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+\usepackage{multirow}
+
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-\begin{document} \RCE{Titre a confirmer.} \title{Comparative performance
-analysis of simulated grid-enabled numerical iterative algorithms}
+\begin{document}
+\title{Grid-enabled simulation of large-scale linear iterative solvers}
%\itshape{\journalnamelc}\footnotemark[2]}
-\author{ Charles Emile Ramamonjisoa and
- David Laiymani and
- Arnaud Giersch and
- Lilia Ziane Khodja and
- Raphaël Couturier
+\author{Charles Emile Ramamonjisoa\affil{1},
+ David Laiymani\affil{1},
+ Arnaud Giersch\affil{1},
+ Lilia Ziane Khodja\affil{2} and
+ Raphaël Couturier\affil{1}
}
\address{
- \centering
- Femto-ST Institute - DISC Department\\
- Université de Franche-Comté\\
- Belfort\\
- Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
+ \affilnum{1}%
+ Femto-ST Institute, DISC Department,
+ University of Franche-Comté,
+ Belfort, France.
+ Email:~\email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}\break
+ \affilnum{2}
+ Department of Aerospace \& Mechanical Engineering,
+ Non Linear Computational Mechanics,
+ University of Liege, Liege, Belgium.
+ Email:~\email{l.zianekhodja@ulg.ac.be}
}
-%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
-
-\begin{abstract} The behavior of multi-core applications is always a challenge
-to predict, especially with a new architecture for which no experiment has been
-performed. With some applications, it is difficult, if not impossible, to build
-accurate performance models. That is why another solution is to use a simulation
-tool which allows us to change many parameters of the architecture (network
-bandwidth, latency, number of processors) and to simulate the execution of such
-applications. We have decided to use SimGrid as it enables to benchmark MPI
-applications.
-
-In this paper, we focus our attention on two parallel iterative algorithms based
-on the Multisplitting algorithm and we compare them to the GMRES algorithm.
-These algorithms are used to solve libear systems. Two different variants of
-the Multisplitting are studied: one using synchronoous iterations and another
-one with asynchronous iterations. For each algorithm we have tested different
-parameters to see their influence. We strongly recommend people interested
-by investing into a new expensive hardware architecture to benchmark
-their applications using a simulation tool before.
+\begin{abstract} %% The behavior of multi-core applications is always a challenge
+%% to predict, especially with a new architecture for which no experiment has been
+%% performed. With some applications, it is difficult, if not impossible, to build
+%% accurate performance models. That is why another solution is to use a simulation
+%% tool which allows us to change many parameters of the architecture (network
+%% bandwidth, latency, number of processors) and to simulate the execution of such
+%% applications. The main contribution of this paper is to show that the use of a
+%% simulation tool (here we have decided to use the SimGrid toolkit) can really
+%% help developers to better tune their applications for a given multi-core
+%% architecture.
+%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations.
+%% For each algorithm we have simulated
+%% different architecture parameters to evaluate their influence on the overall
+%% execution time.
+%% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
+The behavior of multi-core applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.
+In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.
\end{abstract}
given application for a given architecture. In this way and in order to reduce
the access cost to these computing resources it seems very interesting to use a
simulation environment. The advantages are numerous: development life cycle,
-code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly\dots{} In counterpart, the simulation results need to be consistent with the real ones.
In this paper we focus on a class of highly efficient parallel algorithms called
\emph{iterative algorithms}. The parallel scheme of iterative methods is quite
from its neighbors. We say that the iteration computation follows a
\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
iteration without having to wait for the data dependencies coming from its
-neighbors. Both communication and computations are \textit{asynchronous}
+neighbors. Both communications and computations are \textit{asynchronous}
inducing that there is no more idle time, due to synchronizations, between two
iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
-that we detail in section~\ref{sec:asynchro} but even if the number of
+that we detail in Section~\ref{sec:asynchro} but even if the number of
iterations required to converge is generally greater than for the synchronous
case, it appears that the asynchronous iterative scheme can significantly
reduce overall execution times by suppressing idle times due to
allocations policies under varying CPU power, network speeds and loads is very
challenging and labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
problematic is even more difficult for the asynchronous scheme where a small
-parameter variation of the execution platform can lead to very different numbers
-of iterations to reach the converge and so to very different execution times. In
-this challenging context we think that the use of a simulation tool can greatly
-leverage the possibility of testing various platform scenarios.
-
-The main contribution of this paper is to show that the use of a simulation tool
-(i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real parallel
-applications (i.e. large linear system solvers) can help developers to better
-tune their application for a given multi-core architecture. To show the validity
-of this approach we first compare the simulated execution of the multisplitting
-algorithm with the GMRES (Generalized Minimal Residual)
-solver~\cite{saad86} in synchronous mode. The obtained results on different
-simulated multi-core architectures confirm the real results previously obtained
-on non simulated architectures. We also confirm the efficiency of the
-asynchronous multisplitting algorithm compared to the synchronous GMRES. In
-this way and with a simple computing architecture (a laptop) SimGrid allows us
+parameter variation of the execution platform and of the application data can
+lead to very different numbers of iterations to reach the convergence and so to
+very different execution times. In this challenging context we think that the
+use of a simulation tool can greatly leverage the possibility of testing various
+platform scenarios.
+
+The {\bf main contribution of this paper} is to show that the use of a
+simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real
+parallel applications (i.e. large linear system solvers) can help developers to
+better tune their applications for a given multi-core architecture. To show the
+validity of this approach we first compare the simulated execution of the Krylov
+multisplitting algorithm with the GMRES (Generalized Minimal RESidual)
+solver~\cite{saad86} in synchronous mode. The simulation results allow us to
+determine which method to choose for a given multi-core architecture.
+Moreover the obtained results on different simulated multi-core architectures
+confirm the real results previously obtained on non simulated architectures.
+More precisely the simulated results are in accordance (i.e. with the same order
+of magnitude) with the works presented in~\cite{couturier15}, which show that
+the synchronous Krylov multisplitting method is more efficient than GMRES for large
+scale clusters. Simulated results also confirm the efficiency of the
+asynchronous multisplitting algorithm compared to the synchronous GMRES
+especially in case of geographically distant clusters.
+
+In this way and with a simple computing architecture (a laptop) SimGrid allows us
to run a test campaign of a real parallel iterative applications on
different simulated multi-core architectures. To our knowledge, there is no
related work on the large-scale multi-core simulation of a real synchronous and
This paper is organized as follows. Section~\ref{sec:asynchro} presents the
iteration model we use and more particularly the asynchronous scheme. In
-section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
+Section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
Section~\ref{sec:04} details the different solvers that we use. Finally our
-experimental results are presented in section~\ref{sec:expe} followed by some
+experimental results are presented in Section~\ref{sec:expe} followed by some
concluding remarks and perspectives.
-\section{The asynchronous iteration model}
+\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
-Asynchronous iterative methods have been studied for many years theoritecally and
+Asynchronous iterative methods have been studied for many years theoretically and
practically. Many methods have been considered and convergence results have been
proved. These methods can be used to solve, in parallel, fixed point problems
(i.e. problems for which the solution is $x^\star =f(x^\star)$. In practice,
-asynchronous iterations methods can be used to solve, for example, linear and
+asynchronous iteration methods can be used to solve, for example, linear and
non-linear systems of equations or optimization problems, interested readers are
invited to read~\cite{BT89,bahi07}.
requires very few modifications to be able to be executed in both variants. In
practice, only the communications and convergence detection are different. In
the synchronous mode, iterations are synchronized whereas in the asynchronous
-one, they are not. It should be noticed that non blocking communications can be
+one, they are not. It should be noticed that non-blocking communications can be
used in both modes. Concerning the convergence detection, synchronous variants
can use a global convergence procedure which acts as a global synchronization
point. In the asynchronous model, the convergence detection is more tricky as
it must not synchronize all the processors. Interested readers can
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
+The number of iterations required to reach the convergence is generally greater
+for the asynchronous scheme (this number depends on the delay of the
+messages). Note that, it is not the case in the synchronous mode where the
+number of iterations is the same than in the sequential mode. In this way, the
+set of the parameters of the platform (number of nodes, power of nodes,
+inter and intra clusters bandwidth and latency,~\ldots) and of the
+application can drastically change the number of iterations required to get the
+convergence. It follows that asynchronous iterative algorithms are difficult to
+optimize since the financial and deployment costs on large scale multi-core
+architectures are often very important. So, prior to deployment and tests it
+seems very promising to be able to simulate the behavior of asynchronous
+iterative algorithms. The problematic is then to show that the results produced
+by simulation are in accordance with reality i.e. of the same order of
+magnitude. To our knowledge, there is no study on this problematic.
+
\section{SimGrid}
- \label{sec:simgrid}
+\label{sec:simgrid}
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software.
%%%%%%%%%%%%%%%%%%%%%%%%%
+% SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
+% is a simulation framework to study the behavior of large-scale distributed
+% systems. As its name suggests, it emanates from the grid computing community,
+% but is nowadays used to study grids, clouds, HPC or peer-to-peer systems. The
+% early versions of SimGrid date back from 1999, but it is still actively
+% developed and distributed as an open source software. Today, it is one of the
+% major generic tools in the field of simulation for large-scale distributed
+% systems.
+
+SimGrid provides several programming interfaces: MSG to simulate Concurrent
+Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
+run real applications written in MPI~\cite{MPI}. Apart from the native C
+interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
+languages. SMPI is the interface that has been used for the work described in
+this paper. The SMPI interface implements about \np[\%]{80} of the MPI 2.0
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+applications written in C or Fortran, with little or no modifications (cf Section IV - paragraph B).
+
+Within SimGrid, the execution of a distributed application is simulated by a
+single process. The application code is really executed, but some operations,
+like 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
+means of an XML file. It describes the properties of the platform, such as
+the computing nodes with their computing power, the interconnection links with
+their bandwidth and latency, and the routing strategy. The scheduling of the
+simulated processes, as well as the simulated running time of the application
+are 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. bandwidth sharing between competing
+communications), SimGrid uses a fluid model. This allows users to run relatively fast
+simulations, while still keeping accurate
+results~\cite{bedaride+degomme+genaud+al.2013.toward,
+ velho+schnorr+casanova+al.2013.validity}. Moreover, depending on the
+simulated application, SimGrid/SMPI allows to skip long lasting computations and
+to only take their duration into account. When the real computations cannot be
+skipped, but the results are unimportant for the simulation results, it is
+also possible to share dynamically allocated data structures between
+several simulated processes, and thus to reduce the whole memory consumption.
+These two techniques can help to run simulations on a very large scale.
+
+The validity of simulations with SimGrid has been asserted by several studies.
+See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
+referenced therein for the validity of the network models. Comparisons between
+real execution of MPI applications on the one hand, and their simulation with
+SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
+ clauss+stillwell+genaud+al.2011.single,
+ bedaride+degomme+genaud+al.2013.toward}. All these works conclude that
+SimGrid is able to simulate pretty accurately the real behavior of the
+applications.
%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two-stage multisplitting methods}
\label{sec:04}
\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
\label{sec:04.01}
-In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
\begin{equation}
Ax=b,
\label{eq:01}
\end{equation}
-where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows:
\begin{equation}
x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
\label{eq:02}
\end{equation}
-where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
\begin{equation}
A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Bru95,bahi07}.
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
-\begin{figure}[t]
+\begin{figure}[htpb]
%\begin{algorithm}[t]
%\caption{Block Jacobi two-stage multisplitting method}
\begin{algorithmic}[1]
%\end{algorithm}
\end{figure}
-In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous model which allows communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged:
\begin{equation}
k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
\label{eq:04}
\end{equation}
where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
-The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
+The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration:
\begin{equation}
S=[x^1,x^2,\ldots,x^s],~s\ll n.
\label{eq:05}
\end{equation}
-At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual:
\begin{equation}
\min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
\label{eq:06}
\end{equation}
The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
-\begin{figure}[t]
+\begin{figure}[htbp]
%\begin{algorithm}[t]
%\caption{Krylov two-stage method using block Jacobi multisplitting}
\begin{algorithmic}[1]
%\end{algorithm}
\end{figure}
-\subsection{Simulation of two-stage methods using SimGrid framework}
+\subsection{Simulation of the two-stage methods using SimGrid toolkit}
\label{sec:04.02}
-One of our objectives when simulating the application in Simgrid is, as in real
-life, to get accurate results (solutions of the problem) but also ensure the
+One of our objectives when simulating the application in SimGrid is, as in real
+life, to get accurate results (solutions of the problem) but also to ensure the
test reproducibility under the same conditions. According to our experience,
-very few modifications are required to adapt a MPI program for the Simgrid
+very few modifications are required to adapt a MPI program for the SimGrid
simulator using SMPI (Simulator MPI). The first modification is to include SMPI
-libraries and related header files (smpi.h). The second modification is to
+libraries and related header files (\verb+smpi.h+). The second modification is to
suppress all global variables by replacing them with local variables or using a
-Simgrid selector called "runtime automatic switching"
+SimGrid selector called "runtime automatic switching"
(smpi/privatize\_global\_variables). Indeed, global variables can generate side
-effects on runtime between the threads running in the same process, generated by
-Simgrid to simulate the grid environment.
-
-%\RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
-%last modification on the MPI program pointed out for some cases, the review of
-%the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which
-%might cause an infinite loop.
-
+effects on runtime between the threads running in the same process and generated by
+SimGrid to simulate the grid environment.
-\paragraph{Simgrid Simulator parameters}
-\ \\ \noindent Before running a Simgrid benchmark, many parameters for the
+\paragraph{Parameters of the simulation in SimGrid}
+\ \\ \noindent Before running a SimGrid benchmark, many parameters for the
computation platform must be defined. For our experiments, we consider platforms
in which several clusters are geographically distant, so there are intra and
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
- \item hostfile: hosts description file.
+ \item hostfile: hosts description file,
\item platform: file describing the platform architecture: clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-latency lat, \dots{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}),
\item archi : grid computational description (number of clusters, number of
-nodes/processors for each cluster).
+nodes/processors in each cluster).
\end{itemize}
\noindent
In addition, the following arguments are given to the programs at runtime:
\begin{itemize}
- \item maximum number of inner and outer iterations;
- \item inner and outer precisions;
- \item maximum number of the gmres's restarts in the Arnorldi process;
- \item maximum number of iterations qnd the tolerance threshold in classical GMRES;
- \item tolerance threshold for outer and inner-iterations;
- \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on x, y, z axis;
- \item matrix diagonal value = 6.0 for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
- \item matrix off-diagonal value;
- \item execution mode: synchronous or asynchronous;
- \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
- \item Size of matrix S;
- \item Maximum number of iterations and tolerance threshold for CGLS.
+ \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
+ \item inner precision $\TOLG$ and outer precision $\TOLM$,
+ \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+ \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
+ \item matrix off-diagonal value is fixed to $-1.0$,
+ \item number of vectors in matrix $S$ (i.e. value of $s$),
+ \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
+ \item maximum number of iterations and precision for the classical GMRES method,
+ \item maximum number of restarts for the Arnorldi process in GMRES method,
+ \item execution mode: synchronous or asynchronous.
\end{itemize}
-It should also be noticed that both solvers have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Experimental Results}
+\section{Experimental results}
\label{sec:expe}
-In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
-
-\subsection{3D Poisson}
-
+In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
-We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
+\subsection{The 3D Poisson problem}
+\label{3dpoisson}
+We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
\begin{equation}
\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
\label{eq:07}
\end{equation}
-such that
+such that:
\begin{equation*}
\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
\end{equation*}
-where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that:
\begin{equation}
\begin{array}{ll}
\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
\end{equation}
until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
-In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries.
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$
+sub-problems such that $L$ is the number of clusters and $p$ is the number of
+processors in each cluster. We apply the three-dimensional partitioning instead
+of the row-by-row one in order to reduce the size of the data shared at the
+sub-problems boundaries. In this case, each processor is in charge of
+parallelepipedic block of the problem and has at most six neighbors in the same
+cluster or in distant clusters with which it shares data at boundaries.
-\subsection{Study setup and Simulation Methodology}
+\subsection{Study setup and simulation methodology}
First, to conduct our study, we propose the following methodology
which can be reused for any grid-enabled applications.\\
the application to be tested. Numerical parallel iterative algorithms
have been chosen for the study in this paper. \\
-\textbf{Step 2}: Collect the software materials needed for the
-experimentation. In our case, we have two variants algorithms for the
-resolution of the 3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting method. In addition, the Simgrid simulator has been chosen to simulate the behaviors of the
-distributed applications. Simgrid is running on the Mesocentre datacenter in the University of Franche-Comte and also in a virtual machine on a simple laptop. \\
+\textbf{Step 2}: Collect the software materials needed for the experimentation.
+In our case, we have two variants algorithms for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+method. In addition, the SimGrid simulator has been chosen to simulate the
+behaviors of the distributed applications. SimGrid is running in a virtual
+machine on a simple laptop. \\
\textbf{Step 3}: Fix the criteria which will be used for the future
results comparison and analysis. In the scope of this study, we retain
on the one hand the algorithm execution mode (synchronous and asynchronous)
and on the other hand the execution time and the number of iterations to reach the convergence. \\
-\textbf{Step 4 }: Set up the different grid testbed environments that will be
-simulated in the simulator tool to run the program. The following architecture
-has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+\textbf{Step 4}: Set up the different grid testbed environments that will be
+simulated in the simulator tool to run the program. The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
represents the number of clusters in the grid and the second number represents
-the number of hosts (processors/cores) in each cluster. The network has been
-designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a
-latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links
-(resp. inter-clusters backbone links). \\
+the number of hosts (processors/cores) in each cluster. \\
\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations by varying the key parameters, especially
\textbf{Step 6} : Collect and analyze the output results.
-\subsection{Factors impacting distributed applications performance in
-a grid environment}
+\subsection{Factors impacting distributed applications performance in a grid environment}
When running a distributed application in a computational grid, many factors may
-have a strong impact on the performances. First of all, the architecture of the
+have a strong impact on the performance. First of all, the architecture of the
grid itself can obviously influence the performance results of the program. The
performance gain might be important theoretically when the number of clusters
and/or the number of nodes (processors/cores) in each individual cluster
increase.
-Another important factor impacting the overall performances of the application
+Another important factor impacting the overall performance of the application
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
-\item the network bandwidth (bw=bits/s) also known as "the data-carrying
+\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying
capacity" of the network is defined as the maximum of data that can transit
from one point to another in a unit of time.
-\item the network latency (lat : microsecond) defined as the delay from the
- start time to send the data from a source and the final time the destination
- have finished to receive it.
+\item the network latency ($lat$ in microseconds) defined as the delay from the
+ start time to send a simple data from a source to a destination.
\end{enumerate}
-Upon the network characteristics, another impacting factor is the
-application dependent volume of data exchanged between the nodes in the cluster
-and between distant clusters. Large volume of data can be transferred and
-transit between the clusters and nodes during the code execution.
-
- In a grid environment, it is common to distinguish, on the one hand, the
- "intra-network" which refers to the links between nodes within a cluster and,
- on the other hand, the "inter-network" which is the backbone link between
- clusters. In practice, these two networks have different speeds. The
- intra-network generally works like a high speed local network with a high
- bandwith and very low latency. In opposite, the inter-network connects clusters
- sometime via heterogeneous networks components throuth internet with a lower
- speed. The network between distant clusters might be a bottleneck for the
- global performance of the application.
-
-\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
-
-In the scope of this paper, our first objective is to analyze when the Krylov
-Multisplitting method has better performances than the classical GMRES
-method. With an iterative method, better performances mean a smaller number of
-iterations and execution time before reaching the convergence. For a systematic
-study, the experiments should figure out that, for various grid parameters
-values, the simulator will confirm the targeted outcomes, particularly for poor
-and slow networks, focusing on the impact on the communication performance on
-the chosen class of algorithm.
-
-The following paragraphs present the test conditions, the output results
-and our comments.\\
-
-
-\subsubsection{Execution of the the algorithms on various computational grid
-architecture and scaling up the input matrix size}
-\ \\
-% environment
-
-\begin{figure} [ht!]
+Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
+and between distant clusters. This parameter is application dependent.
+
+ In a grid environment, it is common to distinguish, on one hand, the
+ \textit{intra-network} which refers to the links between nodes within a
+ cluster and on the other hand, the \textit{inter-network} which is the
+ backbone link between clusters. In practice, these two networks have
+ different speeds. The intra-network generally works like a high speed
+ local network with a high bandwidth and very low latency. In opposite, the
+ inter-network connects clusters sometime via heterogeneous networks components
+ through internet with a lower speed. The network between distant clusters
+ might be a bottleneck for the global performance of the application.
+
+
+\subsection{Comparison between GMRES and two-stage multisplitting algorithms in
+synchronous mode}
+In the scope of this paper, our first objective is to analyze
+when the synchronous Krylov two-stage method has better performance than the
+classical GMRES method. With a synchronous iterative method, better performance
+means a smaller number of iterations and execution time before reaching the
+convergence.
+
+Table~\ref{tab:01} summarizes the parameters used in the different simulations:
+the grid architectures (i.e. the number of clusters and the number of nodes per
+cluster), the network of inter-clusters backbone links and the matrix sizes of
+the 3D Poisson problem. However, for all simulations we fix the network
+parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency
+$lat=8\mu$s. In what follows, we will present the test conditions, the output
+results and our comments.
+
+\begin{table} [ht!]
\begin{center}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
- - & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline
- \end{tabular}
-\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\begin{tabular}{ll}
+\hline
+Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+ & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
+ & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
+\end{tabular}
+\caption{Parameters for the different simulations}
+\label{tab:01}
\end{center}
-\end{figure}
-
-
-
-
-%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-
-
-In this section, we analyze the performences of algorithms running on various
-grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
-show for all grid configuration the non-variation of the number of iterations of
-classical GMRES for a given input matrix size; it is not the case for the
-multisplitting method.
-
-\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
-\RC{Les légendes ne sont pas explicites...}
-
-
-\begin{figure} [ht!]
- \begin{center}
- \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
- \end{center}
- \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
- \label{fig:01}
-\end{figure}
+\end{table}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
-The execution times between the two algorithms is significant with different
-grid architectures, even with the same number of processors (for example, 2x16
-and 4x8). We can observ the low sensitivity of the Krylov multisplitting method
-(compared with the classical GMRES) when scaling up the number of the processors
-in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
-40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.
+In this section, we analyze the simulations conducted on various grid
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of the network between clusters is fixed to $N2$ (see
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and
+a given matrix size of 170$^3$ elements, a non-variation in the number of
+iterations for the classical GMRES algorithm, which is not the case of the
+Krylov two-stage algorithm. In fact, with multisplitting algorithms, the number
+of splitting (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number of splitting is, the slower the
+convergence of the algorithm is (see the output results obtained from
+configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
+8$\times$8).
-\subsubsection{Running on two different speed cluster inter-networks}
-\ \\
+The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16).
-\begin{figure} [ht!]
+\begin{figure}[ht]
\begin{center}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
- - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
- \end{tabular}
-\caption{Clusters x Nodes - Networks N1 x N2}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
\end{figure}
-
-
-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\subsubsection{Simulations for two different inter-clusters network speeds\\}
+In Figure~\ref{fig:02} we present the execution times of both algorithms to
+solve a 3D Poisson problem of size $150^3$ on two different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure that the Krylov two-stage algorithm is sensitive to the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an interesting behavior of the Krylov two-stage algorithm. It is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the GMRES algorithms. This means that the multisplitting methods are more
+efficient for distributed systems with high latency networks.
+
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Cluster x Nodes N1 x N2}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+\RCE{ok}
\label{fig:02}
\end{figure}
-%\end{wrapfigure}
-These experiments compare the behavior of the algorithms running first on a
-speed inter-cluster network (N1) and also on a less performant network (N2).
-Figure~\ref{fig:02} shows that end users will gain to reduce the execution time
-for both algorithms in using a grid architecture like 4x16 or 8x8: the
-performance was increased in a factor of 2. The results depict also that when
-the network speed drops down (12.5\%), the difference between the execution
-times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi et quoi?}
+\subsubsection{Network latency impacts on performances\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
-\subsubsection{Network latency impacts on performance}
-\ \\
-\begin{figure} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid & 2x16\\ %\hline
- Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
- \end{tabular}
-\caption{Network latency impact}
-\end{figure}
-
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impact on execution time}
+\caption{Network latency impacts on performances}
\label{fig:03}
\end{figure}
+\subsubsection{Network bandwidth impacts on performances\\}
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.
-According the results in Figure~\ref{fig:03}, a degradation of the network
-latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time increase more
-than 75\% (resp. 82\%) of the execution for the classical GMRES (resp. Krylov
-multisplitting) algorithm. In addition, it appears that the Krylov
-multisplitting method tolerates more the network latency variation with a less
-rate increase of the execution time. Consequently, in the worst case
-(lat=6.10$^{-5 }$), the execution time for GMRES is almost the double than the
-time of the Krylov multisplitting, even though, the performance was on the same
-order of magnitude with a latency of 8.10$^{-6}$.
-
-\subsubsection{Network bandwidth impacts on performance}
-\ \\
-\begin{figure} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid & 2x16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
- \end{tabular}
-\caption{Network bandwidth impact}
-\end{figure}
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impact on execution time}
+\caption{Network bandwith impacts on performances}
\label{fig:04}
\end{figure}
+\subsubsection{Matrix size impacts on performances\\}
+In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes. For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
+These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
-
-The results of increasing the network bandwidth show the improvement of the
-performance for both algorithms by reducing the execution time (see
-Figure~\ref{fig:04}). However, in this case, the Krylov multisplitting method
-presents a better performance in the considered bandwidth interval with a gain
-of 40\% which is only around 24\% for classical GMRES.
-
-\subsubsection{Input matrix size impacts on performance}
-\ \\
-\begin{figure} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
- \end{tabular}
-\caption{Input matrix size impact}
-\end{figure}
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Problem size impact on execution time}
+\caption{Problem size impacts on performances}
\label{fig:05}
\end{figure}
-In these experiments, the input matrix size has been set from N$_{x}$ = N$_{y}$
-= N$_{z}$ = 40 to 200 side elements that is from 40$^{3}$ = 64.000 to 200$^{3}$
-= 8,000,000 points. Obviously, as shown in Figure~\ref{fig:05}, the execution
-time for both algorithms increases when the input matrix size also increases.
-But the interesting results are:
-\begin{enumerate}
- \item the drastic increase (300 times) \RC{Je ne vois pas cela sur la figure}
-of the number of iterations needed to reach the convergence for the classical
-GMRES algorithm when the matrix size go beyond N$_{x}$=150;
-\item the classical GMRES execution time is almost the double for N$_{x}$=140
- compared with the Krylov multisplitting method.
-\end{enumerate}
-
-These findings may help a lot end users to setup the best and the optimal
-targeted environment for the application deployment when focusing on the problem
-size scale up. It should be noticed that the same test has been done with the
-grid 2x16 leading to the same conclusion.
-
-\subsubsection{CPU Power impact on performance}
+\subsubsection{CPU power impacts on performances\\}
+Using the SimGrid simulator flexibility, we have tried to determine the impact of the CPU power of the processors in the different clusters on performances of both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The simulation is conducted in a grid of 2$\times$16 processors interconnected by the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size $150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance gain, about $95\%$ for both algorithms, after improving the CPU power of processors.
-\begin{figure} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
- \end{tabular}
-\caption{CPU Power impact}
-\end{figure}
-
-\begin{figure} [ht!]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
-\caption{CPU Power impact on execution time}
+\caption{CPU Power impacts on performances}
\label{fig:06}
\end{figure}
+\ \\
+To conclude these series of experiments, with SimGrid we have been able to make
+many simulations with many parameters variations. Doing all these experiments
+with a real platform is most of the time not possible. Moreover the behavior of
+both GMRES and Krylov two-stage algorithms is in accordance with larger real
+executions on large scale supercomputers~\cite{couturier15}.
-Using the Simgrid simulator flexibility, we have tried to determine the impact
-on the algorithms performance in varying the CPU power of the clusters nodes
-from 1 to 19 GFlops. The outputs depicted in Figure~\ref{fig:06} confirm the
-performance gain, around 95\% for both of the two methods, after adding more
-powerful CPU.
-\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
+\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
The previous paragraphs put in evidence the interests to simulate the behavior
-of the application before any deployment in a real environment. We have focused
-the study on analyzing the performance in varying the key factors impacting the
-results. The study compares the performance of the two proposed algorithms both
-in \textit{synchronous mode }. In this section, following the same previous
-methodology, the goal is to demonstrate the efficiency of the multisplitting
-method in \textit{ asynchronous mode} compared with the classical GMRES staying
-in \textit{synchronous mode}.
-
-Note that the interest of using the asynchronous mode for data exchange
-is mainly, in opposite of the synchronous mode, the non-wait aspects of
-the current computation after a communication operation like sending
-some data between nodes. Each processor can continue their local
-calculation without waiting for the end of the communication. Thus, the
-asynchronous may theoretically reduce the overall execution time and can
-improve the algorithm performance.
-
-As stated supra, Simgrid simulator tool has been used to prove the
-efficiency of the multisplitting in asynchronous mode and to find the
-best combination of the grid resources (CPU, Network, input matrix size,
-\ldots ) to get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
-
-
-The test conditions are summarized in the table below : \\
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
+of the application before any deployment in a real environment. In this
+section, following the same previous methodology, our goal is to compare the
+efficiency of the multisplitting method in \textit{ asynchronous mode} compared with the
+classical GMRES in \textit{synchronous mode}.
+
+The interest of using an asynchronous algorithm is that there is no more
+synchronization. With geographically distant clusters, this may be essential.
+In this case, each processor can compute its iteration freely without any
+synchronization with the other processors. Thus, the asynchronous may
+theoretically reduce the overall execution time and can improve the algorithm
+performance.
+
+In this section, the SimGrid simulator is used to compare the behavior of the
+two-stage algorithm in asynchronous mode with GMRES in synchronous mode. Several
+benchmarks have been performed with various combinations of the grid resources
+(CPU, Network, matrix size, \ldots). The test conditions are summarized
+in Table~\ref{tab:02}. In order to compare the execution times, Table~\ref{tab:03}
+reports the relative gain between both algorithms. It is defined by the ratio
+between the execution time of GMRES and the execution time of the
+multisplitting.
+\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
+\RCE{Table III avec la nouvelle numerotation}
+The ratio is greater than one because the asynchronous
+multisplitting version is faster than GMRES.
+
+\begin{table}[htbp]
+\centering
+\begin{tabular}{ll}
\hline
- Grid & 2x50 totaling 100 processors\\ %\hline
- Processors Power & 1 GFlops to 1.5 GFlops\\
- Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
- Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
- Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
- Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
+ Grid architecture & 2$\times$50 totaling 100 processors\\
+ Processors Power & 1 GFlops to 1.5 GFlops \\
+ \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\
+ & $bw$=5 Mbits, $lat=20ms$s\\
+ Matrix size & from $62^3$ to $150^3$\\
+ Residual error precision & $10^{-5}$ to $10^{-9}$\\ \hline \\
\end{tabular}
-\end{footnotesize}
+\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\label{tab:02}
+\end{table}
-Again, comprehensive and extensive tests have been conducted varying the
-CPU power and the network parameters (bandwidth and latency) in the
-simulator tool with different problem size. The relative gains greater
-than 1 between the two algorithms have been captured after each step of
-the test. Table 7 below has recorded the best grid configurations
-allowing the multisplitting method execution time more performant 2.5 times than
-the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\begin{table}[!t]
- \centering
+\centering
+%\begin{table}
% \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
% \label{"Table 7"}
-Table 7. Relative gain of the multisplitting algorithm compared with
-the classical GMRES \\
-
- \begin{mytable}{11}
+ \begin{mytable}{11}
\hline
bandwidth (Mbit/s)
& 5 & 5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 \\
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
\hline
- size (N)
+ size ($N^3$)
& 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\
\hline
Precision
& 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\
\hline
\end{mytable}
+%\end{table}
+ \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \label{tab:03}
\end{table}
-\section{Conclusion}
-CONCLUSION
+Again, comprehensive and extensive tests have been conducted with different
+parameters as the CPU power, the network parameters (bandwidth and latency)
+and with different problem size. The relative gains greater than $1$ between the
+two algorithms have been captured after each step of the test. In
+Table~\ref{tab:08} are reported the best grid configurations allowing
+the two-stage multisplitting algorithm to be more than $2.5$ times faster than the
+classical GMRES. These experiments also show the relative tolerance of the
+multisplitting algorithm when using a low speed network as usually observed with
+geographically distant clusters through the internet.
-\section*{Acknowledgment}
-
+\section{Conclusion}
+In this paper we have presented the simulation of the execution of three
+different parallel solvers on some multi-core architectures. We have shown that
+the SimGrid toolkit is an interesting simulation tool that has allowed us to
+determine which method to choose given a specified multi-core architecture.
+Moreover the simulated results are in accordance (i.e. with the same order of
+magnitude) with the works presented in~\cite{couturier15}. Simulated results
+also confirm the efficiency of the asynchronous multisplitting
+algorithm compared to the synchronous GMRES especially in case of
+geographically distant clusters.
+
+These results are important since it is very time consuming to find optimal
+configuration and deployment requirements for a given application on a given
+multi-core architecture. Finding good resource allocations policies under
+varying CPU power, network speeds and loads is very challenging and labor
+intensive. This problematic is even more difficult for the asynchronous
+scheme where a small parameter variation of the execution platform and of the
+application data can lead to very different numbers of iterations to reach the
+converge and so to very different execution times.
+
+
+In future works, we plan to investigate how to simulate the behavior of really
+large scale applications. For example, if we are interested to simulate the
+execution of the solvers of this paper with thousand or even dozens of thousands
+of cores, it is not possible to do that with SimGrid. In fact, this tool will
+make the real computation. So we plan to focus our research on that problematic.
+
+
+
+%\section*{Acknowledgment}
+\ack
This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
-
\bibliographystyle{wileyj}
\bibliography{biblio}
+
\end{document}
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