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24 \title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
26 \author{\IEEEauthorblockN{Kahina Guidouche, Abderrahmane Sider }
27 \IEEEauthorblockA{Laboratoire LIMED\\
28 Faculté des sciences exactes\\
29 Université de Bejaia, 06000, Algeria\\
30 Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
32 \IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
33 \IEEEauthorblockA{FEMTO-ST Institute\\
34 University of Bourgogne Franche-Comte, France\\
35 Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
41 Finding roots of polynomials is a very important part of solving
42 real-life problems but it is not so easy for polynomials of high
43 degrees. In this paper, we present two different parallel algorithms
44 of the Ehrlich-Aberth method to find roots of sparse and fully defined
45 polynomials of high degrees. Both algorithms are based on CUDA
46 technology to be implemented on multi-GPU computing platforms but each
47 using different parallel paradigms: OpenMP or MPI. The experiments
48 show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
49 GPU to find roots of polynomials of degree up-to 1.4
50 million. Moreover, other experiments show it is possible to find roots
51 of polynomials of degree up-to 5 millions.
55 \LZK{Faut pas mettre des keywords?}
58 \IEEEpeerreviewmaketitle
61 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
62 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
63 \section{Introduction}
66 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:
68 p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0,
70 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 :
72 p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
75 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}.
77 The main problem of the simultaneous methods is that the necessary
78 time needed for the convergence increases with the increasing of the
79 polynomial's degree. Many authors have treated the problem of
80 implementing simultaneous methods in
81 parallel. Freeman~\cite{Freeman89} implemented and compared
82 Durand-Kerner method, Ehrlich-Aberth method and another method of the
83 fourth order of convergence proposed by Farmer and
84 Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials
85 of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83}
86 often diverges, but the first two methods (Durand-Kerner and
87 Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman
88 and Bane~\cite{Freemanall90} considered asynchronous algorithms in
89 which each processor continues to update its approximations even
90 though the latest values of other approximations $z^{k}_{i}$ have not
91 been received from the other processors, in contrast with synchronous
92 algorithms where it would wait those values before making a new
93 iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods
94 of parallelization for a shared memory architecture with OpenMP and
95 for a distributed memory one with MPI. They are able to compute the
96 roots of sparse polynomials of degree 10,000 in 116 seconds with
97 OpenMP and 135 seconds with MPI only by using 8 personal computers and
98 2 communications per iteration. The authors showed an interesting
99 speedup comparing to the sequential implementation which takes up-to
100 3,300 seconds to obtain same results.
101 \RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?}
102 \LZK{Supprimons ces détails et mettons une référence s'il y en a une}
104 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.
106 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:
107 \LZK{J'ai ajouté une phrase pour justifier notre choix de la méthode Ehrlich-Aberth. A revérifier.}
110 \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.
111 \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.
113 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.
114 \LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.}
116 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.
119 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
120 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
122 \section{Parallel programming models}
124 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.
129 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.
134 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.
139 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.
142 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
143 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
145 \section{The Ehrlich-Aberth algorithm on a GPU}
148 \subsection{The Ehrlich-Aberth method}
150 The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
153 z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
154 {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
157 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 $\epsilon$.
160 \label{eq:Aberth-Conv-Cond}
161 \forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
164 \subsection{Improving Ehrlich-Aberth method}
165 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.
168 In order to solve this problem, we propose to modify the iterative
169 function by using the logarithm and the exponential of a complex and
170 we propose a new version of the Ehrlich-Aberth method. This method
171 allows us to exceed the computation of the polynomials of degree
172 100,000 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 $i=1,\dots,n$:
176 z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
183 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})).
188 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}.
191 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}
193 Our objective consists in implementing a root finding polynomial
194 algorithm on multiple GPUs. To this end, it is primordial to know how
195 to manage CUDA contexts of different GPUs. A direct method for
196 controlling the various GPUs is to use as many threads or processes as
197 GPU devices. We can choose the GPU index based on the identifier of
198 OpenMP thread or the rank of the MPI process. Both approaches will be
199 investigated. \LZK{Répétition! Le même texte est déjà écrit comme
200 intro dans la section II. Sinon ici on parle seulement de
201 l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
202 revoir après, quand les 2 parties suivantes seront plus stables}}
207 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.
209 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
210 named \textit{update} and is used to update the $Z^{K}$ vector. For
211 step 4, a kernel tests the convergence of the method. In order to
212 compute the function H, we have two possibilities: either to use the
213 Jacobi mode, or the Gauss-Seidel mode of iterating which uses the most
214 recent computed roots. It is well known that the Gauss-Seidel mode
215 converges more quickly. So, we use Gauss-Seidel iterations. To
216 parallelize the code, we create kernels and many functions to be
217 executed on the GPU for all the operations dealing with the
218 computation on complex numbers and the evaluation of the
219 polynomials. As said previously, we manage both functions of
220 evaluation: the normal method, based on the method of
221 Horner and the method based on the logarithm of the polynomial. All
222 these methods were rather long to implement, as the development of
223 corresponding kernels with CUDA is longer than on a CPU host. This
224 comes in particular from the fact that it is very difficult to debug
225 CUDA running threads like threads on a CPU host. In the following
226 paragraph Algorithm~\ref{alg1-cuda} shows the GPU parallel
227 implementation of Ehrlich-Aberth method.
228 \LZK{Vaut mieux expliquer l'implémentation en faisant référence à l'algo séquentiel que de parler des différentes steps.}
232 \begin{algorithm}[htpb]
235 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
236 \KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold)}
237 \KwOut{$Z$ (solution vector of roots)}
238 Initialize the polynomial $P$ and its derivative $P'$\;
239 Set the initial values of vector $Z$\;
240 Copy $P$, $P'$ and $Z$ from CPU to GPU\;
241 \While{$\Delta Z_{max} > \epsilon$}{
242 $Z^{prev}$ = KernelSave($Z,n$)\;
243 $Z$ = KernelUpdate($P,P',Z,n$)\;
244 $\Delta Z$ = KernelComputeError($Z,Z^{prev},n$)\;
245 $\Delta Z_{max}$ = CudaMaxFunction($\Delta Z,n$)\;
247 Copy $Z$ from GPU to CPU\;
249 \RC{La ligne avec TestConvergence ca fait une ligne de plus.\LZK{Oui j'ai hésité à l'ajouter. On peut faire le test dans la condition de while mais quelle est la valeur initiale de $\Delta Z_{max}$?! Ou bien on s'en fiche?}}
253 \section{The EA algorithm on Multiple GPUs}
255 \subsection{an OpenMP-CUDA approach}
256 Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
257 OpenMP and CUDA programming model. All the data are shared with
258 OpenMP amoung all the OpenMP threads. The shared data are the solution
259 vector $Z$, the polynomial to solve $P$, and the error vector $\Delta
260 z$. The number of OpenMP threads is equal to the number of GPUs, each
261 OpenMP thread binds to one GPU, and it controls a part of the shared
262 memory. More precisely each OpenMP thread will be responsible to
263 update its owns part of the vector Z. This part is call $Z_{loc}$ in
264 the following. Then all GPUs will have a grid of computation organized
265 according to the device performance and the size of data on which it
266 runs the computation kernels.
268 To compute one iteration of the EA method each GPU performs the
269 followings steps. First roots are shared with OpenMP and the
270 computation of the local size for each GPU is performed (lines 5-7 in
271 Algo\ref{alg2-cuda-openmp}). Each thread starts by copying all the
272 previous roots inside its GPU (line 9). Then each GPU will copy the
273 previous roots (line 10) and it will compute an iteration of the EA
274 method on its own roots (line 11). For that all the other roots are
275 used. The convergence is checked on the new roots (line 12). At the end
276 of an iteration, the updated roots are copied from the GPU to the
277 CPU (line 14) by direcly updating its own roots in the shared memory
278 arrays containing all the roots.
282 \begin{algorithm}[htpb]
285 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
286 \KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold), $ngpu$ (number of GPUs)}
287 \KwOut{$Z$ (solution vector of roots)}
288 Initialize the polynomial $P$ and its derivative $P'$\;
289 Set the initial values of vector $Z$\;
290 Start of a parallel part with OpenMP ($Z$, $\Delta Z$, $\Delta
291 Z_{max}$, $P$, $P'$ are shared variables)\;
292 $id_{gpu}$ = cudaGetDevice()\;
293 $n_{loc}$ = $n/ngpu$ (local size)\;
294 %$idx$ = $id_{gpu}\times n_{loc}$ (local offset)\;
295 Copy $P$, $P'$ from CPU to GPU\;
296 \While{\emph{not convergence}}{
297 Copy $Z$ from CPU to GPU\;
298 $Z^{prev}$ = KernelSave($Z,n$)\;
299 $Z_{loc}$ = KernelUpdate($P,P',Z^{prev},n_{loc}$)\;
300 $\Delta Z_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc},n_{loc}$)\;
301 $\Delta Z_{max}[id_{gpu}]$ = CudaMaxFunction($\Delta Z_{loc},n_{loc}$)\;
302 Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
303 $max$ = MaxFunction($\Delta Z_{max},ngpu$)\;
304 TestConvergence($max,\epsilon$)\;
306 \label{alg2-cuda-openmp}
307 \LZK{J'ai modifié l'algo. Le $P$ est mis shared. Qu'en est-il pour
308 $P'$?}\RC{Je l'ai rajouté. Bon sinon le n\_loc ne remplace pas
309 vraiment un offset et une taille mais bon... et là il y a 4 lignes
310 pour la convergence, c'est bcp ... Zloc, Zmax, max et
311 testconvergence. On pourrait faire mieux}
318 \subsection{a MPI-CUDA approach}
320 Our parallel implementation of EA to find root of polynomials using a
321 CUDA-MPI approach follows a similar computing approach to the one used
322 in CUDA-OpenMP. Each process is responsible to compute its own part of
323 roots using all the roots computed by other processors at the previous
324 iteration. The difference between both approaches lies in the way
325 processes communicate and exchange data. With MPI processors need to
326 send and receive data explicitely. So in
327 Algorithm~\ref{alg2-cuda-mpi}, after the initialization all the
328 processors have the same $Z$ vector. Then they need to compute the
329 parameters used by the $MPI\_AlltoAll$ routines (line 4). In practise,
330 each processor needs to compute its offset and its local size. Then
331 processors need to allocate memory on their GPU (line 5). At the
332 beginning of each iteration, a processor starts by transfering the
333 whole vector Z from the CPU to the GPU (line 7). Then only the local
334 part of $Z^{prev}$ is saved (line 8). After that, a processor is able
335 to compute its own roots (line 9). Next, the local error can be
336 computed (ligne 10) and the global error (line 11). Then the local
337 roots are transfered from the GPU memory to the CPU memory (line 12)
338 before being exchanged between all processors (linge 13) in order to
339 give to all processors the last version of the roots. If the
340 convergence is not statisfied, an new iteration is executed.
344 \begin{algorithm}[htpb]
345 \label{alg2-cuda-mpi}
347 \caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
349 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
350 threshold), P (Polynomial to solve), P' (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition)}
352 \KwOut {$Z$ (Solution root's vector)}
355 Initialization of P\;
356 Initialization of Pu\;
357 Initialization of the solution vector $Z^{0}$\;
358 Computation of the parameters for the $MPI\_AlltoAll$\;
359 Allocate memory to GPU\;
360 \While {$error > \epsilon$}{
361 copy Z from CPU to GPU\;
362 $Z^{Prev}_{loc}=kernel\_save(Z_{loc})$\;
363 $Z_{loc}=kernel\_update(Z,P,P')$\;
364 $\Delta z=kernel\_testConv(Z_{loc},Z^{prev}_{loc})$\;
365 $error=MPI\_Reduce(\Delta z)$\;
366 Copy $Z_{loc}$ from GPU to CPU\;
367 $Z=MPI\_AlltoAll(Z_{loc})$\;
369 \RC{A uniformiser avec les autres algos, mais les grandes lignes sont là}
373 \section{Experiments}
375 We study two categories of polynomials: sparse polynomials and full polynomials.\\
376 {\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:
378 \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})
379 \end{equation}\noindent
380 {\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
382 %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
386 {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
389 For our test, 4 cards GPU tesla Kepler K40 are used. In order to
390 evaluate both the GPU and Multi-GPU approaches, we performed a set of
391 experiments on a single GPU and multiple GPUs using OpenMP or MPI with
392 the EA algorithm, for both sparse and full polynomials of different
393 sizes. All experimental results obtained are perfomed with double
394 precision float data and the convergence threshold of the EA method is
395 set to $10^{-7}$. The initialization values of the vector solution of
396 the methods are given by Guggenheimer method~\cite{Gugg86}.
399 \subsection{Evaluation of the CUDA-OpenMP approach}
401 Here we report some experiments witt full and sparse polynomials of
402 different degrees with multiple GPUs.
403 \subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
405 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.
409 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
410 \caption{Execution time in seconds of the Ehrlich-Aberth method to
411 solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
415 Figure~\ref{fig:01} shows that the CUDA-OpenMP approach scales well
416 with multiple GPUs. This version allows us to solve sparse polynomials
417 of very high degrees.
419 \subsubsection{Execution times of the EA method to solve full polynomials on multiple GPUs}
421 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.
425 \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
426 \caption{Execution time in seconds of the Ehrlich-Aberth method to
427 solve full polynomials on multiple GPUs with CUDA-OpenMP.}
431 In Figure~\ref{fig:02}, we can observe that with full polynomials the EA version with
432 CUDA-OpenMP scales also well. Using 4 GPUs allows us to achieve a
433 quasi-linear speedup.
435 \subsection{Evaluation of the CUDA-MPI approach}
436 In this part we perform some experiments to evaluate the CUDA-MPI
437 approach to solve full and sparse polynomials of degrees ranging from
438 100,000 to 1,400,000.
440 \subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
444 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
445 \caption{Execution time in seconds of the Ehrlich-Aberth method to
446 solve sparse polynomials on multiple GPUs with CUDA-MPI.}
449 Figure~\ref{fig:03} shows the execution times of te EA algorithm,
450 for a single GPU, and multiple GPUs (2, 3, 4) with the CUDA-MPI approach.
452 \subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}
456 \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
457 \caption{Execution times in seconds of the Ehrlich-Aberth method for
458 full polynomials on multiple GPUs with CUDA-MPI.}
462 In Figure~\ref{fig:04}, we can also observe that the CUDA-MPI approach
463 is also efficient to solve full polynimails on multiple GPUs.
465 \subsection{Comparison of the CUDA-OpenMP and the CUDA-MPI approaches}
467 In the previuos section we saw that both approches are very effecient
468 to reduce the execution times the sparse and full polynomials. In
469 this section we try to compare these two approaches.
471 \subsubsection{Solving sparse polynomials}
472 In this experiment three sparse polynomials of size 200K, 800K and 1,4M are investigated.
475 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
476 \caption{Execution times to solvs sparse polynomials of three
477 distinct sizes on multiple GPUs using MPI and OpenMP with the
478 Ehrlich-Aberth method}
481 In Figure~\ref{fig:05} there is one curve for CUDA-MPI and another one
482 for CUDA-OpenMP. We can see that the results are quite similar between
483 OpenMP and MPI for the polynomials size of 200K. For the size of 800K,
484 the MPI version is a little bit slower than the OpenMP approach but for
485 the 1,4 millions size, there is a slight advantage for the MPI
488 \subsubsection{Solving full polynomials}
491 \includegraphics[angle=-90,width=0.5\textwidth]{Full}
492 \caption{Execution time for solving full polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
495 In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
497 \subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
499 In this experiment we compare the execution time of the EA algorithm
500 according to the number of GPUs to solve sparse and full
501 polynomials on multiples GPUs using MPI. We chose three sparse and full
502 polynomials of size 200K, 800K and 1,4M.
505 \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
506 \caption{Execution times to solve sparse and full polynomials of three distinct sizes on multiple GPUs using MPI.}
509 In Figure~\ref{fig:07} we can see that CUDA-MPI can solve sparse and
510 full polynomials of high degrees, the execution times with sparse
511 polynomial are very low compared to full polynomials. With sparse
512 polynomials the number of monomials is reduced, consequently the number
513 of operations is reduced and the execution time decreases.
515 \subsubsection{Solving sparse and full polynomials of the same size with CUDA-OpenMP}
519 \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
520 \caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
524 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.
526 \subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
527 These experiments report the execution times of the EA method for
528 sparse and full polynomials ranging from 1,000,000 to 5,000,000.
531 \includegraphics[angle=-90,width=0.5\textwidth]{big}
532 \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}
535 In Figure~\ref{fig:09} we can see that both approaches are scalable
536 and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both
537 approaches give very similar results.
539 %SIDER JE viens de virer \c ca For sparse polynomials here are a noticeable difference in favour of MPI when the degree is
540 %above 4 millions. Between 1 and 3 millions, OpenMP is more effecient.
541 %Under 1 million, OpenMPI and MPI are almost equivalent.
543 %SIDER : il faut une explication sur les différences ici aussi.
548 In this paper, we have presented a parallel implementation of
549 Ehrlich-Aberth algorithm to solve full and sparse polynomials, on
550 single GPU with CUDA and on multiple GPUs using two parallel
551 paradigms: shared memory with OpenMP and distributed memory with
552 MPI. These architectures were addressed by a CUDA-OpenMP approach and
553 CUDA-MPI approach, respectively. Experiments show that, using
554 parallel programming model like (OpenMP, MPI). We can efficiently
555 manage multiple graphics cards to solve the same
556 problem and accelerate the parallel execution with 4 GPUs and solve a
557 polynomial of degree up to 5,000,000, four times faster than on single
561 Our next objective is to extend the model presented here with clusters
562 of GPU nodes, with a three-level scheme: inter-node communication via
563 MPI processes (distributed memory), management of multi-GPU node by
564 OpenMP threads (shared memory).
567 \section*{Acknowledgment}
569 Computations have been performed on the supercomputer facilities of
570 the Mésocentre de calcul de Franche-Comté. We also would like to thank
571 Nvidia for hardware donation under CUDA Research Center 2014.
575 \bibliography{mybibfile}
