPUs using two parallel paradigms : shared memory with OpenMP and distributed memory with MPI. These architectures were addressed by a CUDA-OpenMP approach and CUDA-MPI approach, respectively.
+The experiments show that, using parallel programming model like (OpenMP, MPI), we can efficiently manage multiple graphics cards to work together to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree 1,000,000, four times faster than on single GPU, that is a quasi-linear speedup.
+
+
+%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 at clusters of nodes featuring multiple GPUs, with a three-level scheme: inter-node communication via MPI processes (distributed memory), management of multi-GPU node by OpenMP threads (shared memory).
+
+%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.
