-\documentclass[times]{cpeauth}
+ \documentclass[times]{cpeauth}
\usepackage{moreverb}
\usepackage{algpseudocode}
%\usepackage{amsthm}
\usepackage{graphicx}
-\usepackage[american]{babel}
% Extension pour les liens intra-documents (tagged PDF)
% et l'affichage correct des URL (commande \url{http://example.com})
%\usepackage{hyperref}
+\usepackage{multirow}
+
\usepackage{url}
\DeclareUrlCommand\email{\urlstyle{same}}
\todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
\newcommand{\RCE}[2][inline]{%
\todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}
+\newcommand{\DL}[2][inline]{%
+ \todo[color=pink!10,#1]{\sffamily\textbf{DL:} #2}\xspace}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\newcommand{\MIG}{\mathit{maxit_{gmres}}}
\newcommand{\TOLM}{\mathit{tol_{multi}}}
\newcommand{\MIM}{\mathit{maxit_{multi}}}
+\newcommand{\TOLC}{\mathit{tol_{cgls}}}
+\newcommand{\MIC}{\mathit{maxit_{cgls}}}
\usepackage{array}
\usepackage{color, colortbl}
\begin{document}
-\RCE{Titre a confirmer.}
-\title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
+\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},
+ Lilia Ziane Khodja\affil{2},
+ David Laiymani\affil{1},
+ Raphaël Couturier\affil{1} and
+ Arnaud Giersch\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,raphael.couturier,arnaud.giersch}@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. 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 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.
-\begin{abstract}
-ABSTRACT
\end{abstract}
-\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
+%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
+%performance}
+\keywords{ Performance evaluation, Simulation, SimGrid, Synchronous and asynchronous iterations, Multisplitting algorithms}
\maketitle
-\section{Introduction}
-
-\section{The asynchronous iteration model}
+\section{Introduction} The use of multi-core architectures to solve large
+scientific problems seems to become imperative in many situations.
+Whatever the scale of these architectures (distributed clusters, computational
+grids, embedded multi-core,~\ldots) they are generally well adapted to execute
+complex parallel applications operating on a large amount of data.
+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
+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: 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
+simple. It generally involves the division of the problem into several
+\emph{blocks} that will be solved in parallel on multiple processing
+units. Each processing unit has to compute an iteration to send/receive some
+data dependencies to/from its neighbors and to iterate this process until the
+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. 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}. 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 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 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.
+
+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 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
+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.
+
+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.
+
+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:04} details the different solvers that we use. Finally our
+experimental results are presented in Section~\ref{sec:expe} followed by some
+concluding remarks and perspectives.
+
+
+\section{The asynchronous iteration model and the motivations of our work}
+\label{sec:asynchro}
+
+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,
+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}.
+
+Before using an asynchronous iterative method, the convergence must be
+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 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
+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 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
+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}
+
+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 splitting methods}
+\section{Two-stage multisplitting methods}
\label{sec:04}
-\subsection{Multisplitting methods for sparse linear systems}
+\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
\label{sec:04.01}
-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}
\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. The 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^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
+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 a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$~\cite{O'leary85,White86}, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting 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}
-M_\ell y_\ell = c_\ell^k,\mbox{~such that~} c_\ell^k = N_\ell x^k + b,
+A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
-then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
+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]
+%\caption{Block Jacobi two-stage multisplitting method}
+\begin{algorithmic}[1]
+ \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+ \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence}
+ \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+ \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
+ \State Send $x_\ell^k$ to neighboring clusters\label{send}
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
+ \EndFor
+\end{algorithmic}
+\caption{Block Jacobi two-stage multisplitting method}
+\label{alg:01}
+%\end{algorithm}
+\end{figure}
+
+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}
-x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y_\ell.
+k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
\label{eq:04}
\end{equation}
-The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{O'leary85,bahi97,Bai99,bahi07}. %It is dependent on the condition
-%\begin{equation}
-%\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
-%\label{eq:05}
-%\end{equation}
-%where $\rho$ is the spectral radius of the square matrix.
-The multisplitting methods are convergent:
-\begin{itemize}
-\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular (i.e. $M^{-1}\geq 0$ and $M^{-1}N\geq 0$) when the iterations are synchronous, or
-\item if $A$ is M-matrix and its splittings are regular (i.e. $M^{-1}\geq 0$ and $N\geq 0$) when the iterations are asynchronous.
-\end{itemize}
-The solutions of the different linear sub-systems~(\ref{eq:03}) arising from the multisplitting of matrix $A$ can be either computed exactly with a direct method or approximated with an iterative method. In the latter case, the multisplitting methods are called {\it inner-outer iterative methods} or {\it two-stage multisplitting methods}. This kind of methods uses two nested iterations: the outer iteration and the inner iteration (that of the iterative method).
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
-In this paper we are focused on two-stage multisplitting methods, in their both versions synchronous and asynchronous, where the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} is used as an inner iteration. Furthermore, 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. weighting matrices $E_\ell$ have only zero and one factors). In this case, the iteration of the multisplitting method presented by (\ref{eq:03}) and~(\ref{eq:04}) can be rewritten in the following form
+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}
-A_{\ell\ell} x_\ell^{k+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
+S=[x^1,x^2,\ldots,x^s],~s\ll n.
\label{eq:05}
\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. In each outer iteration $k$ until the convergence, each sub-system arising from the block Jacobi multisplitting
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual:
\begin{equation}
-A_{\ell\ell} x_\ell = c_\ell,
+\min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
\label{eq:06}
\end{equation}
-is solved iteratively using GMRES method and independently from other sub-systems by a cluster of processors. The right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. Algorithm~\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:06}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold of GMRES respectively.
+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{algorithm}[t]
-\caption{Block Jacobi two-stage multisplitting method}
+\begin{figure}[htbp]
+%\begin{algorithm}[t]
+%\caption{Krylov two-stage method using block Jacobi multisplitting}
\begin{algorithmic}[1]
\Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
\Output $x_\ell$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x^0$
\For {$k=1,2,3,\ldots$ until convergence}
\State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
- \State $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
- \State Send $x_\ell^k$ to neighboring clusters\label{send}
- \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
+ \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$
+ \State $S_{\ell,k\mod s}=x_\ell^k$
+ \If{$k\mod s = 0$}
+ \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
+ \State $\tilde{x_\ell}=S_\ell\alpha$
+ \State Send $\tilde{x_\ell}$ to neighboring clusters
+ \Else
+ \State Send $x_\ell^k$ to neighboring clusters
+ \EndIf
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
\EndFor
\end{algorithmic}
-\label{alg:01}
-\end{algorithm}
-
-Multisplitting methods are more advantageous for large distributed computing platforms composed of hundreds or even thousands of processors interconnected by high latency networks. In this context, the parallel asynchronous model is preferred to the synchronous one to reduce overall execution times of the algorithms, even if it generally requires more iterations to converge. The asynchronous model allows the communications to be overlapped by computations which suppresses the idle times resulting from the synchronizations. So in 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 Algorithm~\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:07}
-\end{equation}
-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold of the two-stage algorithm. The procedure of the convergence detection is implemented as follows. All clusters are interconnected by a virtual unidirectional ring network around which a Boolean token circulates from a cluster to another.
-
-
-
+\caption{Krylov two-stage method using block Jacobi multisplitting}
+\label{alg:02}
+%\end{algorithm}
+\end{figure}
+\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 to ensure the
+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 (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"
+(smpi/privatize\_global\_variables). Indeed, global variables can generate side
+effects on runtime between the threads running in the same process and generated by
+SimGrid to simulate the grid environment.
+
+\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 that there are intra and
+inter-cluster communications. In the following, these parameters are described:
+\begin{itemize}
+ \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{}),
+ \item archi : grid computational description (number of clusters, number of
+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 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}
-\subsection{Simulation of two-stage methods using SimGrid framework}
+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 and Comments}
-
-
-\textbf{V.1. Setup study and Methodology}
-
-To conduct our study, we have put in place the following methodology
-which can be reused with any grid-enabled applications.
-
-\textbf{Step 1} : Choose with the end users the class of algorithms or
-the application to be tested. Numerical parallel iterative algorithms
-have been chosen for the study in the paper.
-
-\textbf{Step 2} : Collect the software materials needed for the
-experimentation. In our case, we have three variants algorithms for the
-resolution of three 3D-Poisson problem: (1) using the classical GMRES alias Algo-1 in this
-paper, (2) using the multisplitting method alias Algo-2 and (3) an
-enhanced version of the multisplitting method as Algo-3. In addition,
-SIMGRID simulator has been chosen to simulate the behaviors of the
-distributed applications. SIMGRID is running on the Mesocentre
-datacenter in Franche-Comte University $[$10$]$ but also in a virtual
-machine on a 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
-in one hand the algorithm execution mode (synchronous and asynchronous)
-and in the other hand the execution time and the number of iterations of
-the application before obtaining the convergence.
-
-\textbf{Step 4 }: Setup up the different grid testbeds environment
-which will be simulated in the simulator tool to run the program. The
-following architecture has been configured in Simgrid : 2x16 - that is a
-grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8,
-4x16, 8x8 and 2x50. The network has been designed to operate with a
-bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8E-6
-microseconds (resp. 5E-5) for the intra-clusters links (resp.
-inter-clusters backbone links).
-
-\textbf{Step 5}: Process an extensive and comprehensive testings
-within these configurations in varying the key parameters, especially
-the CPU power capacity, the network parameters and also the size of the
-input matrix. Note that some parameters should be invariant to allow the
-comparison like some program input arguments.
-
-\textbf{Step 6} : Collect and analyze the output results.
-
-\textbf{ V.2. Factors impacting distributed applications performance in
-a grid environment}
-
-From our previous experience on running distributed application in a
-computational grid, many factors are identified to have an impact on the
-program behavior and performance on this specific environment. Mainly,
-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 performance of the
-application is the network configuration. Two main network parameters
-can modify drastically the program output results : (i) the network
-bandwidth (bw=bits/s) also known as "the data-carrying capacity"
-$[$13$]$ of the network is defined as the maximum of data that can pass
-from one point to another in a unit of time. (ii) 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. 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 in transit between the clusters and nodes during the code
-execution.
-
- In a grid environment, it is common to distinguish in one hand, the
-"\,intra-network" which refers to the links between nodes within a
-cluster and in the other hand, the "\,inter-network" which is the
-backbone link between clusters. By design, these two networks perform
-with different speed. 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 thru internet with a lower speed. The network
-between distant clusters might be a bottleneck for the global
-performance of the application.
-
-\textbf{V.3 Comparing GMRES and Multisplitting algorithms in
-synchronous mode}
-
-In the scope of this paper, our first objective is to demonstrate the
-Algo-2 (Multisplitting method) shows a better performance in grid
-architecture compared with Algo-1 (Classical GMRES) both running in
-\textbf{\textit{synchronous mode}}. Better algorithm performance
-should mean a less number of iterations output and a less 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 $[$12$]$.
-
-The following paragraphs present the test conditions, the output results
-and our comments.
-
-
-\textit{3.a Executing the algorithms on various computational grid
-architecture scaling up the input matrix size}
-\\
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbs-lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150 and\\ %\hline
- - & N$_{x}$ =170 x 170 x 170 \\ \hline
- \end{tabular}
-\end{footnotesize}
+\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.
- Table 1 : Clusters x Nodes with NX=150 or NX=170
+\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:
+\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:
+\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{array}
+\label{eq:08}
+\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.
+
+\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.\\
+
+\textbf{Step 1}: Choose with the end users the class of algorithms or
+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 for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) using 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 architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
+represents the number of clusters in the grid and the second number represents
+the number of hosts (processors/cores) in each cluster. \\
+
+\textbf{Step 5}: Conduct extensive and comprehensive testings
+within these configurations by varying the key parameters, especially
+the CPU power capacity, the network parameters and also the size of the
+input data. \\
-\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
+\textbf{Step 6} : Collect and analyze the output results.
+\subsection{Factors impacting distributed applications performance in a grid environment}
+
+When running a distributed application in a computational grid, many factors may
+have a strong impact on the performance. First of all, the architecture of the
+grid itself can obviously influence the performance results of the program. The
+performance gain might be important theoretically when the number of clusters
+and/or the number of nodes (processors/cores) in each individual cluster
+increase.
+
+Another important factor impacting the overall performance of the application
+is the network configuration. Two main network parameters can modify drastically
+the program output results:
+\begin{enumerate}
+\item the network bandwidth ($bw$ in Gbits/s) also known as "the data-carrying
+ capacity" of the network is defined as the maximum amount of data that can transit
+ from one point to another in a unit of time.
+\item the network latency ($lat$ in microseconds) defined as the delay from the
+ starting time to send a simple data from a source to a destination.
+\end{enumerate}
+Among 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
+ \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. On the contrary, the
+ inter-network connects clusters sometimes via heterogeneous networks components the 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 performances than the
+classical GMRES method. With a synchronous iterative method, better performances
+mean 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 $bw$=10Gbit/s 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\\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbit/s, $lat=8\mu$s \\
+ & $N2$: $bw$=1Gbit/s, $lat=50\mu$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{table}
-The results in figure 1 show 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.
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
+
+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 splittings (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number of splittings 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 are 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 (it is about
+$32\%$ more efficient on a grid of 8$\times$8 than the 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 two matrix sizes: $150^3$ and $170^3$}
+\label{fig:01}
+\end{figure}
-%\begin{wrapfigure}{l}{60mm}
-\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=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-\caption{Cluster x Nodes NX=150 and NX=170}
-%\label{overflow}}
+\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}
+\label{fig:02}
\end{figure}
-%\end{wrapfigure}
-
-Unless the 8x8 cluster, the time
-execution difference between the two algorithms is important when
-comparing between different grid architectures, even with the same number of
-processors (like 2x16 and 4x8 = 32 processors for example). The
-experiment concludes the low sensitivity of the multisplitting method
-(compared with the classical GMRES) when scaling up to higher input
-matrix size.
-
-\textit{3.b Running on various computational grid architecture}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline
- - & N2 : bw=1Gbs-lat=5E-05 \\
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\
- \end{tabular}
-\end{footnotesize}
-%Table 2 : Clusters x Nodes - Networks N1 x N2
-%\RCE{idem pour tous les tableaux de donnees}
+\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 $bw$=1Gbit/s 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{wrapfigure}{l}{60mm}
-\begin{figure} [ht!]
+\begin{figure}[ht]
\centering
-\includegraphics[width=60mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Cluster x Nodes N1 x N2}
-%\label{overflow}}
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impacts on performances}
+\label{fig:03}
\end{figure}
-%\end{wrapfigure}
-
-The experiments compare the behavior of the algorithms running first on
-speed inter- cluster network (N1) and a less performant network (N2).
-The figure 2 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, the difference between the execution
-times can reach more than 25\%.
-
-\textit{\\\\\\\\\\\\\\\\\\3.c Network latency impacts on performance}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16\\ %\hline
- Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\
- \end{tabular}
-\end{footnotesize}
-Table 3 : Network latency impact
+\subsubsection{Network bandwidth impacts on performances\\}
+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 $150^3$. Increasing the
+network bandwidth from $1$Gbit/s to $10$Gbit/s results in improving the performances of 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!]
+\begin{figure}[ht]
\centering
-\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impact on execution time}
-%\label{overflow}}
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
+\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 varies 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 greatly 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
+\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
+\caption{Problem size impacts on performances}
+\label{fig:05}
+\end{figure}
-According the results in table and figure 3, 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. multisplitting) algorithm. In addition, it appears that the
-multisplitting method tolerates more the network latency variation with
-a less rate increase. Consequently, in the worst case (lat=6.10$^{-5
-}$), the execution time for GMRES is almost the double of the time for
-the multisplitting, even though, the performance was on the same order
-of magnitude with a latency of 8.10$^{-6}$.
-
-\textit{3.d Network bandwidth impacts on performance}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
- \end{tabular}
-\end{footnotesize}
+\subsubsection{CPU power impacts on performances\\}
-Table 4 : Network bandwidth impact
+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!]
+\begin{figure}[ht]
\centering
-\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impact on execution time}
-%\label{overflow}
+\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
+\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 impossible 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}.
-The results of increasing the network bandwidth depict the improvement
-of the performance by reducing the execution time for both of the two
-algorithms. However, and again in this case, the multisplitting method
-presents a better performance in the considered bandwidth interval with
-a gain of 40\% which is only around 24\% for classical GMRES.
+\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
-\textit{3.e Input matrix size impacts on performance}
+The previous paragraphs put in evidence the interest to simulate the behavior
+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}.
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
- \end{tabular}
-\end{footnotesize}
+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 iterations freely without any
+synchronization with the other processors. Thus, an asynchronous algorithm may
+theoretically reduce the overall execution time and can also improve the algorithm
+performance.
-Table 5 : Input matrix size impact
+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}.
-\begin{figure} [ht!]
-\centering
-\includegraphics[width=60mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Pb size impact on execution time}
-%\label{overflow}}
-\end{figure}
-In this experimentation, the input matrix size has been set from
-Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to
-200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 5,
-the execution time for the algorithms convergence increases with the
-input matrix size. But the interesting result here direct on (i) the
-drastic increase (300 times) of the number of iterations needed before
-the convergence for the classical GMRES algorithm when the matrix size
-go beyond Nx=150; (ii) the classical GMRES execution time also almost
-the double from Nx=140 compared with the convergence time of the
-multisplitting method. 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. Note that the
-same test has been done with the grid 2x16 getting the same conclusion.
-
-\textit{3.f CPU Power impact on performance}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
- \end{tabular}
-\end{footnotesize}
-Table 6 : CPU Power impact
+%\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
+%\RCE{Table III avec la nouvelle numerotation}
-\begin{figure} [ht!]
-\centering
-\includegraphics[width=60mm]{cpu_power_impact_on_execution_time.pdf}
-\caption{CPU Power impact on execution time}
-%\label{overflow}}
-\end{figure}
-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 the figure 6
-confirm the performance gain, around 95\% for both of the two methods,
-after adding more powerful CPU. Note that the execution time axis in the
-figure is in logarithmic scale.
-
- \textbf{V.4 Comparing GMRES in native synchronous mode and
-Multisplitting algorithms in asynchronous mode}
-
-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. In the same line, the study compares
-the performance of the two proposed methods in \textbf{synchronous mode
-}. In this section, with the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textbf{
-asynchronous mode} compare with the classical GMRES staying in the
-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 "\,relative gain" in comparison with the
-classical GMRES time.
-
-
-The test conditions are summarized in the table below :
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline
- Grid & 2x50 totaling 100 processors\\ %\hline
- Processors & 1 GFlops to 1.5 GFlops\\
- Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
- Inter-Network & bw=5 Mbits - lat=2E-02\\
- Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
- Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline
+\begin{table}[htbp]
+\centering
+\begin{tabular}{ll}
+ \hline
+ Grid architecture & 2$\times$50 totaling 100 processors\\
+ Processors Power & 1 GFlops to 1.5 GFlops \\
+ \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 $10^{-11}$\\ \hline \\
\end{tabular}
-\end{footnotesize}
-
-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 I below has recorded the best grid configurations
-allowing a multiplitting method time more than 2.5 times lower than
-classical GMRES execution and convergence time. The finding thru this
-experimentation is the tolerance of the multisplitting method under a
-low speed network that we encounter usually with distant clusters thru the
-internet.
+\caption{Test conditions: GMRES in synchronous mode vs. two-stage multisplitting in asynchronous mode}
+\label{tab:02}
+\end{table}
+
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
|*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
\end{tabular}}
-\begin{table}[!t]
- \centering
- \caption{Relative gain of the multisplitting algorithm compared with
-the classical GMRES}
- \label{tab.cluster.2x50}
-
- \begin{mytable}{6}
- \hline
- bw
- & 5 & 5 & 5 & 5 & 5 & 50 \\
- \hline
- lat
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
- \hline
- power
- & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 \\
- \hline
- size
- & 62 & 62 & 62 & 100 & 100 & 110 \\
- \hline
- Prec/Eprec
- & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
- \hline
- speedup
- & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
- \hline
- \end{mytable}
-
- \smallskip
- \begin{mytable}{6}
+\begin{table}[!t]
+\centering
+%\begin{table}
+% \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+% \label{"Table 7"}
+ \begin{mytable}{11}
\hline
- bw
- & 50 & 50 & 50 & 50 & 10 & 10 \\
+ bandwidth (Mbit/s)
+ & 5 & 5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 \\
\hline
- lat
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\
+ latency (ms)
+ & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 \\
\hline
- power
- & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 \\
+ power (GFlops)
+ & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
\hline
- size
- & 120 & 130 & 140 & 150 & 171 & 171 \\
+ size ($N^3$)
+ & 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\
\hline
- Prec/Eprec
- & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\
+ Precision
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
\hline
- speedup
- & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\
+ Relative gain
+ & 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 asynchronous two-stage multisplitting algorithm compared to the classical synchronous GMRES algorithm}
+ \label{tab:03}
\end{table}
-\section{Conclusion}
-CONCLUSION
+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. On average, the two-stage
+multisplitting algorithm is 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}
-
-
-The authors would like to thank\dots{}
+\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 allocation 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
+convergeence and consequently 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 in simulating 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. That is why, 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}
%%% Local Variables:
%%% 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
