new

diff --git a/krylov_multi.tex b/krylov_multi.tex
index a8184a11fa006b50fe3324d618ad614c893a1551..5a736703be2507a04a0fa4cc485d8d7423b9c473 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -106,10 +106,7 @@ performed large scale experiments up-to  32,768 cores and they conclude that
 asynchronous  multisplitting algorithm  could be more  efficient  than traditional
 solvers on an exascale architecture with hundreds of thousands of cores.
 
 asynchronous  multisplitting algorithm  could be more  efficient  than traditional
 solvers on an exascale architecture with hundreds of thousands of cores.
 

-
-So compared to these works, we  propose in this paper a practical multisplitting
-which is  based on parallel iterative  blocks and which give  better result than
-GMRES for the 3D Poisson problem we considered.
+So compared to these works, we propose in this paper a practical multisplitting method based on parallel iterative blocks and gives better results than classical GMRES method for the 3D Poisson problem we considered.

 \\
 
 The key idea of a multisplitting method to solve a large system of linear equations $Ax=b$ is defined as follows. The first step consists in partitioning the matrix $A$ in $L$ several ways 
 \\
 
 The key idea of a multisplitting method to solve a large system of linear equations $Ax=b$ is defined as follows. The first step consists in partitioning the matrix $A$ in $L$ several ways 


@@ -163,7 +160,7 @@ where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\e
 \end{equation} 
 where $A_{\ell m}$ is a sub-block of size $n_\ell\times  n_m$ of the rectangular matrix $A_\ell$, $X_m\neq  X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.
 
 \end{equation} 
 where $A_{\ell m}$ is a sub-block of size $n_\ell\times  n_m$ of the rectangular matrix $A_\ell$, $X_m\neq  X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.
 

-Our multisplitting method proceeds by iteration for solving the linear system in such a way each sub-system
+Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system

 \begin{equation}
 \left\{
 \begin{array}{l}
 \begin{equation}
 \left\{
 \begin{array}{l}


@@ -175,7 +172,16 @@ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_
 \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}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.  
 
 \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}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.  
 

-It should be noted that the convergence of the inner iterative solver for the different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the convergence of the multisplitting method. It strongly depends on the properties of the global sparse linear system to be solved~\cite{o1985multi,ref18}. Furthermore, the multisplitting of the linear system among several clusters of processors increases the spectral radius of the iteration matrix, thereby slowing the convergence. In this paper, we based on the work presented in~\cite{huang1993krylov} to increase the convergence and improve the scalability of the multisplitting methods.
+It should  be noted that the convergence  of the inner iterative  solver for the
+different  sub-systems~(\ref{sec03:eq03})  does   not  necessarily  involve  the
+convergence of the multisplitting method.  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
+iteration  matrix, thereby  slowing the  convergence.  In  fact, the  larger the
+number of  splitting is, the larger the  spectral radius is.  In  this paper, we
+based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
+convergence and improve the scalability of the multisplitting methods.

 
 In order to accelerate the convergence, we implemented the outer iteration of the multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
 \begin{equation}
 
 In order to accelerate the convergence, we implemented the outer iteration of the multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
 \begin{equation}


@@ -269,9 +275,8 @@ obtained for  a 3D Poisson  problem, the number  of iterations is high.  Using a
 preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
 preconditioners are not scalable when using many cores.
 
 preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
 preconditioners are not scalable when using many cores.
 

-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 presented  some experiments we could achieved out on the
+%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 achieved out on the

 Hector architecture,  the previous UK's  high-end computing resource,  funded by
 the UK Research Councils, which has been stopped in the early 2014.
 
 Hector architecture,  the previous UK's  high-end computing resource,  funded by
 the UK Research Councils, which has been stopped in the early 2014.
 


@@ -326,9 +331,9 @@ neglectable.
 \section{Conclusion and perspectives}
 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 and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a virtual multi-cluster environment based on processors of the computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. 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 multisplitting method.
 
 \section{Conclusion and perspectives}
 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 and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a virtual multi-cluster environment based on processors of the computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. 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 multisplitting method.
 

-We have tested our multisplitting method for solving 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 2048 to 8192 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 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 communications.
+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 2048 to 8192 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 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 communications.

 
 

-In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances for solving other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks.    
+In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances to solve other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks.    

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