-\documentclass[times]{cpeauth}
+ \documentclass[times]{cpeauth}
\usepackage{moreverb}
%\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{
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
+ Email:~\email{{charles.ramamonjisoa,david.laiymani,raphael.couturier,arnaud.giersch}@univ-fcomte.fr}\break
\affilnum{2}
Department of Aerospace \& Mechanical Engineering,
Non Linear Computational Mechanics,
%% 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 always proves quite challenging to predict, especially with a new architecture for which no experiment has yet 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.
Unfortunately, users (industrials or scientists), who need such computational
resources, may not have an easy access to such efficient architectures. The cost
of using the platform and/or the cost of testing and deploying an application
-are often very important. So, in this context it is difficult to optimize a
+are often very important. So, in this context, it is difficult to optimize a
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\dots{} In counterpart, the simulation results need to be consistent with the real ones.
+simulation environment. The advantages are numerous: life cycle development,
+code debugging, ability to obtain results quickly\dots{} In return, 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
convergence of the method. Several well-known studies demonstrate the
convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
task cannot begin a new iteration while it has not received data dependencies
-from its neighbors. We say that the iteration computation follows a
+from its neighbors. The iteration computation is said to follow 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 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
synchronizations~(see~\cite{bahi07} for more details).
-Nevertheless, in both cases (synchronous or asynchronous) it is very time
-consuming to find optimal configuration and deployment requirements for a given
+Nevertheless, in both cases (synchronous or asynchronous) it is extremely time
+consuming to find optimal configurations 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~\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 and of the application data can
-lead to very different numbers of iterations to reach the convergence and so to
+lead to very different numbers of iterations to reach the convergence and consequently 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.
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.
+Moreover, the obtained results on different simulated multi-core 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 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
+Thus, with a simple computing architecture (a laptop) SimGrid allows us
+to run a test campaign of 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
asynchronous iterative application.
\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
-Asynchronous iterative methods have been studied for many years theoretically and
+Asynchronous iterative methods have been studied for many years both 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
+non-linear systems of equations or optimization problems. Interested readers are
invited to read~\cite{BT89,bahi07}.
Before using an asynchronous iterative method, the convergence must be
-studied. Otherwise, the application is not ensure to reach the convergence. An
+studied. Otherwise, there is no guarantee that the application will reach the convergence. An
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
+practice, only the communications management and the 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
used in both modes. Concerning the convergence detection, synchronous variants
can use a global convergence procedure which acts as a global synchronization
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
+number of iterations is the same as in the sequential mode. Thus, 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
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.
-%%%%%%%%%%%%%%%%%%%%%%%%%
-% 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.
+In the scope of this paper, we have chosen the SimGrid
+toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.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 application-oriented, 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 provides four user interfaces which can be convenient for different
+distributed applications. In this paper we are interested on the SMPI
+(Simulated MPI) user interface which implements about \np[\%]{80} of the MPI 2.0
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and allows minor
+modifications of the initial code (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 an XML input file 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 employs a fluid model to simulate the use of
+these resources along the program execution. This model produces accurate
+results while still running relatively
+fast~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}.
+During the simulation, the computations are really executed, but the communications
+are intercepted and their execution time 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, when applicable, the
+application can be run 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}
\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 both their 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}
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 so 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~\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}.
+where the 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. Line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using the 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}[htpb]
%\begin{algorithm}[t]
%\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 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 scheme 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}
\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}).
+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 the CGLS method~\cite{Hestenes52} sosuch 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}[htbp]
%\begin{algorithm}[t]
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,
+test reproducibility under similar conditions. According to our experience,
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
+simulator using SMPI (Simulated MPI). The first modification is to include SMPI
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"
effects on runtime between the threads running in the same process and generated by
SimGrid to simulate the grid environment.
-\paragraph{Parameters of the simulation in SimGrid}
+\paragraph{Simulation parameters for 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
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
-\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
-\RCE{ok}
+%\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+%\RCE{ok}
\label{fig:02}
\end{figure}
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 gains 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
\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, $lat=20ms$\\
+ \multirow{2}{*}{Network inter-clusters} & $bw$: 5 Mbits to 50 Mbits\\
+ & $lat$: 20 ms\\
Matrix size & from $62^3$ to $150^3$\\
- Residual error precision & $10^{-5}$ to $10^{-9}$\\ \hline \\
+ Residual error precision & $10^{-5}$ to $10^{-11}$\\ \hline \\
\end{tabular}
-\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\caption{Test conditions: GMRES in synchronous mode vs. two-stage multisplitting in asynchronous mode}
\label{tab:02}
\end{table}
\hline
\end{mytable}
%\end{table}
- \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \caption{Relative gains of the asynchronous two-stage multisplitting algorithm compared to the classical synchronous GMRES algorithm}
\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}
%%% fill-column: 80
%%% ispell-local-dictionary: "american"
%%% End:
+
+% LocalWords: Ramamonjisoa Ziane Khodja Laiymani Raphaël Arnaud Giersch Femto
+% LocalWords: Franche Comté Belfort GMRES multisplitting SimGrid Krylov SMPI
+% LocalWords: MPI
