%\itshape{\journalnamelc}\footnotemark[2]}
\author{Charles Emile Ramamonjisoa\affil{1},
- David Laiymani\affil{1},
- Arnaud Giersch\affil{1},
- Lilia Ziane Khodja\affil{2} and
- Raphaël Couturier\affil{1}
+ Lilia Ziane Khodja\affil{2},
+ David Laiymani\affil{1},
+ Raphaël Couturier\affil{1} and
+ Arnaud Giersch\affil{1}
}
\address{
%% 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.
+%% 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.
+%% 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.
+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.
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}. 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
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.
+confirm the real results previously obtained on real physical 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
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,
+(i.e. problems for which the solution is $x^\star =f(x^\star)$). In practice,
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}.
algorithm that supports both the synchronous or the asynchronous iteration model
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
+the synchronous mode iterations are synchronized, whereas in the asynchronous
+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
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.
+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}
-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.
+In the scope of this paper, we have chosen the SimGrid toolkit~\cite{SimGrid,casanova+giersch+legrand+al.2014.versatile} to simulate the behavior of parallel iterative linear solvers on different computational grid configurations. In opposite to most of the simulators which are stayed very oriented-application, the SimGrid framework is designed to study the behavior of many large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds or High Performance Computation systems. 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.
+SimGrid provides four user interfaces which can be convenient for different distributed applications~\cite{casanova+legrand+quinson.2008.simgrid}. In this paper we are interested on the SMPI user interface (Simulator MPI) which implements about \np[\%]{80} of the MPI 2.0 standard and allows minor modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward} (see Section~\ref{sec:04.02}). SMPI enables the direct simulation of the execution, as in the real life, of an unmodified MPI distributed application, and gets accurate results with the detailed resources consumption.
+
+SimGrid simulator uses at least three XML input files describing the computational grid resources: the number of clusters in the grid, the number of processors/cores in each cluster, the detailed description of the intra and inter networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). SimGrid uses a fluid model to simulate the program execution. It allows several simulation modes which produce accurate results~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode really executes the computation but "intercepts" the communications (the execution time is then evaluated according to the parameters of the simulated platform). It is also possible for SimGrid/SMPI to only keep the duration of large computations by skipping them. Moreover the application can be run "in vitro" mode by sharing some in-memory structures between the simulated processes and thus allowing the use of very large-scale data.
+
+The choice of SimGrid/SMPI as a simulator tool in this study has been emphasized by the results obtained by several studies to validate, in the real environments, the behavior of different network models simulated in SimGrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline the comparison between the real MPI application executions and the SimGrid/SMPI ones~\cite{guermouche+renard.2010.first,clauss+stillwell+genaud+al.2011.single,bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of SimGrid simulations compared to the executions on real physical architectures.
+
+
+
+
+
+
+
+
+
+
+
+
+
+%% In the scope of this paper, the SimGrid toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile},
+%% an open source framework actively developed by its scientific community, has been chosen to simulate the behavior of iterative linear solvers in different computational grid configurations. SimGrid pretends to be non-specialized in opposite to some other simulators which stayed to be very specific oriented-application. One of the well-known SimGrid advantage is its SMPI (Simulated MPI) user interface. SMPI purpose is to execute by simulation in a similar way as in real life, an MPI distributed application and to get accurate results with the detailed resources
+%% consumption.Several studies have demonstrated the accuracy of the simulation
+%% compared with execution on real physical architectures. In addition of SMPI,
+%% Simgrid provides other API which can be convienent for different distrbuted
+%% applications: computational grid applications, High Performance Computing (HPC),
+%% P2P but also clouds applications. In this paper we use the SMPI API. It
+%% implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
+%% modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}
+%% (see Section~\ref{sec:04.02}).
+
+
+%% Provided as an input to the simulator, at least $3$ XML files describe the
+%% computational grid resources: number of clusters in the grid, number of
+%% processors/cores in each cluster, detailed description of the intra and inter
+%% networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). Simgrid uses a fluid model to simulate the program execution.
+%% This gives several simulation modes which produce accurate
+%% results~\cite{bedaride+degomme+genaud+al.2013.toward,
+%% velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode
+%% really executes the computation but "intercepts" the communications (running
+%% time is then evaluated according to the parameters of the simulated platform).
+%% It is also possible for SimGrid/SMPI to only keep duration of large
+%% computations by skipping them. Moreover the application can be run "in vitro"
+%% by sharing some in-memory structures between the simulated processes and
+%% thus allowing the use of very large data scale.
+
+
+%% The choice of Simgrid/SMPI as a simulator tool in this study has been emphasized
+%% by the results obtained by several studies to validate, in real environments,
+%% the behavior of different network models simulated in
+%% Simgrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
+%% the comparison between real MPI executions and SimGrid/SMPI
+%% ones\cite{guermouche+renard.2010.first, clauss+stillwell+genaud+al.2011.single,
+%% bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of
+%% SimGrid simulations.
+
+
+
+
+
+
+% 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}
\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}
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 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 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, 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.
+ 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}{ll}
\hline
-Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
+Grid architecture & 2$\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 \\
+ & $N2$: $bw$=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}
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 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 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).
-
-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$16).
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and
+a given matrix size of 170$^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).
+
+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$16).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
-\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
\label{fig:01}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Various grid configurations with networks $N1$ vs. $N2$}
-\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
-\RCE{ok}
+\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
+%\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+%\RCE{ok}
\label{fig:02}
\end{figure}
\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.
+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.
\begin{figure}[ht]
\centering
\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.
+
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ 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 $1$Gbs to $10$Gbs show the performances improvement for
+both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain in the considered bandwidth
+interval with a gain of $40\%$ compared to only about $24\%$ for the classical
+GMRES algorithm.
\begin{figure}[ht]
\centering
\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.
+
+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}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes. For all problem
+sizes, the 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.
\begin{figure}[ht]
\centering
\end{figure}
\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.
+
+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 on a grid of $2\times16$ 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
\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}.
+with a real platform is most of the time not possible or very costly. Moreover
+the behavior of both GMRES and Krylov two-stage algorithms is in accordance
+with larger real executions on large scale supercomputers~\cite{couturier15}.
\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
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
+In this case, each processor can compute its iterations 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.
+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}.
+
+
+
+%\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
+%\RCE{Table III avec la nouvelle numerotation}
+
\begin{table}[htbp]
\centering
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\\
+ & $bw$=5 Mbits, $lat=20ms$\\
Matrix size & from $62^3$ to $150^3$\\
Residual error precision & $10^{-5}$ to $10^{-9}$\\ \hline \\
\end{tabular}
\label{tab:03}
\end{table}
-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.
+
+Table~\ref{tab:03} reports the relative gains between both algorithms. It is
+defined by the ratio between the execution time of GMRES and the execution time
+of the multisplitting. The ratio is greater than one because the asynchronous
+multisplitting version is faster than GMRES. In average, 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{Conclusion}
