27-04-2014b

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index d296dcf35b8ffd43b590059731674e38adedecc7..dfb67e45d77e92d4b0670078a3e3874d84fe0e61 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -17,10 +17,10 @@
 \newcommand{\Prec}{\mathit{prec}}
 \newcommand{\Ratio}{\mathit{Ratio}}
 
 \newcommand{\Prec}{\mathit{prec}}
 \newcommand{\Ratio}{\mathit{Ratio}}
 

-\usepackage{xspace}
-\usepackage[textsize=footnotesize]{todonotes}
-\newcommand{\LZK}[2][inline]{%
-\todo[color=green!40,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+%\usepackage{xspace}
+%\usepackage[textsize=footnotesize]{todonotes}
+%\newcommand{\LZK}[2][inline]{%
+%\todo[color=green!40,#1]{\sffamily\textbf{LZK:} #2}\xspace}

 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 
 \date{}
 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 
 \date{}


@@ -74,7 +74,6 @@ traditionnal  multisplitting  method  that  suffer  from  slow  convergence,  as
 proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
 drastically improve the convergence.
 
 proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
 drastically improve the convergence.
 

-\LZK[]{Suite\dots}

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


@@ -96,9 +95,9 @@ increase the convergence, then the other tasks receive the updated solution unti
 convergence of the global system. 
 
 In~\cite{couturier2008gremlins}, the  authors proposed practical implementations
 convergence of the global system. 
 
 In~\cite{couturier2008gremlins}, the  authors proposed practical implementations

-of multisplitting algorithms that take benefit from multisplitting algorithms\LZK[]{répétition ???} to
+of multisplitting algorithms that take benefit from multisplitting algorithms to

 solve large scale linear systems. Inner  solvers could be based on scalar direct
 solve large scale linear systems. Inner  solvers could be based on scalar direct

-method with the LU method or scalar iterative one with GMRES.\LZK[]{lu et gmres par exemple}
+method with the LU method or scalar iterative one with GMRES.

 
 In~\cite{prace-multi},  the  authors have  proposed a  parallel  multisplitting
 algorithm in which large blocks are solved using a GMRES solver. The authors have
 
 In~\cite{prace-multi},  the  authors have  proposed a  parallel  multisplitting
 algorithm in which large blocks are solved using a GMRES solver. The authors have


@@ -106,8 +105,6 @@ performed large scale experiments up-to  32,768 cores and they conclude that
 asynchronous  multisplitting algorithm  could be more  efficient  than traditional
 solvers on exascale architecture with hundreds of thousands of cores.
 
 asynchronous  multisplitting algorithm  could be more  efficient  than traditional
 solvers on exascale architecture with hundreds of thousands of cores.
 

-\LZK[]{Peut-être autres related works\ldots}\\
-

 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 
 \begin{equation}
 A = M_l - N_l,
 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 
 \begin{equation}
 A = M_l - N_l,


@@ -289,7 +286,6 @@ fixed a  maximum number of  iterations for each  internal step. In  practise, we
 limit the  number of 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.
 limit the  number of 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.

-\LZK{J'ai ajouté les paramètres concernant la résolution du problème de moindres carrés. Confirmer leur valeurs.}

 
 
 
 
 
 


@@ -326,9 +322,9 @@ 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 multi-cluster environment based on processors of the large-scale 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 multi-cluster environment based on processors of the large-scale 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 the 3D Poisson problem. We have compared its performances to 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 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 communications.
+We have tested our multisplitting method for solving the sparse linear system issued from the discretization of the 3D Poisson problem. We have compared its performances to those of 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 by using preconditioning techniques 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 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.    

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