X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/d9a3251a5ed3fb85594afc884b06d5bc106ae9f6..5933ab9497ff790a931894f60fa16b9949111ff9:/paper.tex

diff --git a/paper.tex b/paper.tex
index eb8020a..4309369 100644
--- a/paper.tex
+++ b/paper.tex
@@ -88,7 +88,7 @@
   Femto-ST Institute, DISC Department,
   University of Franche-ComtÃ©,
   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,
@@ -215,7 +215,7 @@ 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 garantee that the  application will 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 management and  the convergence detection are  different. In
@@ -230,7 +230,7 @@ consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 The number of iterations required to reach the convergence is generally greater
 for the asynchronous scheme (this number depends on  the delay of the
 messages). Note that, it is not the case in the synchronous mode where the
-number of iterations is the same than in the sequential mode. In this way, the
+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
@@ -246,38 +246,38 @@ magnitude). To our knowledge, there is no study on this problematic.
 \label{sec:simgrid}
 
 In the scope of this paper, we have chosen the SimGrid
-toolkit~\cite{SimGrid,casanova+giersch+legrand+al.2014.versatile}
+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 oriented-application, the SimGrid framework is designed to study
+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~\cite{casanova+legrand+quinson.2008.simgrid}. In this
-paper we are interested on the SMPI user interface (Simulator MPI) which
-implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
-modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}
-(see Section~\ref{sec:04.02}). SMPI enables the direct simulation of the
-execution, as in the real life, of an unmodified MPI distributed application,
-and gets accurate results with the detailed resources consumption.
-
-SimGrid simulator uses at least three XML input files describing the
-computational grid resources: the number of clusters in the grid, the number of
-processors/cores in each cluster, the detailed description of the intra and
-inter networks and the list of the hosts in each cluster (see the details in
-Section~\ref{sec:expe}). SimGrid uses a fluid model to simulate the program
-execution. It allows several simulation modes which produce accurate
-results~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}. For
-instance, the "in vivo" mode really executes the computation but "intercepts"
-the communications (the execution time is then evaluated according to the
-parameters of the simulated platform). It is also possible for SimGrid/SMPI to
-only keep the duration of large computations by skipping them. Moreover the
-application can be run "in vitro" mode by sharing some in-memory structures
-between the simulated processes and thus allowing the use of very large-scale
-data.
+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
@@ -389,7 +389,7 @@ real physical architectures.
 \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}
@@ -399,12 +399,12 @@ where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side
 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]
@@ -442,7 +442,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde
 \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]
@@ -475,9 +475,9 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 
 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"
@@ -485,10 +485,10 @@ 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
+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}
@@ -563,8 +563,8 @@ the application to be tested. Numerical parallel iterative algorithms
 have been chosen for the study in this paper. \\
 
 \textbf{Step 2}: Collect the software materials needed for the experimentation.
-In our case, we have two variants algorithms for the resolution of the
-3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+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. \\
@@ -580,7 +580,7 @@ have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times
 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 an extensive and comprehensive testings
+\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.  \\
@@ -600,39 +600,38 @@ 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 bits/s) also  known  as "the  data-carrying
-    capacity" of the network is defined as  the maximum of data that can transit
+\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
-  start time to send  a simple data from a source to a destination.
+  starting time to send  a simple data from a source to a destination.
 \end{enumerate}
-Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
+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 one hand,  the
+ 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. In opposite, the
- inter-network connects clusters sometime via  heterogeneous networks components
- through internet with a lower speed.  The network  between distant  clusters
+ 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 performance than the
-classical GMRES method. With a synchronous iterative method, better performance
-means a smaller number of iterations and execution time before reaching the
+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 $bw$=10Gbs and the latency
+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.
 
@@ -641,8 +640,8 @@ results and our comments.
 \begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
-                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{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}
@@ -705,7 +704,7 @@ efficient for distributed systems with high latency networks.
 \end{figure}
 
 \subsubsection{Network latency impacts on performances\\}
-Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=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{figure}[ht]
 \centering
@@ -719,7 +718,7 @@ Figure~\ref{fig:03} shows the impact of the network latency on the 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 $150^3$. The results of increasing the
-network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
+network bandwidth from $1$Gbit/s to $10$Gbit/s show the performances improvement for
 both algorithms by reducing the execution times. However, the Krylov two-stage
 algorithm presents a better performance gain in the considered bandwidth
 interval with a gain of $40\%$ compared to only about $24\%$ for the classical
@@ -919,3 +918,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:
+
+%  LocalWords:  Ramamonjisoa Ziane Khodja Laiymani RaphaÃ«l Arnaud Giersch Femto
+%  LocalWords:  Franche ComtÃ© Belfort GMRES multisplitting SimGrid Krylov SMPI
+%  LocalWords:  MPI
