renomme un fichier figure

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index a5c7827164382f23aa4c32b953e0db3249ff9a3b..79345018f4a9c66e356086ab5e35794a212b5d93 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -84,22 +84,29 @@ thousands of cores are used.
 
 Traditional parallel iterative solvers are based on fine-grain computations that
 frequently  require  data exchanges  between  computing  nodes  and have  global
 
 Traditional parallel iterative solvers are based on fine-grain computations that
 frequently  require  data exchanges  between  computing  nodes  and have  global

-synchronizations  that penalize  the  scalability. Particularly,  they are  more
-penalized on large  scale architectures or on distributed  platforms composed of
-distant  clusters interconnected  by  a high-latency  network.  It is  therefore
-imperative to develop coarse-grain based algorithms to reduce the communications
-in the  parallel iterative  solvers. Two possible  solutions consists  either in
-using  asynchronous  iterative  methods~\cite{ref18}  or in  using  multisplitting
-algorithmss.  In this  paper,  we will  reconsider  the use  of a  multisplitting
-method. In opposition to traditional multisplitting method that suffer from slow
-convergence, as  proposed in~\cite{huang1993krylov},  the use of  a minimization
-process can drastically improve the convergence.\\
+synchronizations that penalize the scalability~\cite{zkcgb+14:ij}. Particularly,
+they are more penalized on large scale architectures or on distributed platforms
+composed of  distant clusters interconnected  by a high-latency network.   It is
+therefore  imperative to  develop coarse-grain  based algorithms  to  reduce the
+communications  in  the  parallel  iterative  solvers.  Two  possible  solutions
+consists either in using asynchronous iterative methods~\cite{ref18} or in using
+multisplitting  algorithms.  In  this paper,  we will  reconsider the  use  of a
+multisplitting method.  In opposition to traditional  multisplitting method that
+suffer from slow convergence,  as proposed in~\cite{huang1993krylov}, the use of
+a minimization process can drastically improve the convergence.\\

 
 
 %%% AJOUTE************************
 %%%*******************************
 
 
 %%% AJOUTE************************
 %%%*******************************

-\noindent {\bf Contributions:}\\ 
-In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to others iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm.\\
+\noindent  {\bf  Contributions:}\\  In  this  work we  develop  a  new  parallel
+two-stage algorithm for large-scale clusters.   Our objective is to create a mix
+between Krylov based iterative methods  and the multisplitting method to improve
+scalability.   In fact  Krylov subspace  methods are  well-known for  their good
+convergence compared to  other iterative methods.  So, our  main contribution is
+to  use  the  multisplitting method  which  splits  the  problem to  solve  into
+different blocks in  order to reduce the large amount  of communications and, to
+implement both inner and outer iterations as Krylov subspace iterations in order
+to improve the convergence of the multisplitting algorithm.\\

 %%%*******************************
 %%%*******************************
 
 %%%*******************************
 %%%*******************************
 


@@ -183,7 +190,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 
 %%% MODIFIE ************************
 %%%*********************************
 
 %%% MODIFIE ************************
 %%%*********************************

-Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting method to the linear system as follows

 %%%*********************************
 %%%*********************************
 
 %%%*********************************
 %%%*********************************
 


@@ -201,7 +208,7 @@ b & = & [B_{1}, \ldots, B_{L}]
 where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. 
 %%% MODIFIE ***********************
 %%%********************************
 where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. 
 %%% MODIFIE ***********************
 %%%********************************

-The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. 
+The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a block of processors. 

 %%%********************************
 %%%********************************
 So, the multisplitting format of the linear system is defined as follows
 %%%********************************
 %%%********************************
 So, the multisplitting format of the linear system is defined as follows


@@ -221,7 +228,7 @@ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_
 \right.
 \label{sec03:eq03}
 \end{equation}
 \right.
 \label{sec03:eq03}
 \end{equation}

-is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). 
+is solved independently by a {\it block of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the blocks. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). 

 %%% MODIFIE ***********************
 %%%********************************
 GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. 
 %%% MODIFIE ***********************
 %%%********************************
 GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. 


@@ -234,9 +241,9 @@ different  sub-systems~(\ref{sec03:eq03})  does   not  necessarily  involve  the
 convergence of the multisplitting algorithm.  It strongly depends on the properties
 of       the       global      sparse       linear       system      to       be
 solved~\cite{o1985multi,ref18}. Furthermore, the  splitting of the linear system
 convergence of the multisplitting algorithm.  It strongly depends on the properties
 of       the       global      sparse       linear       system      to       be
 solved~\cite{o1985multi,ref18}. Furthermore, the  splitting of the linear system

-among  several clusters  of  processors  increases the  spectral  radius of  the
+among  several blocks  of  processors  increases the  spectral  radius of  the

 iteration  matrix, thereby  slowing the  convergence.  In  fact, the  larger the
 iteration  matrix, thereby  slowing the  convergence.  In  fact, the  larger the

-number of  splitting is, the larger the  spectral radius is.  In  this paper, our
+number of  splittings is, the larger the  spectral radius is.  In  this paper, our

 work is based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
 convergence and improve the scalability of the multisplitting methods.
 
 work is based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
 convergence and improve the scalability of the multisplitting methods.
 


@@ -253,11 +260,12 @@ S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
 where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 
 %%% MODIFIE ***********************
 %%%********************************
 where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 
 %%% MODIFIE ***********************
 %%%********************************

-The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+
+The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system.

 %%%********************************
 %%%********************************
 
 %%%********************************
 %%%********************************
 

-The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error function $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system

 \begin{equation}
 R\alpha=b,
 \label{sec03:eq05}
 \begin{equation}
 R\alpha=b,
 \label{sec03:eq05}


@@ -272,7 +280,7 @@ which is associated with the least squares problem
 \text{minimize}~\|b-R\alpha\|_2,
 \label{sec03:eq07}
 \end{equation}  
 \text{minimize}~\|b-R\alpha\|_2,
 \label{sec03:eq07}
 \end{equation}  

-where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel Conjugate Gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
+where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ blocks, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel Conjugate Gradient method for the normal equations CGNR~\cite{S96,refCGNR}.

 
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
 
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}


@@ -287,12 +295,12 @@ where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ a
 \For {$j=1,2,\ldots,s$}
 \State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
 \State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
 \For {$j=1,2,\ldots,s$}
 \State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
 \State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$

-\State Exchange local values of $X_\ell^j$ with the neighboring clusters
+\State Exchange local values of $X_\ell^j$ with the neighboring blocks

 \State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$ 
 \EndFor 
 \State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
 \State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
 \State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$ 
 \EndFor 
 \State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
 \State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$

-\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
+\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring blocks

 \EndFor
 
 \Statex
 \EndFor
 
 \Statex


@@ -306,7 +314,7 @@ where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ a
 \Statex
 
 \Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
 \Statex
 
 \Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}

-\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
+\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ blocks using parallel CGNR method

 \State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
 \State \Return $\tilde{X}_\ell^k$
 \EndFunction
 \State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
 \State \Return $\tilde{X}_\ell^k$
 \EndFunction


@@ -314,7 +322,7 @@ where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ a
 \label{algo:01}
 \end{algorithm}
 
 \label{algo:01}
 \end{algorithm}
 

-The main key points of our Krylov multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each cluster of processors. Matrices and vectors with the subscript $\ell$ represent the local data for cluster $\ell$, where $\ell\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: the GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and the CGNR method, executed periodically in parallel by all clusters to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ clusters. The first one is performed  line~\ref{line12} in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~\ref{line7} in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.
+The main key points of our Krylov multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each block of processors. Matrices and vectors with the subscript $\ell$ represent the local data for block $\ell$, where $\ell\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: the GMRES method to solve each splitting~(\ref{sec03:eq03}) on a block of processors, and the CGNR method, executed periodically in parallel by all blocks to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ blocks. The first one is performed  line~\ref{line12} in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~\ref{line7} in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.

 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%


@@ -346,54 +354,76 @@ preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
 preconditioners are not scalable when using many cores.
 
 
 preconditioners are not scalable when using many cores.
 
 

-%%% MODIFIE ***********************
+%%% AJOUTE ************************

 %%%********************************
 %%%********************************

-We have performed some experiments on an infiniband cluster of 3 nodes of Intel Xeon quad-core CPU E5620 2.40 GHz and 12 GB of memory. For the GMRES code (alone and in both multisplitting versions) the restart parameter is fixed to 16. The precision of the GMRES version is fixed to 1e-6. For the multisplitting versions, there are two precisions, one for the external solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It should be noted that a high precision is used but we also fixed a maximum number of iterations for each internal step. In practice, we limit the number of iterations in the internal step to 10. So an internal iteration is finished when the precision is reached or when the maximum internal number of iterations is reached. The precision and the maximum number of iterations of CGNR method used by our Krylov multisplitting algorithm are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis S is fixed to 10 vectors.
+We have performed some experiments on  an infiniband cluster of three Intel Xeon
+quad-core E5620 CPUs of 2.40 GHz and  12 GB of memory. For the GMRES code (alone
+and in both  multisplitting versions) the restart parameter is  fixed to 16. The
+precision  of  the  GMRES version  is  fixed  to  1e-6. For  the  multisplitting
+versions, there are  two precisions, one for the external  solver which is fixed
+to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It
+should be noted that a high precision is used but we also fixed a maximum number
+of  iterations for  each internal  step.  In practice,  we limit  the number  of
+iterations in the internal step to 10. So an internal iteration is finished when
+the precision  is reached or when  the maximum internal number  of iterations is
+reached. The precision and the maximum  number of iterations of CGNR method used
+by   our  Krylov   multisplitting  algorithm   are   fixed  to   1e-25  and   20
+respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.

 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{strong_scaling_150x150x150}
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{strong_scaling_150x150x150}

-\caption{Strong scaling with 3 blocks of cores}
+\caption{Strong scaling with 3 blocks of 4 cores each to solve a 3D Poisson problem of size $150^3$ components}

 \label{fig:001}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
 \begin{tabular}{c}
 \label{fig:001}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
 \begin{tabular}{c}

-\includegraphics[width=0.8\textwidth]{weak_scaling_280k} \\ \includegraphics[width=0.8\textwidth]{weak_scaling_280K}\\
+\includegraphics[width=0.8\textwidth]{weak_scaling_280k} \\ \includegraphics[width=0.8\textwidth]{weak_scaling_280K2}\\

 \end{tabular}
 \end{tabular}

-\caption{Weak scaling with 3 blocks of cores}
+\caption{Weak scaling with 3 blocks of 4 cores each to solve a 3D Poisson problem with approximately 280K components per core}

 \label{fig:002}
 \end{figure}
 
 \label{fig:002}
 \end{figure}
 

+%%The experiments are performed on 3 different clusters of cores interconnected by an infiniband network (each cluster is a quad-core CPU). 
+Figures~\ref{fig:001}  and~\ref{fig:002} show  the  scalability performances  of
+GMRES, classical  multisplitting and  Krylov multisplitting methods:  strong and
+weak scaling are  presented respectively. We can remark  from these figures that
+the performances  of our Krylov multisplitting  method are better  than those of
+GMRES and classical multisplitting methods. In the experiments conducted in this
+work, our method is approximately twice faster than the GMRES method and about 9
+times faster than the classical multisplitting method. Our multisplitting method
+uses a  minimization step  over a  Krylov subspace which  reduces the  number of
+iterations  and  accelerates the  convergence.   We  can  also remark  that  the
+performances of the  classical block Jacobi multisplitting method  are the worst
+compared with  those of  the other two  methods.  This  is why in  the following
+experiments we compare the performances of our Krylov multisplitting method with
+only those of the GMRES method.

 %%%********************************
 %%%********************************
 
 
 %%%********************************
 %%%********************************
 
 

+%%% MODIFIE ************************
+%%%*********************************

 %Doing many experiments  with many cores is  not easy and requires to  access to a supercomputer  with several  hours for  developing  a code  and then  improving it. 
 In the following we present some experiments we could achieve out on the Hector
 %Doing many experiments  with many cores is  not easy and requires to  access to a supercomputer  with several  hours for  developing  a code  and then  improving it. 
 In the following we present some experiments we could achieve out on the Hector

-architecture,  a UK's  high-end computing  resource, funded  by the  UK Research
+architecture,  a UK  high-end computing  resource, funded  by the  UK Research

 Councils~\cite{hector}.  This is  a Cray  XE6 supercomputer,  equipped  with two
 16-core AMD  Opteron 2.3 GHz  and 32 GB  of memory. Machines  are interconnected
 Councils~\cite{hector}.  This is  a Cray  XE6 supercomputer,  equipped  with two
 16-core AMD  Opteron 2.3 GHz  and 32 GB  of memory. Machines  are interconnected

-with a 3D torus.
+with a 3D torus. The different parameters used by the GMRES and the Krylov multisplitting codes are as those previously mentioned. 

 
 Table~\ref{tab1} shows  the result of  the experiments.  The first  column shows
 
 Table~\ref{tab1} shows  the result of  the experiments.  The first  column shows

-the  size of  the  3D Poisson  problem.  The size  is chosen  in  order to  have
+the  size of  the 3D  Poisson problem.   The  size is  chosen in  order to  have

 approximately  50,000 components  per core.   The second  column  represents the
 approximately  50,000 components  per core.   The second  column  represents the

-number of  cores used. In brackets,  one can find the decomposition  used for the
-Krylov multisplitting. The  third column and the sixth  column respectively show
-the execution time for the GMRES  and the Krylov multisplitting codes. The fourth
-and  the   seventh  column  describe   the  number  of  iterations.    For  the
-multisplitting  code, the  total number  of inner  iterations is  represented in
-brackets. For  the GMRES code (alone  and in the  multisplitting version) the
-restart parameter is fixed to 16. The precision of the GMRES version is fixed to
-1e-6. For  the multisplitting,  there are two  precisions, one for  the external
-solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which
-is fixed to 1e-10. It should be noted  that a high precision is used but we also
-fixed  a maximum number of  iterations for each  internal step. In  practice, we
-limit the  number of iterations in the internal step to  10. So an internal  iteration is finished
-when the precision is reached or  when the maximum internal number of iterations
-is reached. The precision and the maximum number of iterations of CGNR method are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
+number of cores used. Between brackets,  one can find the decomposition used for
+the Krylov  multisplitting. The third  column and the sixth  column respectively
+show the execution  time for the GMRES and the  Krylov multisplitting codes. The
+fourth  and the  seventh  column describe  the  number of  iterations.  For  the
+multisplitting code, the total number of inner iterations is represented between
+brackets.
+%%%********************************
+%%%******************************** 

 
 \begin{table}[htbp]
 \begin{center}
 
 \begin{table}[htbp]
 \begin{center}


@@ -422,17 +452,13 @@ $743^3$ & 8,192 (4x2,048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   &
 \end{center}
 \end{table}
 
 \end{center}
 \end{table}
 

-
-
-
-

 From these  experiments, it can be  observed that the  multisplitting version is
 always  faster   than  the  GMRES   version.   The  acceleration  gain   of  the
 multisplitting version ranges between 4 and 6.  It can be noticed that the number of
 iterations is drastically reduced with the multisplitting version even it is not
 From these  experiments, it can be  observed that the  multisplitting version is
 always  faster   than  the  GMRES   version.   The  acceleration  gain   of  the
 multisplitting version ranges between 4 and 6.  It can be noticed that the number of
 iterations is drastically reduced with the multisplitting version even it is not

-negligible. Moreover, with 8,192 cores, we  can see that using 4 clusters gives a
-better performance than simply using 2 clusters. In fact, we can notice that the
-precision with 2 clusters is slightly  better but in both cases the precision is
+negligible. Moreover, with 8,192 cores, we  can see that using 4 blocks of cores gives a
+better performance than simply using 2 blocks. In fact, we can notice that the
+precision with 2 blocks is slightly  better but in both cases the precision is

 under the specified threshold.
 
 
 under the specified threshold.
 
 


@@ -440,32 +466,31 @@ under the specified threshold.
 %%%*******************************
 In Figure~\ref{fig:01}, the number of iterations per second is reported for both
 GMRES and the  multisplitting methods. It should be noted that  we took only the
 %%%*******************************
 In Figure~\ref{fig:01}, the number of iterations per second is reported for both
 GMRES and the  multisplitting methods. It should be noted that  we took only the

-inner number  of iterations (i.e.  the GMRES iterations) for  the multisplitting
+inner number of  iterations (i.e.  the GMRES iterations)  for the multisplitting

 method. Iterations of CGNR are not  taken into account. From this figure, it can
 method. Iterations of CGNR are not  taken into account. From this figure, it can

-be seen that the  number of iterations per second is higher  with GMRES but it is
-not  so different  with the  multisplitting method.  For the  case  with $8,192$
-cores,  the number of  iterations per  second with  4 clusters  is approximately
-equals to 115. So it is not different from GMRES.
-
+be seen that the number of iterations  per second is higher with GMRES but it is
+not  so different with  the multisplitting  method.  For  the case  with $8,192$
+cores, the number of iterations per second with 4 blocks is approximately equal
+to 115. So it is not different from GMRES.

 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.7\textwidth]{nb_iter_sec}
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.7\textwidth]{nb_iter_sec}

-\caption{Number of iterations per second  with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters}
+\caption{Number of iterations per second  with the same parameters as in Table~\ref{tab1} (weak scaling) with only  blocks of cores}

 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 

-
-\noindent {\bf Final remarks:}\\
-It should  be noted, on  the one  hand, that the  development of a  complete new
-method usable with any  kind of problem is a really long  and fastidious task if
-one is working from  scratch. On the other hand, using an  existing tool for the
-inner solver is also not easy because it requires to make link between the inner
-solver  and the outer  one.  We  plan to  do that  later with  engineers working
-specifically on  that point.  Moreover,  we think that  it is very  important to
-analyze the convergence  of this method compared to other  method. In this work,
-we have  focused on the  description of this  method and its performance  with a
-typical application. Many other investigations are required for this method as explained in the next section.
+\noindent {\bf Final  remarks:}\\ It should be noted, on the  one hand, that the
+development of a complete  new code usable with any kind of  problem is a really
+long and  fastidious task  if one is  working from  scratch. On the  other hand,
+using an existing  tool for the inner solver is also  quite difficult because it
+requires to  establish a link  between the inner  solver and the outer  one.  We
+plan  to  do that  later  with engineers  working  specifically  on that  point.
+Moreover, we think that it is  very important to analyze the convergence of this
+method  compared  to  other methods.  In  this  work,  we  have focused  on  the
+description of this method and its performances with a typical application.  Many
+other  investigations are  required for  this method  as explained  in  the next
+section.

 %%%*******************************
 %%%*******************************
 
 %%%*******************************
 %%%*******************************
 


@@ -474,8 +499,8 @@ We  have implemented  a  Krylov  multisplitting method  to  solve sparse  linear
 systems  on large-scale computing  platforms.  We  have developed  a synchronous
 two-stage  method based  on the  block Jacobi  multisplitting which  uses GMRES
 iterative  method as  an inner  iteration.  Our  contribution in  this  paper is
 systems  on large-scale computing  platforms.  We  have developed  a synchronous
 two-stage  method based  on the  block Jacobi  multisplitting which  uses GMRES
 iterative  method as  an inner  iteration.  Our  contribution in  this  paper is

-twofold. First we provide a multi cluster decomposition that allows us to choose
-the  appropriate size  of  the clusters  according  to the  architecures of  the
+twofold. First we provide a multi block decomposition that allows us to choose
+the  appropriate size  of  the blocks  according  to the  architecures of  the

 supercomputer.  Second,   we  have  implemented  the  outer   iteration  of  the
 multisplitting method  as a  Krylov subspace method  which minimizes  some error
 function.  This  increases the convergence  and improves the scalability  of the
 supercomputer.  Second,   we  have  implemented  the  outer   iteration  of  the
 multisplitting method  as a  Krylov subspace method  which minimizes  some error
 function.  This  increases the convergence  and improves the scalability  of the


@@ -484,11 +509,11 @@ multisplitting method.
 We  have tested  our multisplitting  method to  solve the  sparse  linear system
 issued from  the discretization of  a 3D Poisson  problem. We have  compared its
 performances to the  classical GMRES method on a  supercomputer composed of 2,048
 We  have tested  our multisplitting  method to  solve the  sparse  linear system
 issued from  the discretization of  a 3D Poisson  problem. We have  compared its
 performances to the  classical GMRES method on a  supercomputer composed of 2,048

-to 8,192 cores. The experimental results showed that the multisplitting method is
+up-to 8,192 cores. The experimental results showed that the multisplitting method is

 about 4  to 6  times faster  than the GMRES  method for  different sizes  of the
 problem split into  2 or 4 blocks when using the  multisplitting method. Indeed, the
 GMRES  method  has  difficulties to  scale  with  many  cores while  the  Krylov
 about 4  to 6  times faster  than the GMRES  method for  different sizes  of the
 problem split into  2 or 4 blocks when using the  multisplitting method. Indeed, the
 GMRES  method  has  difficulties to  scale  with  many  cores while  the  Krylov

-multisplitting  method  allows to  hide  latency  and  reduce the  inter-cluster
+multisplitting  method  allows to  hide  latency  and  reduce the  inter-block

 communications.
 
 In future  works, we plan to conduct  experiments on larger numbers  of cores and
 communications.
 
 In future  works, we plan to conduct  experiments on larger numbers  of cores and


@@ -503,7 +528,9 @@ preconditioning  techniques  for  Krylov  iterative methods  and  multisplitting
 methods with overlapping blocks.
 
 \section{Acknowledgement}
 methods with overlapping blocks.
 
 \section{Acknowledgement}

-The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR. This work has been partially supported by the Labex 
+ACTION project (contract “ANR-11-LABX-01-01”). 
+

 
 %Other applications (=> other matrices)\\
 %Larger experiments\\
 
 %Other applications (=> other matrices)\\
 %Larger experiments\\