english correct

[kahina_paper2.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 308e42aeaff23891f47a874157ba13ed831b7a66..00c96254c9a97c003afc362f20382e82f2874d27 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -39,17 +39,17 @@ Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \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
-degrees. In this paper, we present two different parallel algorithms
-of the Ehrlich-Aberth method to find roots of sparse and fully defined
-polynomials of high degrees. Both algorithms are based on CUDA
-technology to be implemented on multi-GPU computing platforms but each
-using different parallel paradigms: OpenMP or MPI. The experiments
-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.
+Finding the roots of polynomials is a very important part of solving
+real-life problems but the higher the degree of the polynomials is,
+the less easy it becomes. In this paper, we present two different
+parallel algorithms of the Ehrlich-Aberth method to find roots of
+sparse and fully defined polynomials of high degrees. Both algorithms
+are based on CUDA technology to be implemented on multi-GPU computing
+platforms but each use different parallel paradigms: OpenMP or
+MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU
+devices compared to 1 GPU to find the roots of polynomials of degree up-to
+1.4 million. Moreover, other experiments show it is possible to find the
+roots of polynomials of degree up-to 5 million.

 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -64,7 +64,7 @@ of polynomials of degree up-to 5 millions.
 \section{Introduction}
 
 
 \section{Introduction}
 
 

-Finding roots of polynomials of very high degrees arises in many complex problems of 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:
+Finding the roots of polynomials of very high degrees arises in many complex problems of 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}{\alpha_ix^i},\alpha_n\neq 0, 
 \end{equation}
 \begin{equation}
 p(x) = \displaystyle\sum^n_{i=0}{\alpha_ix^i},\alpha_n\neq 0, 
 \end{equation}


@@ -73,14 +73,25 @@ where $\{\alpha_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high
  p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
 \end{equation}
 
  p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
 \end{equation}
 

-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}.
-
-
-The convergence time of simultaneous methods drastically increases with the increasing of the polynomial's degree. the great challenges with simultaneous methods is the primordial need to parallelize it and will improve the convergence time. Many authors have treated to
-implement simultaneous methods in parallel.
-Many authors have treated to implement simultaneous methods in parallel(~\cite{Freeman89},~\cite{Loizou83}, ~\cite{Freemanall90}, ~\cite{Raphaelall01},~\cite{Couturier02}), using several paradigms of parallelization   (synchronous or asynchronous calculus, mechanism of shared or distributed memory,...).
-%but they can not able to solve very high polynomial's degree exceed to 100,000.
-They are able to solve only polynomial's degree not exceed 20,000.
+Most of the numerical methods that deal with the polynomial
+root-finding problems are simultaneous methods, \textit{i.e.} the
+iterative methods to find simultaneous approximations of the $n$
+polynomial roots. These methods start from the initial approximation
+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
+the Durand-Kerner method~\cite{Durand60,Kerner66} and the Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
+
+
+The convergence time of simultaneous methods drastically increases
+with the increasing of the polynomial's degree. The great challenge
+with simultaneous methods is to parallelize them and to improve their
+convergence. Many authors have proposed parallel simultaneous
+methods~\cite{Freeman89,Loizou83,Freemanall90,bini96,cs01:nj,Couturier02},
+using several paradigms of parallelization (synchronous or
+asynchronous computations, mechanism of shared or distributed memory,
+etc). However, so fat until now, only polynomials not exceeding
+degrees of less than 100,000 have been solved.

 
 %The main problem of the simultaneous methods is that the necessary
 %time needed for the convergence increases with the increasing of the
 
 %The main problem of the simultaneous methods is that the necessary
 %time needed for the convergence increases with the increasing of the


@@ -103,25 +114,26 @@ They are able to solve only polynomial's degree not exceed 20,000.
 %for a distributed memory one with MPI. They are able to compute the
 %roots of sparse polynomials of degree 10,000. The authors showed an interesting
 %speedup that is 20 times as fast as the sequential implementation. 
 %for a distributed memory one with MPI. They are able to compute the
 %roots of sparse polynomials of degree 10,000. The authors showed an interesting
 %speedup that is 20 times as fast as the sequential implementation. 

-\KG{je viens de modifier le paragraphe ci-dessus pour eviter des similarité avec d'autre article, merci de le relire....}

 
 

-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{Y a pas d'autres travaux pour la résolution de polynômes sur GPUs?}
+With the recent advent of the Compute Unified Device Architecture
+(CUDA)~\cite{CUDA15}, a parallel computing platform and a programming
+model invented by NVIDIA had revived parallel programming interest for
+this problem. Indeed, the computing power of GPUs (Graphics Processing
+Units) has exceeded that of traditional  CPUs processors, which makes it very appealing to the research community to investigate new parallel implementations for a whole set of scientific problems in the reasonable hope to solve bigger instances of well known computationally demanding issues such as the one beforehand. 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. 

 
 

-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:
+In this paper we propose the parallelization of the Ehrlich-Aberth (EA) method which has a cubic convergence rate which is much better than the quadratic rate of the Durand-Kerner method which has already been investigated in \cite{Kahinall14}. In the other hand,  EA is suitable to be implemented in parallel computers according to the data-parallel paradigm. In this model, computing elements carry computations on the data they are assigned and communicate with other computing elements in order to get fresh data or to synchronise. Classically, two parallel programming paradigms OpenMP and MPI are used to code such solutions. But in our case, computing elements are CUDA multi-GPU platforms. This architectural setting poses new programming challenges but offers also new opportunities to efficiently solve huge problems, otherwise considered intractable until recently. To the best of our knowledge, our CUDA-MPI and CUDA-OpenMP codes are the first implementations of EA method with multiple GPUs for finding roots of polynomials. Our major contributions include:

  \begin{itemize}
 
  \begin{itemize}
 

-\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
+\item The parallel implementation of EA 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 EA 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
   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.
+  exchange their data by message-passing communications. This 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
   Our method is efficient to compute the roots of sparse and full
   polynomials of degree up to 5 millions.
  \end{itemize}
 \item
   Our method is efficient to compute the roots of sparse and full
   polynomials of degree up to 5 millions.
  \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.
-%\AS{La résolution du problème pour des polynomes pleins de degré 6M est une contribution aussi à mon avis}}
+

 
 The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA. In Section~\ref{sec3} we present the implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on multiple GPUs using the OpenMP and MPI approaches. In section~\ref{sec5} we present our experiments and discuss them. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic. 
 
 
 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. 
 


@@ -134,18 +146,12 @@ The paper is organized as follows. In Section~\ref{sec2} we present three differ
 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}
 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}

-
-

 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} 
 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} 

-
-

 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}
 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 (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. 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.
 
 
 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. 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.
 
 


@@ -184,7 +190,6 @@ root modules are lower than a fixed value $\epsilon$.
 \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.
  
 \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.
  

-

 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
 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


@@ -203,25 +208,20 @@ where:
 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}
 
 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}
 

-
-

 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}. In practice, the  exponential and
 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}. In practice, the  exponential and

-logarithm mode is used a root excepts the circle unit, represented by the radius $R$ evaluated in C language as :
+logarithm mode is used when a root is outisde the circle unit represented by the radius $R$ evaluated in C language with:

 \begin{equation}
 \label{R.EL}
 R = exp(log(DBL\_MAX)/(2*n) );
 \end{equation}
 \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.
+where \verb=DBL_MAX= stands for the maximum representable
+\verb=double= value and $n$ is the degree of the polynimal.

 
 
 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}
 
 
 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}

-
-
-
-

 The code is organized as kernels which are parts of code that are run
 on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
 implementation of the Ehrlich-Aberth on a GPU.  This algorithms starts
 The code is organized as kernels which are parts of code that are run
 on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
 implementation of the Ehrlich-Aberth on a GPU.  This algorithms starts


@@ -238,7 +238,7 @@ long since it performs all the operations with complex numbers with
 the normal mode of the EA method but also with the
 logarithm-exponential one. Then the error is computed with a final
 kernel (line 7). Finally when the EA method has converged, the roots
 the normal mode of the EA method but also with the
 logarithm-exponential one. Then the error is computed with a final
 kernel (line 7). Finally when the EA method has converged, the roots

-are transferred from the GPU to the CPU.%\LZK{Quand est ce qu'on utilise un normal mode ou logarithm-exponential mode?}
+are transferred from the GPU to the CPU.

 
 \begin{algorithm}[htpb]
 \label{alg1-cuda}
 
 \begin{algorithm}[htpb]
 \label{alg1-cuda}


@@ -329,7 +329,6 @@ Copy $P$, $P'$ from CPU to GPU\;
 
 
 \subsection{A MPI-CUDA approach}
 
 
 \subsection{A MPI-CUDA approach}

-

 Our parallel implementation of EA to find roots of polynomials using a
 CUDA-MPI approach follows a similar approach to the one used in
 CUDA-OpenMP. Each process is responsible to compute its own part of
 Our parallel implementation of EA to find roots of polynomials using a
 CUDA-MPI approach follows a similar approach to the one used in
 CUDA-OpenMP. Each process is responsible to compute its own part of


@@ -484,9 +483,9 @@ These experiments report the execution times of the EA method for sparse and ful
 
 \section{Conclusion}
 \label{sec6}
 
 \section{Conclusion}
 \label{sec6}

-In this paper, we have presented parallel implementations of the Ehrlich-Aberth algorithm to solve full and sparse polynomials, on a 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. Experiments show that, using parallel programming model like (OpenMP or MPI), we can efficiently manage multiple graphics cards to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree up-to 5,000,000 four times faster than on single GPU. 
+In this paper, we have presented parallel implementations of the Ehrlich-Aberth algorithm to solve full and sparse polynomials, on a 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. Experiments show that, using parallel programming model like (OpenMP or MPI), we can efficiently manage multiple graphics cards to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree up-to 5,000,000 four times faster than on a single GPU. 

 
 

-Our next objective is to extend the model presented here with clusters of GPU nodes, with a three-level scheme: inter-node communications via MPI processes (distributed memory), management of multi-GPU nodes by OpenMP threads (shared memory).
+Our next objective is to extend the model presented here to clusters of GPU nodes, with a three-level scheme: inter-node communications via MPI processes (distributed memory), management of multi-GPU nodes by OpenMP threads (shared memory). Actual platforms may probably also contain purely multi-core nodes without any GPU. This heterogeneous setting may lead to the integration of load balancing algorithms so as to allow an optimal use of hardware ressources. 

 
 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}