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.
+ polynomials of degree up to 5 million.
\end{itemize}
\section{Parallel programming models}
\label{sec2}
-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.
+Our objective consists in implementing a root-finding algorithm of
+polynomials on multiple GPUs. To this end, it is essential to know how
+to manage the CUDA contexts of different GPUs. A direct method to control 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.
+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 a relatively 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.
+MPI (Message Passing Interface) is a portable message passing style of
+the parallel programming designed specifically for 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 a way that 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 the 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 latter. 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 with CUDA programming,
+it is necessary to design carefully the arrangement of the thread
+blocks in order to ensure a low latency and a proper use of the shared
+memory. As for the global memory accesses, it should also be minimized.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
-The code is organized as kernels which are parts of code that are run
+The code is organized as kernels which are parts of codes that are run
on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
implementation of the Ehrlich-Aberth on a GPU. This algorithms starts
by initializing the polynomial and its derivative (line 1). The
initialization of the roots is performed (line 2). Data are transferred
-from the CPU to the GPU (after allocation of the required memory on
+from the CPU to the GPU (after the allocation of the required memory on
the GPU) (line 3). Then at each iteration, if the error is greater
-than a threshold, the following operations are performed. The previous
+than the threshold, the following operations are performed. The previous
roots are saved using a kernel (line 5). Then the new roots with the
new iterations are computed using the EA method with a Gauss-Seidel
-iteration mode in order to use the lastest updated roots (line
+iteration mode in order to use the latest updated roots (line
6). This improves the convergence. This kernel is, in practice, very
long since it performs all the operations with complex numbers with
the normal mode of the EA method but also with the
of corresponding kernels with CUDA is longer than on a CPU host. This
comes in particular from the fact that it is very difficult to debug
CUDA running threads like threads on a CPU host. In the following
-section the GPU parallel implementation of Ehrlich-Aberth method with
+section the GPU parallel implementation of the Ehrlich-Aberth method with
OpenMP and MPI is presented.
vector $error$. The number of OpenMP threads is equal to the number of
GPUs, each OpenMP thread binds to one GPU, and it controls a part of
the shared memory. More precisely each OpenMP thread will be
-responsible to update its owns part of the vector $Z$. This part is
+responsible for updating its own part of the vector $Z$. This part is
called $Z_{loc}$ in the following. Then all GPUs will have a grid of
computation organized according to the device performance and the size
of data on which it runs the computation kernels.
To compute one iteration of the EA method each GPU performs the
-followings steps. First roots are shared with OpenMP and the
+followings steps. First, roots are shared with OpenMP and the
computation of the local size for each GPU is performed (line 4). Each
thread starts by copying all the previous roots inside its GPU (line
5). At each iteration, the following operations are performed. First
GPU copies the previous roots (line 8) and it computes an iteration of
the EA method on its own roots (line 9). For that all the other roots
are used. The local error is computed on the new roots (line 10) and
-the max of the local errors is computed on all OpenMP threads (lien 11). At
+the maximum of the local errors is computed on all OpenMP threads (line 11). At
the end of an iteration, the updated roots are copied from the GPU to
-the CPU (line 12) by directly updating its own roots in the shared
+the CPU (line 12) and each CPU directly updates its own roots in the shared
memory arrays containing all the roots.
\subsection{A MPI-CUDA approach}
-Our parallel implementation of EA to find roots of polynomials using a
+Our parallel implementation of EA to find the 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
+CUDA-OpenMP. Each processor is responsible for computing its own part of
roots using all the roots computed by other processors at the previous
iteration. The difference between both approaches lies in the way
-processes communicate and exchange data. With MPI, processors need to
+processors communicate and exchange data. With MPI, processors need to
send and receive data explicitly. So in
-Algorithm~\ref{alg2-cuda-mpi}, after the initialization all the
+Algorithm~\ref{alg2-cuda-mpi}, after the initialization phase all the
processors have the same $Z$ vector. Then they need to compute the
parameters used by the $MPI\_AlltoAll$ routines (line 4). In practice,
each processor needs to compute its offset and its local
GPU (line 7). Only the local part of $Z^{prev}$ is saved (line
8). After that, a processor is able to compute an updated version of
its own roots (line 9) with the EA method. The local error is computed
-(line 10) and the global error using $MPI\_Reduce$ (line 11). Then
+(line 10) and the global error is also computed using $MPI\_Reduce$ (line 11). Then
the local roots are transferred from the GPU memory to the CPU memory
(line 12) before being exchanged between all processors (line 13) in
order to give to all processors the last version of the roots (with
-the MPI\_AlltoAll routine). If the convergence is not satisfied, an
+the MPI\_AlltoAll routine). If the convergence is not satisfied, a
new iteration is executed.
\begin{algorithm}[htpb]
{\Large \forall \alpha_{i} \in \mathbb{C}, i\in \mathbb{N}; p(x)=\sum^{n}_{i=0} \alpha_{i}.x^{i}}
\end{equation}
-For our tests, 4 cards GPU Tesla Kepler K40 are used. In order to evaluate both the GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI with the EA algorithm, for both sparse and full polynomials of different sizes. All experimental results obtained are performed with double precision float data and the convergence threshold of the EA method is set to $10^{-7}$. The initialization values of the vector solution of the methods are given by Guggenheimer method~\cite{Gugg86}.
+For our tests, 4 GPU cards Tesla Kepler K40 are used. In order to
+evaluate both the GPU and Multi-GPU approaches, we performed a set of
+experiments on a single GPU and multiple GPUs using OpenMP or MPI with
+the EA algorithm, for both sparse and full polynomials of different
+sizes. All experimental results obtained are performed with double
+precision float data and the convergence threshold of the EA method is
+set to $10^{-7}$. The initialization values of the vector solution of
+the methods are given by the Guggenheimer method~\cite{Gugg86}.
\subsection{Evaluation of the multi-GPUs approaches}
-Here we evaluate the performances of the CUDA-OpenMP and CUDA-MPI approaches of the EA algorithm on different GPU platforms composed each of 1, 2, 3 or 4 GPUs. In this experiments we report the experimental results of the EA algorithms to find roots of different sparse and full polynomials of high degrees ranging from 100,000 to 1,400,000. Figures~\ref{fig:01} and~\ref{fig:02} show the execution times to solve, respectively, sparse and full polynomials with the CUDA-OpenMP algorithm, and Figures~\ref{fig:03} and~\ref{fig:04} show those to solve, respectively, sparse and full polynomials with the CUDA-MPI algorithm.
+In this part, we evaluate the performances of the CUDA-OpenMP and
+CUDA-MPI approaches of the EA algorithm on different GPU platforms
+composed each of 1, 2, 3 or 4 GPUs. In this experiments we report the
+experimental results of the EA algorithms to find the roots of different sparse and full polynomials of high degrees ranging from 100,000 to 1,400,000. Figures~\ref{fig:01} and~\ref{fig:02} show the execution times to solve, respectively, sparse and full polynomials with the CUDA-OpenMP algorithm, and Figures~\ref{fig:03} and~\ref{fig:04} show those to solve, respectively, sparse and full polynomials with the CUDA-MPI algorithm.
-All these figures show that the CUDA-OpenMP and CUDA-MPI approaches of the EA algorithm, compared to the single GPU version, are efficient and scale well with multiple GPUs. Both approaches allow us to solve sparse and full polynomials of very high degrees. Using 4 GPUs allows us to achieve a quasi-linear speedup.
+All these figures show that the CUDA-OpenMP and the CUDA-MPI approaches of the EA algorithm, compared to the single GPU version, are efficient and scale well with multiple GPUs. Both approaches allow us to solve sparse and full polynomials of very high degrees. Using 4 GPUs allows us to achieve a quasi-linear speedup.
\begin{figure}[htbp]
\centering
\label{fig:06}
\end{figure}
-In Figure~\ref{fig:05} there is one curve for CUDA-OpenMP and another one for CUDA-MPI. We can see that the results are quite similar between OpenMP and MPI for the polynomial degree of 200K. For the degree of 800K, the MPI version is a little bit slower than the OpenMP version but for the degree of 1,4 millions, there is a slight advantage for the MPI version. In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
+In Figure~\ref{fig:05} there is one curve for CUDA-OpenMP and another one for CUDA-MPI. We can see that the results are quite similar between OpenMP and MPI for the polynomial degree of 200K. For the degree of 800K, the MPI version is a little bit slower than the OpenMP version but for the degree of 1,4 million, there is a slight advantage for the MPI version. In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
\subsection{Solving sparse and full polynomials of the same degree on multiple GPUs}
-In this experiment we compare the execution times of the EA algorithm according to the number of GPUs to solve sparse and full polynomials on multiple GPUs using OpenMP or MPI approaches. We chose three sparse and three full polynomials of degrees 200,000, 800,000 and 1,400,000. Figures~\ref{fig:07} and~\ref{fig:08} show the execution times to solve sparse and full polynomials of the same degrees with CUDA-OpenMP version and CUDA-MPI version, respectively.
+In this experiment we compare the execution times of the EA algorithm
+according to the number of GPUs to solve sparse and full polynomials
+on multiple GPUs using OpenMP or MPI approaches. We chose three sparse
+and three full polynomials of degrees 200,000, 800,000 and
+1,400,000. Figures~\ref{fig:07} and~\ref{fig:08} show the execution
+times to solve sparse and full polynomials of the same degrees with
+the CUDA-OpenMP version and the CUDA-MPI version, respectively.
\begin{figure}[htbp]
\centering
\label{fig:08}
\end{figure}
-In Figure ~\ref{fig:07} the execution times of the CUDA-OpenMP version to solve sparse polynomials are very low compared to those to solve full polynomials. With sparse polynomials the number of monomials is reduced, consequently the number of operations is reduced and the execution time decreases. Figure~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-MPI approach. We can see that the impact follows the same pattern, a difference in execution times in favor of the sparse polynomials.
+In Figure~\ref{fig:07} the execution times of the CUDA-OpenMP version to solve sparse polynomials are very low compared to those to solve full polynomials. With sparse polynomials the number of monomials is reduced, consequently the number of operations is reduced and the execution time decreases. Figure~\ref{fig:08} shows the impact of sparsity on the efficiency of the CUDA-MPI approach. We can see that the impact follows the same pattern, a difference in execution times in favor of the sparse polynomials.
\subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
