13-12-2014 v01

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index 40ed640cd517e1588d1f9a57799bbc257d839c0f..f12f3ae7547bafd66a607802d9cc8622434a2207 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -183,7 +183,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 
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-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 the 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 to the linear system as follows

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@@ -201,7 +201,7 @@ 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 one block of processors. 

 %%%********************************
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 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 +221,7 @@ 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. 


@@ -234,7 +234,7 @@ 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
 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
 iteration  matrix, thereby  slowing the  convergence.  In  fact, the  larger the
 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


@@ -253,7 +253,7 @@ 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 is necessary 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 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
 %%%********************************


@@ -273,7 +273,7 @@ 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}


@@ -288,12 +288,12 @@ 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


@@ -307,7 +307,7 @@ 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


@@ -315,7 +315,7 @@ 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.

 
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@@ -425,9 +425,9 @@ 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
 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.
 
 


@@ -439,13 +439,13 @@ 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
 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 $8,192$
 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
+cores,  the number of  iterations per  second with  4 blocks  is approximately

 equals to 115. So it is not different from GMRES.
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.7\textwidth]{nb_iter_sec}
 equals to 115. So it is not different from GMRES.
 
 \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 2 blocks of cores}

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


@@ -467,8 +467,8 @@ 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


@@ -481,7 +481,7 @@ up-to 8,192 cores. The experimental results showed that the multisplitting metho
 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