11-12-2014 v06

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index 936cdb6013d4a99406f5528d5a89b348b1cdb6e7..42a47d09a51efc8c7cb9c369671c009e026eea66 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.\\

 
 
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-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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@@ -345,21 +346,32 @@ 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.
 
 

+

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-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 3 nodes of Intel Xeon quad-core 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.

 
 \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 blocks of cores}
+\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 blocks of cores}
+\label{fig:002}

 \end{figure}
 
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 \end{figure}
 
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+

 %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
 %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