12-12-2014 v01

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index 98b443336a9663165dbd49f94758f9dbedb3ff52..32c2e31117427b4d10d1c01e1ac97d9a77fe644e 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -8,7 +8,8 @@
 \usepackage{multirow}
 \usepackage{authblk}
 
 \usepackage{multirow}
 \usepackage{authblk}
 

-\algnewcommand\algorithmicinput{\textbf{Input:}}
+
+\algnewcommand\algorithmicinput{\textbf{I1nput:}}

 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}


@@ -89,12 +90,18 @@ 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
 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.\\
+

 
 

-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.
+%%% 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 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.\\
+%%%*******************************
+%%%*******************************

 
 The present paper is  organized as follows. First, Section~\ref{sec:02} presents
 some  related  works and  the  principle  of  multisplitting methods.  Then,  in
 
 The present paper is  organized as follows. First, Section~\ref{sec:02} presents
 some  related  works and  the  principle  of  multisplitting methods.  Then,  in


@@ -173,7 +180,14 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 
 \section{A two-stage method with a minimization}
 \label{sec:03}
 
 \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 where $A\in\mathbb{R}^{n\times n}$ is a sparse 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. We use a multisplittig method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+
+%%% MODIFIE ************************
+%%%*********************************
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse 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. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+%%%*********************************
+%%%*********************************
+
+

 \begin{equation}
 \left\{
 \begin{array}{lll}
 \begin{equation}
 \left\{
 \begin{array}{lll}


@@ -184,7 +198,13 @@ b & = & [B_{1}, \ldots, B_{L}]
 \right.
 \label{sec03:eq01}
 \end{equation}  
 \right.
 \label{sec03:eq01}
 \end{equation}  

-where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. So, the multisplitting format of the linear system is defined as follows
+where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. 
+%%% MODIFIE ***********************
+%%%********************************
+The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. 
+%%%********************************
+%%%********************************
+So, the multisplitting format of the linear system is defined as follows

 \begin{equation}
 \forall \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell, 
 \label{sec03:eq02}
 \begin{equation}
 \forall \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell, 
 \label{sec03:eq02}


@@ -201,7 +221,13 @@ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_
 \right.
 \label{sec03:eq03}
 \end{equation}
 \right.
 \label{sec03:eq03}
 \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 by minimizing the residuals over an orthonormal basis of a Krylov subspace. %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.  
+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}). 
+%%% MODIFIE ***********************
+%%%********************************
+GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. 
+%%%********************************
+%%%********************************
+%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
 
 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


@@ -210,18 +236,29 @@ 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.
 

-Krylov subspace methods are well-known for their good convergence compared to other iterative methods. In order to accelerate the convergence, we implemented the outer iteration of our 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})
+%%% AJOUTE ************************
+%%%********************************
+Krylov subspace methods are well-known for their good convergence compared to other iterative methods. 
+%%%********************************
+%%%********************************
+In order to accelerate the convergence, we implemented the outer iteration of our 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}
 S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
 \label{sec03:eq04}
 \end{equation}
 \begin{equation}
 S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
 \label{sec03:eq04}
 \end{equation}

-where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 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.
+where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 
+%%% 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}


@@ -285,10 +322,12 @@ The main key points of our Krylov multisplitting method to solve a large sparse
 
 \section{Experiments}
 \label{sec:04}
 
 \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 commonly  used  method  in  many
-situations.  We have chosen to focus on only one problem which is very simple to
-implement: a 3 dimension Poisson problem.
+%%% MODIFIE ***********************
+%%%********************************
+In order to illustrate the interest of our Krylov multisplitting algorithm, we have compared its performances with those of a classical block Jacobi multisplitting method and those of the GMRES method which is a commonly used method in many situations.
+%%%********************************
+%%%********************************
+ We have chosen to focus on only one problem which is very simple to implement: a 3 dimension Poisson problem.

 
 \begin{equation}
 \left\{
 
 \begin{equation}
 \left\{


@@ -307,12 +346,40 @@ 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.
 

+
+%%% 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.
+
+\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}
+\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}
+
+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.
+%%%********************************
+%%%********************************
+
+
+%%% 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
 %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.
+16-core AMD  Opteron 2.3 GHz  and 32 GB  of memory. Machines  are interconnected
+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


@@ -322,15 +389,9 @@ 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}


@@ -359,7 +420,6 @@ $743^3$ & 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


@@ -369,6 +429,38 @@ better performance than simply using 2 clusters. In fact, we can notice that the
 precision with 2 clusters is slightly  better but in both cases the precision is
 under the specified threshold.
 
 precision with 2 clusters is slightly  better but in both cases the precision is
 under the specified threshold.
 

+
+%%% AJOUTE************************
+%%%*******************************
+In Figure~\ref{fig:01}, the number of iterations per second is reported for both
+GMRES and the  multisplitting methods. It should be noted that  we took only the
+inner number  of iterations (i.e.  the GMRES iterations) for  the multisplitting
+method. Iterations of CGNR are not  taken into account. From this figure, it can
+be seen that the  number of iterations per second is higher  with GMRES but it is
+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.
+
+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.7\textwidth]{nb_iter_sec}
+\caption{Number of iterations per second  with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters}
+\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
+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
+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.
+%%%*******************************
+%%%*******************************
+

 \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
 \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


@@ -384,7 +476,7 @@ 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