11-12-2014 v07

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index 936cdb6013d4a99406f5528d5a89b348b1cdb6e7..cdcaae289dcd92dbd06f9d0c507f6d44e0641826 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -93,12 +93,13 @@ using  asynchronous  iterative  methods~\cite{ref18}  or in  using  multisplitti
 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
 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.
+process can drastically improve the convergence.\\

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

-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 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.\\

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@@ -345,27 +346,39 @@ 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 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
 
 \begin{figure}[htbp]
 \centering

-  \includegraphics[width=0.7\textwidth]{strong_scaling_150x150x150}
-\caption{Number of iterations per second  with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters}
-\label{fig:01}
+  \includegraphics[width=0.8\textwidth]{strong_scaling_150x150x150}
+\caption{Strong scaling with 3 clusters of cores to solve a 3D Poisson problem of size $150^3$ components}
+\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}\\
+\end{tabular}
+\caption{Weak scaling with 3 clusters of cores to solve a 3D Poisson problem with approximately 280K components per core}
+\label{fig:002}

 \end{figure}
 
 \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 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 about 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 two other 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
 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
 %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
 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
 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


@@ -375,15 +388,9 @@ 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
 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.
+brackets.
+%%%********************************
+%%%******************************** 

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


@@ -412,10 +419,6 @@ $743^3$ & 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
 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


@@ -437,7 +440,6 @@ 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.
 
 cores,  the number of  iterations per  second with  4 clusters  is approximately
 equals 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}


@@ -445,7 +447,6 @@ equals to 115. So it is not different from GMRES.
 \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
 \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