% not capitalized unless they are the first or last word of the title.
% Linebreaks \\ can be used within to get better formatting as desired.
% Do not put math or special symbols in the title.
-\title{Two parallel implementations of Ehrlich-Aberth algorithm for root finding of polynomials
-on multiple GPUs with OpenMP and MPI}
+\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
% author names and affiliations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 !}
+%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.
The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called zeros or roots of $p$. If zeros are $\{\alpha_{i}\}_{1\leq i\leq n}$, then $p(x)$ can be written as :
\begin{equation}
- p(x)=a_n\prod_{i=1}^n(x-\alpha_i), a_0 a_n\neq 0.
+ p(x)=\sum_{i=0}^{n}{a_ix^i}=a_n\prod_{i=1}^n(x-\alpha_i), a_0 a_n\neq 0.
\end{equation}
+%\LZK{C'est $a_n\neq 0$ (polynôme de degré $n$) et non pas $a_0 a_n\neq 0$, non?}
+%\LZK{Est-ce $\alpha_i$ sont les $z_i$ définis dans la suite du papier?}
-The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
+%The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous ones, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial zeros. These methods start from the initial approximations of all $n$ polynomial roots and give a sequence of approximations that converge to the roots of the polynomial. The first method of this group is Durand-Kerner method:
\begin{equation}
\label{DK}
%Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
%the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of %convergence.
-The main problem of the simultaneous methods is that the necessary time needed for the convergence increases with the increasing of the polynomial's degree. Many authors have treated the problem of implementing simultaneous methods in parallel. Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order of convergence proposed by Farmer and Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83} often diverges, but the first two methods (DK and EA) have a speed-up equals to 5.5. Later, Freeman and Bane~\cite{Freemanall90} considered asynchronous algorithms in which each processor continues to update its approximations even though the latest values of other $z^{k}_{i}$ have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for a distributed memory one with \textit{MPI}. They are able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only by using 8 personal computers and 2 communications per iteration. \LZK{je suppose que c'est pour la version mpi (only by using 8 personal computers and 2 communications per iteration). A t on utilisé le même nombre de procs pour les deux versions openmp et mpi} The authors showed an interesting speedup comparing to the sequential implementation that takes up-to 3,300 seconds to obtain same results.
+The main problem of the simultaneous methods is that the necessary time needed for the convergence increases with the increasing of the polynomial's degree. Many authors have treated the problem of implementing simultaneous methods in parallel. Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order of convergence proposed by Farmer and Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83} often diverges, but the first two methods (DK and EA) have a speed-up equals to 5.5. Later, Freeman and Bane~\cite{Freemanall90} considered asynchronous algorithms in which each processor continues to update its approximations even though the latest values of other $z^{k}_{i}$ have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for a distributed memory one with \textit{MPI}. They are able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only by using 8 personal computers and 2 communications per iteration.
+\LZK{je suppose que c'est pour la version mpi (only by using 8 personal computers and 2 communications per iteration). A t on utilisé le même nombre de procs pour les deux versions openmp et mpi} The authors showed an interesting speedup comparing to the sequential implementation that takes up-to 3,300 seconds to obtain same results.
Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche and al~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
-%Finding polynomial roots rapidly and accurately is the main objective of our work. We consider two architectures: shared-memory computers with OpenMP API and distributed-memory computers with MPI API. The first approach is based on threads from the same system process, with each thread attached to one GPU and after the various memory allocations, each thread launches its part of computations. To do this we must first load on the GPU required data and after the computations are carried, repatriate the result on the host. The second approach i.e distributed memory with MPI relies on the MPI library which is often used for parallel programming~\cite{Peter96} in cluster systems because it is a message-passing programming language. Each GPU is attached to one MPI process, and a loop is in charge of the distribution of tasks between the MPI processes. This solution can be used on one GPU, or executed on a distributed cluster of GPUs, employing the Message Passing Interface (MPI) to communicate between separate CUDA cards. This solution permits scaling of the problem size to larger classes than would be possible on a single device and demonstrates the performance which users might expect from future HPC architectures where accelerators are deployed.
+%Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. We consider two architectures: shared memory and distributed memory computers. The first parallel algorithm is implemented on shared memory computers by using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory. The second parallel algorithm uses the MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. This latter approach is more used on distributed memory clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
+%\LZK{Cette partie est réécrite. \\ Sinon qu'est ce qui a été fait pour l'accuracy dans ce papier (Finding polynomial roots rapidly and accurately is the main objective of our work.)?}
+%\LZK{Les contributions ne sont pas définies !!}
-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.)?}
+In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. Our (CUDA MPI)and (CUDA OpenMP) codes is the first implementation of Ehrlich-Aberth algorithm with multiple GPUs for finding roots polynomial. Our major contributions include:
+
+\begin{itemize}
+\item The parallel implementation of EA algorithm on multi-GPU platform with 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.
-{\color{red}{This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section~\ref{sec3} we present EA algorithm on single GPU. In section~\ref{sec4} we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In sectioné\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}}\LZK{A revoir toute cette organization}
+\item The parallel implementation of EA algorithm on multi-GPU platform with 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.
+ \end{itemize}
+\LZK{Cette partie est réécrite. \\ Sinon qu'est ce qui a été fait pour l'accuracy dans ce papier (Finding polynomial roots rapidly and accurately is the main objective of our work.)?}
+\LZK{Les contributions ne sont pas définies !!}
+ ? %This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section~\ref{sec3} we present EA algorithm on single GPU. In section~\ref{sec4} we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In sectioné\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}
-The paper is organized as follows. In Section~\ref{sec2} we present three parallel programming models OpenMP, MPI and CUDA.
+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 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.
+
+\LZK{A revoir toute cette organization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Parallel programming models}
\label{sec2}
-In this section we present the parallel programming models OpenMP, MPI and CUDA.
+\LZK{Toute cette section a été réécrite. Donc à relire et à améliorer si possible.}
+
+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 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.
%Sequential natively. Threads share some or all of the available memory and can
%have private memory areas [6].
-OpenMP (Open Multi-processing) is an application programming interface for shared memory parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications.
-\LZK{Cette partie est réécrite. A relire et à améliorer si possible.}
+OpenMP (Open Multi-processing) is an application programming interface for shared memory parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications.
\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.
+%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 (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.
+%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.
+%We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
-\section{The EA algorithm on a single GPU}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{The Ehrlich-Aberth algorithm on a GPU}
\label{sec3}
-\subsection{The EA method}
+\subsection{The EA method}
A cubically convergent iteration method to find zeros of
polynomials was proposed by O. Aberth~\cite{Aberth73}. The
Ehrlich-Aberth (EA is short) method contains 4 main steps, presented in what
\subsection{EA parallel implementation on CUDA}
+We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
+
+
+
+
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
-%\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]{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]{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]{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]{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]{big}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degrees on 4 GPUs.}
+\label{fig:09}
+\end{figure}
% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
