13-12-2014 v00

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index cdcaae289dcd92dbd06f9d0c507f6d44e0641826..40ed640cd517e1588d1f9a57799bbc257d839c0f 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -90,7 +90,7 @@ distant  clusters interconnected  by  a high-latency  network.  It is  therefore
 imperative to develop coarse-grain based algorithms to reduce the communications
 in the  parallel iterative  solvers. Two possible  solutions consists  either in
 using  asynchronous  iterative  methods~\cite{ref18}  or in  using  multisplitting
 imperative to develop coarse-grain based algorithms to reduce the communications
 in the  parallel iterative  solvers. Two possible  solutions consists  either in
 using  asynchronous  iterative  methods~\cite{ref18}  or in  using  multisplitting

-algorithmss.  In this  paper,  we will  reconsider  the use  of a  multisplitting
+algorithms.  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.\\
 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.\\


@@ -99,7 +99,7 @@ process can drastically improve the convergence.\\
 %%% AJOUTE************************
 %%%*******************************
 \noindent {\bf Contributions:}\\ 
 %%% AJOUTE************************
 %%%*******************************
 \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.\\
+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 other 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.\\

 %%%*******************************
 %%%*******************************
 
 %%%*******************************
 %%%*******************************
 


@@ -236,7 +236,7 @@ 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
 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, our
+number of  splittings is, the larger the  spectral radius is.  In  this paper, our

 work is based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
 convergence and improve the scalability of the multisplitting methods.
 
 work is based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
 convergence and improve the scalability of the multisplitting methods.
 


@@ -254,10 +254,11 @@ where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the
 %%% MODIFIE ***********************
 %%%********************************
 The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
 %%% MODIFIE ***********************
 %%%********************************
 The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.

+
+The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system

 %%%********************************
 %%%********************************
 
 %%%********************************
 %%%********************************
 

-The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error function $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system

 \begin{equation}
 R\alpha=b,
 \label{sec03:eq05}
 \begin{equation}
 R\alpha=b,
 \label{sec03:eq05}


@@ -348,12 +349,12 @@ preconditioners are not scalable when using many cores.
 
 %%% AJOUTE ************************
 %%%********************************
 
 %%% AJOUTE ************************
 %%%********************************

-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.
+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
   \includegraphics[width=0.8\textwidth]{strong_scaling_150x150x150}
 
 \begin{figure}[htbp]
 \centering
   \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}
+\caption{Strong scaling with 3 blocks of 4 cores each to solve a 3D Poisson problem of size $150^3$ components}

 \label{fig:001}
 \end{figure}
 
 \label{fig:001}
 \end{figure}
 


@@ -362,11 +363,12 @@ We have performed some experiments on an infiniband cluster of three Intel Xeon
 \begin{tabular}{c}
 \includegraphics[width=0.8\textwidth]{weak_scaling_280k} \\ \includegraphics[width=0.8\textwidth]{weak_scaling_280K}\\
 \end{tabular}
 \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}
+\caption{Weak scaling with 3 blocks of 4 cores each to solve a 3D Poisson problem with approximately 280K components per core}

 \label{fig:002}
 \end{figure}
 
 \label{fig:002}
 \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.
+%%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 are presented 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 the other two methods. This is why in the following experiments we compare the performances of our Krylov multisplitting method with only those of the GMRES method.

 %%%********************************
 %%%********************************
 
 %%%********************************
 %%%********************************
 


@@ -454,7 +456,7 @@ one is working from  scratch. On the other hand, using an  existing tool for the
 inner solver is also not easy because it requires to make link between the inner
 solver  and the outer  one.  We  plan to  do that  later with  engineers working
 specifically on  that point.  Moreover,  we think that  it is very  important to
 inner solver is also not easy because it requires to make link between the inner
 solver  and the outer  one.  We  plan to  do that  later with  engineers working
 specifically on  that point.  Moreover,  we think that  it is very  important to

-analyze the convergence  of this method compared to other  method. In this work,
+analyze the convergence  of this method compared to other  methods. In this work,

 we have  focused on the  description of this  method and its performance  with a
 typical application. Many other investigations are required for this method as explained in the next section.
 %%%*******************************
 we have  focused on the  description of this  method and its performance  with a
 typical application. Many other investigations are required for this method as explained in the next section.
 %%%*******************************


@@ -475,7 +477,7 @@ 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
 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
+up-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 the  multisplitting method. Indeed, the
 GMRES  method  has  difficulties to  scale  with  many  cores while  the  Krylov
 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