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[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 9757efc190abec922a559ce9a804dd8d8321c6dc..2d0bc6994339e973f66571ac17f0008f0b7b822a 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -38,7 +38,7 @@
 
 \begin{abstract}
 In  this paper  we  revisit  the Krylov  multisplitting  algorithm presented  in
 
 \begin{abstract}
 In  this paper  we  revisit  the Krylov  multisplitting  algorithm presented  in

-\cite{huang1993krylov}  which  uses  a  scalar  method to  minimize  the  Krylov
+\cite{huang1993krylov}  which  uses  a  sequential  method to  minimize  the  Krylov

 iterations computed by a multisplitting algorithm. Our new algorithm is based on
 a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a
 parallel GMRES method inside each block and on a parallel Krylov minimization in
 iterations computed by a multisplitting algorithm. Our new algorithm is based on
 a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a
 parallel GMRES method inside each block and on a parallel Krylov minimization in


@@ -75,11 +75,14 @@ traditional  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.
 

+The paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. The, in Section~\ref{sec:03} is presented the algorithm of our Krylov multisplitting method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the parallel experiments on Hector architecture show the performances of the Krylov multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D Poisson problem.
+

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

-\section{Related works and presention of the multisplitting method}
+\section{Related works and presentation of the multisplitting method}
+\label{sec:02}

 A general framework  for studying parallel multisplitting has  been presented in~\cite{o1985multi}
 by O'Leary and White. Convergence conditions are given for the
 most general case.  Many authors improved multisplitting algorithms by proposing
 A general framework  for studying parallel multisplitting has  been presented in~\cite{o1985multi}
 by O'Leary and White. Convergence conditions are given for the
 most general case.  Many authors improved multisplitting algorithms by proposing


@@ -97,7 +100,7 @@ convergence of the global system.
 
 In~\cite{couturier2008gremlins}, the  authors proposed practical implementations
 of multisplitting algorithms to solve  large scale linear systems. Inner solvers
 
 In~\cite{couturier2008gremlins}, the  authors proposed practical implementations
 of multisplitting algorithms to solve  large scale linear systems. Inner solvers

-could be  based on scalar direct method  with the LU method  or scalar iterative
+could be  based on sequential direct method  with the LU method  or sequential iterative

 one with GMRES.
 
 In~\cite{prace-multi},  the  authors have  proposed a  parallel  multisplitting
 one with GMRES.
 
 In~\cite{prace-multi},  the  authors have  proposed a  parallel  multisplitting


@@ -106,10 +109,7 @@ 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 


@@ -145,6 +145,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{A two-stage method with a minimization}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{A two-stage method with a minimization}

+\label{sec:03}

 Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows
 \begin{equation}
 \left\{
 Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows
 \begin{equation}
 \left\{


@@ -163,7 +164,7 @@ 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 +176,16 @@ 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}


@@ -247,6 +257,7 @@ The main key points of our Krylov multisplitting method to solve a large sparse
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experiments}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experiments}

+\label{sec:04}

 In order to illustrate  the interest  of our algorithm. We have  compared our
 algorithm  with  the  GMRES  method  which  is a very  well  used  method  in  many
 situations.  We have chosen to focus on only one problem which is very simple to
 In order to illustrate  the interest  of our algorithm. We have  compared our
 algorithm  with  the  GMRES  method  which  is a very  well  used  method  in  many
 situations.  We have chosen to focus on only one problem which is very simple to


@@ -269,11 +280,12 @@ 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
-Hector architecture,  the previous UK's  high-end computing resource,  funded by
-the UK Research Councils, which has been stopped in the early 2014.
+%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,  a UK's  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
+with a 3D torus.

 
 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
 
 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


@@ -301,13 +313,13 @@ is reached. The precision and the maximum number of iterations of CGNR method ar
  \cline{3-8}
            &                   &  Time (s) & nb Iter. & $\Delta$  &   Time (s)& nb Iter. & $\Delta$ & \\
 \hline
  \cline{3-8}
            &                   &  Time (s) & nb Iter. & $\Delta$  &   Time (s)& nb Iter. & $\Delta$ & \\
 \hline

-$468^3$ & 2048 (2x1024)        &  299.7    & 41,028    & 5.02e-8  &  48.4    & 691(6,146) & 8.24e-08  & 6.19   \\
+$468^3$ & 2,048 (2x1,024)        &  299.7    & 41,028    & 5.02e-8  &  48.4    & 691(6,146) & 8.24e-08  & 6.19   \\

 \hline
 \hline

-$590^3$ & 4096 (2x2048)        &  433.1    & 55,494    & 4.92e-7  &  74.1    & 1,101(8,211) & 6.62e-08  & 5.84   \\
+$590^3$ & 4,096 (2x2,048)        &  433.1    & 55,494    & 4.92e-7  &  74.1    & 1,101(8,211) & 6.62e-08  & 5.84   \\

 \hline
 \hline

-$743^3$ & 8192 (2x4096)        & 704.4     & 87,822    & 4.80e-07 &  151.2   & 3,061(14,914) & 5.87e-08 & 4.65    \\
+$743^3$ & 8,192 (2x4,096)        & 704.4     & 87,822    & 4.80e-07 &  151.2   & 3,061(14,914) & 5.87e-08 & 4.65    \\

 \hline
 \hline

-$743^3$ & 8192 (4x2048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   & 1,531(12,721) & 1.47e-07& 6.39  \\
+$743^3$ & 8,192 (4x2,048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   & 1,531(12,721) & 1.47e-07& 6.39  \\

 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}


@@ -320,16 +332,47 @@ $743^3$ & 8192 (4x2048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   & 1
 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 is between 4 and 6.  It can be noticed that the number of
 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 is between 4 and 6.  It can be noticed that the number of

-iterations is drastically reduced with  the multisplitting version even it is not
-neglectable.
+iterations is drastically reduced with the multisplitting version even it is not
+neglectable. Moreover, with 8,192 cores, we  can see that using 4 clusters gives
+better performance than simply using 2 clusters. In fact, we can remark that the
+precision with 2 clusters is slightly  better but in both cases the precision is
+under the specified threshold.

 
 \section{Conclusion and perspectives}
 
 \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 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 using 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 those of 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.    
-
+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  multisaplitting 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
+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
+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
+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  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 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.
+
+\section{Acknowledgement}
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.

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