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33 \title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
35 \author{\IEEEauthorblockN{Kahina Guidouche, Abderrahmane Sider }
36 \IEEEauthorblockA{Laboratoire LIMED\\
37 Faculté des sciences exactes\\
38 Université de Bejaia, 06000, Algeria\\
39 Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
41 \IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
42 \IEEEauthorblockA{FEMTO-ST Institute\\
43 University of Bourgogne Franche-Comte, France\\
44 Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
50 Finding roots of polynomials is a very important part of solving
51 real-life problems but it is not so easy for polynomials of high
52 degrees. In this paper, we present two different parallel algorithms
53 of the Ehrlich-Aberth method to find roots of sparse and fully defined
54 polynomials of high degrees. Both algorithms are based on CUDA
55 technology to be implemented on multi-GPU computing platforms but each
56 using different parallel paradigms: OpenMP or MPI. The experiments
57 show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
58 GPU to find roots of polynomials of degree up-to 1.4
59 million. Moreover, other experiments show it is possible to find roots
60 of polynomials of degree up-to 5 millions.
64 \LZK{Faut pas mettre des keywords?}
67 \IEEEpeerreviewmaketitle
70 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
71 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
72 \section{Introduction}
75 Finding roots of polynomials of very high degrees arises in many complex problems in various domains such as algebra, biology or physics. A polynomial $p(x)$ in $\mathbb{C}$ in one variable $x$ is an algebraic expression in $x$ of the form:
77 p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0,
79 where $\{a_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $a_n\neq0$ then $n$ is called the degree of the polynomial. The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called roots of $p(x)$. Let $\{z_i\}_{1\leq i\leq n}$ be the roots of polynomial $p(x)$, then $p(x)$ can be written as :
81 p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
84 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}.
86 The main problem of the simultaneous methods is that the necessary
87 time needed for the convergence increases with the increasing of the
88 polynomial's degree. Many authors have treated the problem of
89 implementing simultaneous methods in
90 parallel. Freeman~\cite{Freeman89} implemented and compared
91 Durand-Kerner method, Ehrlich-Aberth method and another method of the
92 fourth order of convergence proposed by Farmer and
93 Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials
94 of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83}
95 often diverges, but the first two methods (Durand-Kerner and
96 Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman
97 and Bane~\cite{Freemanall90} considered asynchronous algorithms in
98 which each processor continues to update its approximations even
99 though the latest values of other approximations $z^{k}_{i}$ have not
100 been received from the other processors, in contrast with synchronous
101 algorithms where it would wait those values before making a new
102 iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods
103 of parallelization for a shared memory architecture with OpenMP and
104 for a distributed memory one with MPI. They are able to compute the
105 roots of sparse polynomials of degree 10,000 in 116 seconds with
106 OpenMP and 135 seconds with MPI only by using 8 personal computers and
107 2 communications per iteration. The authors showed an interesting
108 speedup comparing to the sequential implementation which takes up-to
109 3,300 seconds to obtain same results.
110 \RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?}
111 \LZK{Supprimons ces détails et mettons une référence s'il y en a une}
113 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.
115 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:
116 \LZK{J'ai ajouté une phrase pour justifier notre choix de la méthode Ehrlich-Aberth. A revérifier.}
119 \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.
120 \item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a distributed memory using MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications.
122 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.
123 \LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.}
125 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.
126 %\LZK{A revoir toute cette organization: je viens de la revoir}
128 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
129 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
131 \section{Parallel programming models}
133 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.
138 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.
143 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.
148 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.
151 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
152 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
154 \section{The Ehrlich-Aberth algorithm on a GPU}
157 \subsection{The Ehrlich-Aberth method}
159 The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
162 z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
163 {1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,\ldots,n
166 This method contains 4 steps. The first step consists in the initializing the polynomial. The second step initializes the solution vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all distinct from each other. In step 3, the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is applied. In our case, the Ehrlich-Aberth is applied as in~(\ref{Eq:EA1}). Iterations of the Ehrlich-Aberth method will converge to the roots of the considered polynomial. In order to stop the iterative function, a stop condition is applied, this is the 4th step. This condition checks that all the root modules are lower than a fixed value $\epsilon$.
169 \label{eq:Aberth-Conv-Cond}
170 \forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
173 \subsection{Improving Ehrlich-Aberth method}
174 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.
177 In order to solve this problem, we propose to modify the iterative
178 function by using the logarithm and the exponential of a complex and
179 we propose a new version of the Ehrlich-Aberth method. This method
180 allows us to exceed the computation of the polynomials of degree
181 100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
185 z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
192 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})).
197 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}.
200 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}
202 Our objective consists in implementing a root finding polynomial
203 algorithm on multiple GPUs. To this end, it is primordial to know how
204 to manage CUDA contexts of different GPUs. A direct method for
205 controlling the various GPUs is to use as many threads or processes as
206 GPU devices. We can choose the GPU index based on the identifier of
207 OpenMP thread or the rank of the MPI process. Both approaches will be
208 investigated. \LZK{Répétition! Le même texte est déjà écrit comme
209 intro dans la section II. Sinon ici on parle seulement de
210 l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
211 revoir après, quand les 2 parties suivantes seront plus stables}}
216 Like any parallel code, a GPU parallel implementation first requires to determine the sequential code and the data-parallel operations of a algorithm. In fact, all the operations that are easy to execute in parallel must be made by the GPU to accelerate the execution, like the steps 3 and 4. On the other hand, all the sequential operations and the operations that have data dependencies between CUDA threads or recursive computations must be executed by only one CUDA thread or a CPU thread (the steps 1 and 2).\LZK{La méthode est déjà mal présentée, dans ce cas c'est encore plus difficile de comprendre que représentent ces différentes étapes!} Initially, we specify the organization of parallel threads by specifying the dimension of the grid \verb+Dimgrid+, the number of blocks per grid \verb+DimBlock+ and the number of threads per block.
218 The code is organized as kernels which are parts of code that are run on GPU devices. For step 3, there are two kernels, the first is named \textit{save} is used to save vector $Z^{K-1}$ and the second one is
219 named \textit{update} and is used to update the $Z^{K}$ vector. For
220 step 4, a kernel tests the convergence of the method. In order to
221 compute the function H, we have two possibilities: either to use the
222 Jacobi mode, or the Gauss-Seidel mode of iterating which uses the most
223 recent computed roots. It is well known that the Gauss-Seidel mode
224 converges more quickly. So, we use Gauss-Seidel iterations. To
225 parallelize the code, we create kernels and many functions to be
226 executed on the GPU for all the operations dealing with the
227 computation on complex numbers and the evaluation of the
228 polynomials. As said previously, we manage both functions of
229 evaluation: the normal method, based on the method of
230 Horner and the method based on the logarithm of the polynomial. All
231 these methods were rather long to implement, as the development of
232 corresponding kernels with CUDA is longer than on a CPU host. This
233 comes in particular from the fact that it is very difficult to debug
234 CUDA running threads like threads on a CPU host. In the following
235 paragraph Algorithm~\ref{alg1-cuda} shows the GPU parallel
236 implementation of Ehrlich-Aberth method.
237 \LZK{Vaut mieux expliquer l'implémentation en faisant référence à l'algo séquentiel que de parler des différentes steps.}
241 \begin{algorithm}[htpb]
244 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
245 \KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold)}
246 \KwOut{$Z$ (solution vector of roots)}
247 Initialize the polynomial $P$ and its derivative $P'$\;
248 Set the initial values of vector $Z$\;
249 Copy $P$, $P'$ and $Z$ from CPU to GPU\;
250 \While{\emph{not convergence}}{
251 $Z^{prev}$ = KernelSave($Z,n$)\;
252 $Z$ = KernelUpdate($P,P',Z,n$)\;
253 $\Delta Z$ = KernelComputeError($Z,Z^{prev},n$)\;
254 $\Delta Z_{max}$ = CudaMaxFunction($\Delta Z,n$)\;
255 TestConvergence($\Delta Z_{max},\epsilon$)\;
257 Copy $Z$ from GPU to CPU\;
259 \LZK{J'ai modifié l'algo. Sinon, est ce qu'on doit mettre en paramètre
260 $Z^{prev}$ ou $Z$ tout court (dans le cas où on exploite
261 l'asynchronisme des threads cuda!) pour le Kernel\_Update? }
262 \RC{Le $Z_{prev}$ sert à calculer l'erreur donc j'ai remis Z. La ligne
263 avec TestConvergence ca fait une ligne de plus.}
267 \section{The EA algorithm on Multiple GPUs}
269 \subsection{an OpenMP-CUDA approach}
270 Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
271 OpenMP and CUDA programming model. All the data are shared with
272 OpenMP amoung all the OpenMP threads. The shared data are the solution
273 vector $Z$, the polynomial to solve $P$, and the error vector $\Delta
274 z$. The number of OpenMP threads is equal to the number of GPUs, each
275 OpenMP thread binds to one GPU, and it controls a part of the shared
276 memory. More precisely each OpenMP thread will be responsible to
277 update its owns part of the vector Z. This part is call $Z_{loc}$ in
278 the following. Then all GPUs will have a grid of computation organized
279 according to the device performance and the size of data on which it
280 runs the computation kernels.
282 To compute one iteration of the EA method each GPU performs the
283 followings steps. First roots are shared with OpenMP and the
284 computation of the local size for each GPU is performed (lines 5-7 in
285 Algo\ref{alg2-cuda-openmp}). Each thread starts by copying all the
286 previous roots inside its GPU (line 9). Then each GPU will copy the
287 previous roots (line 10) and it will compute an iteration of the EA
288 method on its own roots (line 11). For that all the other roots are
289 used. The convergence is checked on the new roots (line 12). At the end
290 of an iteration, the updated roots are copied from the GPU to the
291 CPU (line 14) by direcly updating its own roots in the shared memory
292 arrays containing all the roots.
296 \begin{algorithm}[htpb]
299 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
300 \KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold), $ngpu$ (number of GPUs)}
301 \KwOut{$Z$ (solution vector of roots)}
302 Initialize the polynomial $P$ and its derivative $P'$\;
303 Set the initial values of vector $Z$\;
304 Start of a parallel part with OpenMP ($Z$, $\Delta Z$, $\Delta
305 Z_{max}$, $P$, $P'$ are shared variables)\;
306 $id_{gpu}$ = cudaGetDevice()\;
307 $n_{loc}$ = $n/ngpu$ (local size)\;
308 %$idx$ = $id_{gpu}\times n_{loc}$ (local offset)\;
309 Copy $P$, $P'$ from CPU to GPU\;
310 \While{\emph{not convergence}}{
311 Copy $Z$ from CPU to GPU\;
312 $Z^{prev}$ = KernelSave($Z,n$)\;
313 $Z_{loc}$ = KernelUpdate($P,P',Z^{prev},n_{loc}$)\;
314 $\Delta Z_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc},n_{loc}$)\;
315 $\Delta Z_{max}[id_{gpu}]$ = CudaMaxFunction($\Delta Z_{loc},n_{loc}$)\;
316 Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
317 $max$ = MaxFunction($\Delta Z_{max},ngpu$)\;
318 TestConvergence($max,\epsilon$)\;
320 \label{alg2-cuda-openmp}
321 \LZK{J'ai modifié l'algo. Le $P$ est mis shared. Qu'en est-il pour
322 $P'$?}\RC{Je l'ai rajouté. Bon sinon le n\_loc ne remplace pas
323 vraiment un offset et une taille mais bon... et là il y a 4 lignes
324 pour la convergence, c'est bcp ... Zloc, Zmax, max et
325 testconvergence. On pourrait faire mieux}
332 \subsection{an MPI-CUDA approach}
334 Our parallel implementation of EA to find root of polynomials using a CUDA-MPI approach is a data parallel approach. It splits input data of the polynomial to solve among MPI processes. In Algorithm \ref{alg2-cuda-mpi}, input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $ZPrev$, and the value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the degree of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $\lceil n/p \rceil$ roots to find per MPI process, for each $Z$ and $ZPrec$. Consequently, each MPI process of rank $k$ will have its own solution vector $Z_{k}$ and $ZPrec$, the error related to the stop condition $\Delta z_{k}$, enabling each MPI process to compute $\lceil n/p \rceil$ roots.
336 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.
338 \begin{algorithm}[htpb]
339 \label{alg2-cuda-mpi}
341 \caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
343 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
344 threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
346 \KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
349 Initialization of P\;
350 Initialization of Pu\;
351 Initialization of the solution vector $Z^{0}$\;
353 Allocate memory to GPU\;
354 \While {$error > \epsilon$}{
355 copy Z from CPU to GPU\;
356 $ZPrec_{loc}=kernel\_save(Z_{loc})$\;
357 $Z_{loc}=kernel\_update(Z,P,Pu)$\;
358 $\Delta z=kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
359 $error=MPI\_Reduce(\Delta z)$\;
360 Copy $Z_{loc}$ from GPU to CPU\;
361 $Z=MPI\_AlltoAll(Z_{loc})$\;
366 \section{Experiments}
368 We study two categories of polynomials: sparse polynomials and full polynomials.\\
369 {\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:
371 \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
372 \end{equation}\noindent
373 {\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
375 %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
379 {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
382 For our test, 4 cards GPU tesla Kepler K40 are used. In order to
383 evaluate both the GPU and Multi-GPU approaches, we performed a set of
384 experiments on a single GPU and multiple GPUs using OpenMP or MPI with
385 the EA algorithm, for both sparse and full polynomials of different
386 sizes. All experimental results obtained are perfomed with double
387 precision float data and the convergence threshold of the EA method is
388 set to $10^{-7}$. The initialization values of the vector solution of
389 the methods are given by Guggenheimer method~\cite{Gugg86}.
392 \subsection{Evaluation of the CUDA-OpenMP approach}
394 Here we report some experiments witt full and sparse polynomials of
395 different degrees with multiple GPUs.
396 \subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
398 In this experiments we report the execution time of the EA algorithm, on single GPU and multi-GPUs with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
402 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
403 \caption{Execution time in seconds of the Ehrlich-Aberth method to
404 solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
408 Figure~\ref{fig:01} shows that the CUDA-OpenMP approach scales well
409 with multiple GPUs. This version allows us to solve sparse polynomials
410 of very high degrees.
412 \subsubsection{Execution times of the EA method to solve full polynomials on multiple GPUs}
414 These experiments show the execution times 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.
418 \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
419 \caption{Execution time in seconds of the Ehrlich-Aberth method to
420 solve full polynomials on multiple GPUs with CUDA-OpenMP.}
424 In Figure~\ref{fig:02}, we can observe that with full polynomials the EA version with
425 CUDA-OpenMP scales also well. Using 4 GPUs allows us to achieve a
426 quasi-linear speedup.
428 \subsection{Evaluation of the CUDA-MPI approach}
429 In this part we perform some experiments to evaluate the CUDA-MPI
430 approach to solve full and sparse polynomials of degrees ranging from
431 100,000 to 1,400,000.
433 \subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
437 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
438 \caption{Execution time in seconds of the Ehrlich-Aberth method to
439 solve sparse polynomials on multiple GPUs with CUDA-MPI.}
442 Figure~\ref{fig:03} shows the execution times of te EA algorithm,
443 for a single GPU, and multiple GPUs (2, 3, 4) with the CUDA-MPI approach.
445 \subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}
449 \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
450 \caption{Execution times in seconds of the Ehrlich-Aberth method for
451 full polynomials on multiple GPUs with CUDA-MPI.}
455 In Figure~\ref{fig:04}, we can also observe that the CUDA-MPI approach
456 is also efficient to solve full polynimails on multiple GPUs.
458 \subsection{Comparison of the CUDA-OpenMP and the CUDA-MPI approaches}
460 In the previuos section we saw that both approches are very effecient
461 to reduce the execution times the sparse and full polynomials. In
462 this section we try to compare these two approaches.
464 \subsubsection{Solving sparse polynomials}
465 In this experiment three sparse polynomials of size 200K, 800K and 1,4M are investigated.
468 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
469 \caption{Execution times to solvs sparse polynomials of three
470 distinct sizes on multiple GPUs using MPI and OpenMP with the
471 Ehrlich-Aberth method}
474 In Figure~\ref{fig:05} there is one curve for CUDA-MPI and another one
475 for CUDA-OpenMP. We can see that the results are quite similar between
476 OpenMP and MPI for the polynomials size of 200K. For the size of 800K,
477 the MPI version is a little bit slower than the OpenMP approach but for
478 the 1,4 millions size, there is a slight advantage for the MPI
481 \subsubsection{Solving full polynomials}
484 \includegraphics[angle=-90,width=0.5\textwidth]{Full}
485 \caption{Execution time for solving full polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
488 In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
490 \subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
492 In this experiment we compare the execution time of the EA algorithm
493 according to the number of GPUs to solve sparse and full
494 polynomials on multiples GPUs using MPI. We chose three sparse and full
495 polynomials of size 200K, 800K and 1,4M.
498 \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
499 \caption{Execution times to solve sparse and full polynomials of three distinct sizes on multiple GPUs using MPI.}
502 In Figure~\ref{fig:07} we can see that CUDA-MPI can solve sparse and
503 full polynomials of high degrees, the execution times with sparse
504 polynomial are very low compared to full polynomials. With sparse
505 polynomials the number of monomials is reduced, consequently the number
506 of operations is reduced and the execution time decreases.
508 \subsubsection{Solving sparse and full polynomials of the same size with CUDA-OpenMP}
512 \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
513 \caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
517 Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact follows the same pattern, a difference in execution time in favor of the sparse polynomials.
519 \subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
520 These experiments report the execution times of the EA method for
521 sparse and full polynomials ranging from 1,000,000 to 5,000,000.
524 \includegraphics[angle=-90,width=0.5\textwidth]{big}
525 \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degree on 4 GPUs for sizes ranging from 1M to 5M}
528 In Figure~\ref{fig:09} we can see that both approaches are scalable
529 and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both
530 approaches give very similar results.
532 %SIDER JE viens de virer \c ca For sparse polynomials here are a noticeable difference in favour of MPI when the degree is
533 %above 4 millions. Between 1 and 3 millions, OpenMP is more effecient.
534 %Under 1 million, OpenMPI and MPI are almost equivalent.
536 %SIDER : il faut une explication sur les différences ici aussi.
541 In this paper, we have presented a parallel implementation of
542 Ehrlich-Aberth algorithm to solve full and sparse polynomials, on
543 single GPU with CUDA and on multiple GPUs using two parallel
544 paradigms: shared memory with OpenMP and distributed memory with
545 MPI. These architectures were addressed by a CUDA-OpenMP approach and
546 CUDA-MPI approach, respectively. Experiments show that, using
547 parallel programming model like (OpenMP, MPI). We can efficiently
548 manage multiple graphics cards to solve the same
549 problem and accelerate the parallel execution with 4 GPUs and solve a
550 polynomial of degree up to 5,000,000, four times faster than on single
554 Our next objective is to extend the model presented here with clusters
555 of GPU nodes, with a three-level scheme: inter-node communication via
556 MPI processes (distributed memory), management of multi-GPU node by
557 OpenMP threads (shared memory).
560 \section*{Acknowledgment}
562 Computations have been performed on the supercomputer facilities of
563 the Mésocentre de calcul de Franche-Comté. We also would like to thank
564 Nvidia for hardware donation under CUDA Research Center 2014.
568 \bibliography{mybibfile}
