X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/645f3921415c716f3c2185090bd32a5d48861879..9a16c3f8b303f6260ecf3bf14459ee0bd43e6ef1:/hpcc.tex diff --git a/hpcc.tex b/hpcc.tex index 1179edb..e780bbd 100644 --- a/hpcc.tex +++ b/hpcc.tex @@ -41,6 +41,7 @@ \algnewcommand\Output{\item[\algorithmicoutput]} \newcommand{\MI}{\mathit{MaxIter}} +\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}} \begin{document} @@ -97,6 +98,8 @@ a residual precision up to \np{E-11}. Such successful results open perspectives on experimentations for running the algorithm on a simulated large scale growing environment and with larger problem size. +\LZK{Long\ldots} + % no keywords for IEEE conferences % Keywords: Algorithm distributed iterative asynchronous simulation SimGrid \end{abstract} @@ -172,7 +175,7 @@ asynchronous mode. This article is structured as follows: after this introduction, the next section will give a brief description of iterative asynchronous model. Then, the simulation framework SimGrid is presented with the settings to create various -distributed architectures. The algorithm of the multisplitting method used by GMRES written with MPI primitives and +distributed architectures. The algorithm of the multisplitting method used by GMRES \LZK{??? GMRES n'utilise pas la méthode de multisplitting! Sinon ne doit on pas expliquer le choix d'une méthode de multisplitting?} written with MPI primitives and its adaptation to SimGrid with SMPI (Simulated MPI) is detailed in the next section. At last, the experiments results carried out will be presented before some concluding remarks and future works. @@ -196,7 +199,7 @@ times and the arrows the communications. With this algorithmic model, the number of iterations required before the convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially -in a grid computing context. +in a grid computing context.\LZK{Répétition par rapport à l'intro} \begin{figure}[!t] \centering @@ -254,12 +257,12 @@ like the communications are intercepted, and their running time is computed according to the characteristics of the simulated execution platform. The description of this target platform is given as an input for the execution, by the mean of an XML file. It describes the properties of the platform, such as -the computing node with their computing power, the interconnection links with +the computing nodes with their computing power, the interconnection links with their bandwidth and latency, and the routing strategy. The simulated running time of the application is computed according to these properties. To compute the durations of the operations in the simulated world, and to take -into account resource sharing (e.g. bandwith sharing between competiting +into account resource sharing (e.g. bandwidth sharing between competing communications), SimGrid uses a fluid model. This allows to run relatively fast simulations, while still keeping accurate results~\cite{bedaride:hal-00919507,tomacs13}. Moreover, depending on the @@ -293,51 +296,75 @@ Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, whe B_L \end{array} \right) \end{equation*} -in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$. +in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ +are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$ $A_{\ell + m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and +$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each and +$\sum_{\ell} n_\ell=\sum_{m} n_m=n$. The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system \begin{equation} \label{eq:4.1} \left\{ \begin{array}{l} - A_{ll}X_l = Y_l \text{, such that}\\ - Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m + A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\ + Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m \end{array} \right. \end{equation} -is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. +is solved independently by a cluster and communications are required to update +the right-hand side sub-vector $Y_\ell$, such that the sub-vectors $X_m$ +represent the data dependencies between the clusters. As each sub-system +(\ref{eq:4.1}) is solved in parallel by a cluster of processors, our +multisplitting method uses an iterative method as an inner solver which is +easier to parallelize and more scalable than a direct method. In this work, we +use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most +used iterative method by many researchers. \begin{figure}[!t] %%% IEEE instructions forbid to use an algorithm environment here, use figure %%% instead \begin{algorithmic}[1] -\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector) -\Output $X_l$ (solution sub-vector)\vspace{0.2cm} -\State Load $A_l$, $B_l$ +\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector) +\Output $X_\ell$ (solution sub-vector)\medskip + +\State Load $A_\ell$, $B_\ell$ \State Set the initial guess $x^0$ \For {$k=0,1,2,\ldots$ until the global convergence} \State Restart outer iteration with $x^0=x^k$ \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$} -\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters -\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$ +\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters +\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$ \EndFor \Statex \Function {InnerSolver}{$x^0$, $k$} -\State Compute local right-hand side $Y_l$: +\State Compute local right-hand side $Y_\ell$: \begin{equation*} - Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0 + Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0 \end{equation*} -\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method -\State \Return $X_l^k$ +\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method +\State \Return $X_\ell^k$ \EndFunction \end{algorithmic} \caption{A multisplitting solver with GMRES method} \label{algo:01} \end{figure} -Algorithm on Figure~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines. +Algorithm on Figure~\ref{algo:01} shows the main key points of the +multisplitting method to solve a large sparse linear system. This algorithm is +based on an outer-inner iteration method where the parallel synchronous GMRES +method is used to solve the inner iteration. It is executed in parallel by each +cluster of processors. For all $\ell,m\in\{1,\ldots,L\}$, the matrices and +vectors with the subscript $\ell$ represent the local data for cluster $\ell$, +while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix +$A$ and $\{X_m\}_{m\neq \ell}$ contain vector elements of solution $x$ shared +with neighboring clusters. At every outer iteration $k$, asynchronous +communications are performed between processors of the local cluster and those +of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in +Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are +exchanged by message passing using MPI non-blocking communication routines. \begin{figure}[!t] \centering @@ -359,14 +386,14 @@ sets the token to \textit{True} if the local convergence is achieved or to global convergence is detected when the master of cluster 1 receives from the master of cluster $L$ a token set to \textit{True}. In this case, the master of cluster 1 broadcasts a stop message to masters of other clusters. In this work, -the local convergence on each cluster $l$ is detected when the following +the local convergence on each cluster $\ell$ is detected when the following condition is satisfied \begin{equation*} - (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon) + (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon) \end{equation*} where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution -$X_l^k$ and $X_l^{k+1}$. +$X_\ell^k$ and $X_\ell^{k+1}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code @@ -378,9 +405,12 @@ Note here that the use of SMPI functions optimizer for memory footprint and CPU As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. First, all declared global variables have been moved to local variables for each subroutine. In fact, global variables generate side effects arising from the concurrent access of shared memory used by threads simulating each computing unit in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had -also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid. +also to be reviewed. +\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.} + Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid. In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real -environment. We have successfully executed the code in synchronous mode using GMRES algorithm compared with a multisplitting method in asynchrnous mode after few modification. +environment. We have successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications. + \section{Experimental results} @@ -399,8 +429,7 @@ study that the results depend on the following parameters: experimentation of the simulation in having an execution time in asynchronous less than in synchronous mode. The ratio between the execution time of asynchronous compared to the synchronous mode is defined as the "relative gain". So, our objective running the algorithm in SimGrid is to obtain a relative gain greater than 1. \end{itemize} -\LZK{Propositions pour remplacer le terme ``speedup'': acceleration ratio ou relative gain} -\CER{C'est fait. En conséquence, les tableaux et les commentaires ont été aussi modifiés} + A priori, obtaining a relative gain greater than 1 would be difficult in a local area network configuration where the synchronous mode will take advantage on the rapid exchange of information on such high-speed links. Thus, the methodology @@ -447,7 +476,8 @@ containing 50 hosts each, totaling 100 hosts. Various combinations of the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = -\text{\np{5211000}}$ entries. +\text{\np{5000211}}$ entries. +\AG{Expliquer comment lire les tableaux.} % use the same column width for the following three tables \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}} @@ -538,7 +568,7 @@ relative gains greater than 1 with a matrix size from 62 to 100 elements. & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\ \hline Relative gain - & 1.003 & 1,01 & 1,08 & 0.19 & 1.28 & 1.01 \\ + & 1.003 & 1.01 & 1.08 & 1.19 & 1.28 & 1.01 \\ \hline \end{mytable} \end{table} @@ -573,6 +603,7 @@ Note that the program was run with the following parameters: \paragraph*{SMPI parameters} +~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).} \begin{itemize} \item HOSTFILE: Hosts file description. \item PLATFORM: file description of the platform architecture : clusters (CPU power, @@ -588,11 +619,8 @@ lat latency, \dots{}). \item Maximum number of internal and external iterations; \item Internal and external precisions; \item Matrix size $N_x$, $N_y$ and $N_z$; -%<<<<<<< HEAD \item Matrix diagonal value: \np{6.0}; - \item Matrix Off-diagonal value: \np{-1.0}; -%======= -%>>>>>>> 5fb6769d88c1720b6480a28521119ef010462fa6 + \item Matrix off-diagonal value: \np{-1.0}; \item Execution Mode: synchronous or asynchronous. \end{itemize} @@ -631,8 +659,9 @@ with 200 nodes in total. The convergence with a relative gain around 1.1 was obtained with a bandwidth of \np[Mbit/s]{1} as shown in Table~\ref{tab.cluster.3x67}. -\LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants} -\CER{En fait, les résultats ont été obtenus en comparant les temps d'exécution entre l'algo classique GMRES en mode synchrone avec le multisplitting en mode asynchrone, le tout sur un environnement de clusters distants} +\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...} +\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)} +\LZK{Ma question est: le bw et lat sont ceux inter-clusters ou pour les deux inter et intra cluster??} \section{Conclusion} The experimental results on executing a parallel iterative algorithm in